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Defining Decentralization: An Ontological Perspective
Jakub Kacper Szeląg, Aydin Abadi, Mohammad Naseri
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Abstract
Abstract:Decentralization as a concept in computer science has existed for over half a century. Despite its fundamental role across domains such as security, distributed computing, artificial intelligence, cloud infrastructures, and Internet of Things (IoT) architectures, there remains no universally accepted definition of decentralization applicable across computer communication systems. This has become increasingly problematic with the emergence of decentralized AI and machine learning paradigms, including collaborative training, distributed inference, blockchain-based, and agentic AI, where decentralization is often treated as a core design objective. Meanwhile, existing approaches frequently conflate decentralization with related notions such as distribution of trust or specific implementation paradigms. Such ambiguity creates inconsistencies in system analysis, limits comparability between works, and weakens the rigor of formal reasoning surrounding communication architectures and protocol design. In this work, we define this research gap as the Decentralization Problem. We analyze the formal-semantic, epistemological, and pragmatic foundations of decentralization and introduce a graph-based ontology defining it as both relational and subject-specific property of computer communication systems. The framework formally distinguishes decentralization from distribution and supports evaluation through two novel metrics: Void Tolerance and Imperviousness. We also provide a browser-based implementation that enables automated classification and metric computation of arbitrary systems. Instantiations to federated learning and blockchain architectures show consistent, comparable assessments where existing definitions produce incomplete or contradictory conclusions, providing a domain-independent foundation for analysing decentralization across heterogeneous systems.
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- Source: https://arxiv.org/abs/2608.09748v1
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Defining Decentralization: An Ontological Perspective Jakub K. Szeląg Aydin Abadi Mohammad Naseri Abstract Decentralization as a concept in computer science has existed for over half a century. Despite its fundamental role across domains such as security, distributed computing, artificial intelligence, cloud infrastructures, and Internet of Things (IoT) architectures, there remains no universally accepted definition of decentralization applicable across computer communication systems. This absence has become increasingly problematic with the emergence of decentralized AI and machine learning paradigms, including collaborative training, distributed inference, blockchain-based, and agentic AI, where decentralization is often treated as a core design objective. Meanwhile, existing approaches frequently conflate decentralization with related notions such as distribution of trust or specific implementation paradigms. This ambiguity creates inconsistencies in system analysis, limits comparability between works, and weakens the rigor of formal reasoning surrounding communication architectures and protocol design. In this work, we define this research gap as the Decentralization Problem. We systematically analyze the formal-semantic, epistemological, and pragmatic foundations of decentralization, showing why existing definitions remain context-dependent, technology-specific, or conflated with distribution. From this analysis, we derive the requirements for a transferable formal treatment and introduce a graph-based ontology that defines decentralization as a structural, relational, and subject-specific property of computer communication systems. The framework formally distinguishes decentralization from distribution and supports multidimensional evaluation through two novel analytical metrics: Void Tolerance and Imperviousness. We implement the framework as a browser-based application supporting interactive modelling, automated classification, metric computation, and large-scale deterministic simulation. Applications to federated learning and blockchain architectures demonstrate that the approach produces consistent and comparable assessments where existing definitions yield incomplete or contradictory conclusions, providing a domain-independent foundation for analysing decentralization across heterogeneous systems. I Introduction Decentralization has become one of the defining architectural principles in modern computer communication systems. The concept is foundational across security, distributed computing, and increasingly artificial intelligence. Despite that, there exists a striking inconsistency between how decentralization is represented and how it is described in many modern systems. A useful illustration emerges by contrasting one of the earliest and most influential models of decentralization with a widely adopted machine learning paradigm. Baran’s [5] seminal work on distributed communications distinguishes centralized, decentralized, and distributed networks as structurally different graph topologies, with decentralization explicitly characterized through intermediate hierarchical connectivity and distribution represented through fully peer-connected redundancy [5]. This structural framing has shaped decades of systems literature and remains a common conceptual reference point. By contrast, modern decentralized machine learning literature frequently applies the term decentralized to architectures whose structural properties differ substantially from Baran’s taxonomy. Federated Learning (FL) serves as a particularly illustrative example, one we will consistently refer to. In vanilla FL [33], a central server coordinates model aggregation while client devices train locally and periodically transmit updates to the server. This approach is widely described as privacy-preserving and decentralized because data remains distributed across participants, allowing for clients to keep their own datasets private. From a communication-graph perspective, however, the system remains structurally centralized: clients depend on a single aggregation point, communication paths are mediated through the server, and the server represents a coordination bottleneck and a trust authority. This is often represented as a star-topology, which through the lens of [5] is considered to be centralized. Interestingly, despite the fact that the FL as a whole is seen as one of the more promising decentralized ML paradigms, further classifications follow in the FL literature breaking it down further into “Centralized FL” and “Decentralized FL” [52]. This discrepancy highlights a crucial question, central to this work. If a canonical systems model classifies a topology as centralized, while field-specific literature labels an analogous architecture as decentralized, then questions arise as to the exact meaning of decentralization. It is clear that Baran’s structural point of view significantly differs from much of, for example, ML literature where decentralization is inferred from placement of data or model training. As much as these are important properties for ML, they are only specific dimensions that do not fully encapsulate decentralized communication structure even in the field of ML, provided that we may have to consider for other dimensions (e.g., security-enhancing solutions). As a result, identical architectures may be described as decentralized and centralized as a whole in the same context. This ambiguity raises a question regarding the formal semantics of these terms, and how can we universally address them. These questions have become increasingly important with the emergence of decentralized AI systems, including federated learning, swarm intelligence, collaborative model training, distributed inference pipelines, and blockchain-assisted AI [32, 20, 49, 39, 38]. Across these domains, decentralization is frequently presented as a core design objective and as a source of resilience, privacy, autonomy, and trust minimization. However, the term itself is often used interchangeably with distribution, locality, or absence of centralized storage, even when underlying coordination mechanisms remain structurally centralized. This creates an unusual tension, as decentralization is routinely invoked as both a design objective and an analytical property, and systems are often described, compared, or evaluated in terms of how decentralized they are. All while lacking a formal definition that specifies what decentralization is independent of a particular technology or application domain, making these comparisons often incompatible across literature and conclusions from evaluations inconsistent. In practice, decentralization is frequently treated as self-evident, that is, inferred through intuition, approximated through domain-specific metrics, or assumed through the absence of a surface-level, and easily identifiable central authority. While useful in narrow contexts, these interpretations do not generalize and often reinforce previously listed concerns even in the same context. This lack of formalization gives rise to several unresolved research problems. Problem 1: Definitional Ambiguity. There is currently no transferable and context-independent definition of decentralization for arbitrary computer communication systems. Existing work often describes decentralization through examples, system-specific heuristics, or associated properties such as resilience or trust minimization, without addressing decentralization itself but rather its properties in the given scope. As a result, the same system may be considered decentralized under one interpretation and centralized under another. Problem 2: Conceptual Confusion with Distribution. A persistent ambiguity throughout the literature is the interchangeable use of decentralization and distribution. Although the terms are often treated as equivalent, they refer to fundamentally different properties: one concerning relational structure and dependency, the other concerning allocation or placement of components. Without a formal distinction, analytical conclusions derived from one are frequently attributed to the other, which obscures both. Problem 3: Lack of Structural Formalism. Many existing approaches evaluate decentralization through node-centric quantities such as participant counts, ownership concentration, or resource allocation. While informative, these approaches abstract away the structure of relationships between entities. This presents a fundamental limitation, as decentralization is inherently relational, it emerges through dependencies, communication paths, and structural constraints connecting participants rather than through participant attributes in isolation. Problem 4: Absence of Transferable Quantification. Even where decentralization is quantified, existing metrics are typically tied to specific architectures, most notably blockchain systems. This limits their applicability across broader classes of computer communication systems and prevents consistent comparison between fundamentally different architectures such as peer-to-peer networks, federated learning environments, and decentralized AI coordination systems. Addressing these challenges requires a formal treatment of decentralization that is independent of specific technologies, governance models, and application domains. The remainder of this paper develops such a framework. I Our Contributions With that said, this work addresses the Decentralization Problem: the absence of a formal, transferable, and context-independent definition of decentralization applicable across computer communication systems. To do so, we first examine why the concept has remained resistant to formalization, then derive a definition grounded in graph-theoretic ontology and demonstrate how it can be used analytically across heterogeneous systems. Our contributions are summarized as follows: 1. We systematically expose why existing approaches fail to define decentralization universally. We provide a structured analysis of modern decentralization literature from both philosophical and technical perspectives, showing that prevailing approaches are either context-dependent, tied to specific technologies (e.g., blockchain), or conflate decentralization with related notions such as distribution, governance, or resource allocation. This establishes the precise limitations preventing current approaches from serving as transferable definitions. 2. We identify the formal limitations and requirements necessary for a universal definition of decentralization. Building on this analysis, we derive the missing conceptual and technical requirements that a valid definition must satisfy: transferability across domains, explicit treatment of relational structure, compatibility with formal reasoning, and (arguably the most difficult part) the ability to preserve contextual interpretation across differing decentralized systems while retaining the previous qualities. 3. We introduce a graph-based ontological definition of decentralization for computer communication systems. We propose, to the best of our knowledge, the first ontology-level formalization of decentralization designed as a general conceptual foundation independent of any application domain. The ontology defines decentralization as a structural and relational property of interacting computational entities, supports multiple subjects of decentralization (multi-dimensionality of decentralization), and is directly applicable to arbitrary graph-representable systems. 4. We formally separate decentralization from distribution and redefine centralization through this framework. A central result of this work is a rigorous semantic distinction between decentralization and distribution within graph-realistic setting, resolving the ambiguity present throughout the literature. Within the same framework, we show that centralization emerges naturally as the degenerate zero-dimensional case of decentralization, establishing a cleaner and more principled relationship between the two. 5. We extend the definition with analytical metrics that quantify decentralization beyond binary classification. Based on the ontology, we introduce two graph-based subject-specific metrics: Void Tolerance and Imperviousness, which measure the impact of nodes and connections respectively. Together they produce a decentralization vector that enables direct comparison of systems and exposes drawbacks that scalar-only measures cannot capture. On top of that, we base our reasoning in formal semantics to argue for why these metrics (specifically, their aggregate), can be used to describe a system as more or less decentralized from one another. 6. We validate the framework on heterogeneous real-world systems and demonstrate transferability. We instantiate the ontology on fundamentally different communication architectures, including blockchain and federated learning, and show that the framework yields consistent classifications and meaningful analytical insight where existing definitions become incomplete or contradictory. This demonstrates that the proposed ontology can function as a universal formal foundation for decentralization across computer systems. 7. We provide a sandbox implementation that can be used to infer systems level of decentralization. Based on our analytical evaluation metrics and ontological construction. We implement a prototype tool that can be used to evaluate arbitrary systems encoded in graph representations, providing an ontological inference regarding the status of decentralization, as well as exact values for Imperviousness and Void Tolerance. I Related Works I-A Structural Descriptions The study of decentralization in computer systems has produced a substantial body of work spanning network theory, distributed systems, and blockchain research [30, 52, 6]. While these contributions have advanced informal understanding of decentralized architectures and introduced important analytical tools, a persistent limitation remains: decentralization itself is rarely treated as a formally defined concept. Instead, as previously stated, it is typically approached through intuition and contextual interpretation or applications. This has produced a fragmented literature in which decentralization is frequently invoked yet inconsistently formalized. Early foundational work on network topology introduced what remains one of the most widely cited conceptual frameworks in the study of decentralization. Namely, a three-way typology distinguishing centralized, decentralized, and distributed network structures [5]. Baran’s work established the notion of redundancy level as the minimum number of links required to preserve network connectivity and proposed probabilistic models for evaluating resilience under node and link failure. Importantly, the separation between nodes and links already implied that network structure cannot be captured through node enumeration alone. However, despite its foundational role, the work does not provide a formal definition of decentralization itself. The categories are introduced descriptively through representative examples and intuitive reasoning, leaving decentralization as an interpreted structural pattern rather than a rigorously specified property. Moreover, the decentralized case is illustrated hierarchically, implicitly associating decentralization with hierarchy, an assumption not justified formally and not necessarily shared by decentralized systems more broadly. I-B Consensus and Trust Frameworks A similar challenge appears in distributed systems research through “The Generals Problem” and subsequent Byzantine fault tolerance literature [26]. These works introduced rigorous formal reasoning around coordination under adversarial conditions and established that reliable consensus cannot be guaranteed once adversarial participants reach or exceed 13 13 of the total. This marked a major step in connecting decentralization to provable trust and fault tolerance guarantees. However, the formalism concerns the feasibility of consensus under adversarial assumptions rather than the definition of decentralization itself. Decentralization is treated as an operational context in which consensus occurs, but its underlying structural or epistemological meaning remains assumed rather than explicitly defined. Further down the line, CAP theorem demonstrated that distributed systems cannot simultaneously guarantee consistency, availability, and partition tolerance [7]. This formalized the presence of unavoidable trade-offs in network systems and contributed to a more nuanced understanding of distributed architectures. Regardless, CAP theorem does not define decentralization either but constrains system behaviour under partition. Its significance lies in formalizing trade-offs between system properties, not in specifying what makes a system decentralized in the first place. As a result, it does not resolve the definitional ambiguity surrounding decentralization despite the works influence. The emergence of blockchain-based peer-to-peer systems substantially reshaped the academic discourse [35]. Across both the originating work and much of the literature it inspired, decentralization became increasingly associated with the removal of a trusted third party. This framing proved highly influential and effectively established blockchain as the dominant interpretation in which decentralization is discussed and evaluated. However, this is inherently domain-specific and ties decentralization to a particular technology and context. Notably, the originating work itself frames the network primarily in terms of distribution rather than decentralization, introducing a terminological ambiguity that would persist throughout subsequent literature. As a result, decentralization often became treated as synonymous with peer-to-peer distribution or trust minimization, without a formal distinction between these concepts. I-C Quantitative Proxy Measures Recent work has increasingly attempted to quantify decentralization directly. One prominent example is the Nakamoto coefficient, which measures the minimum number of entities whose compromise yields systemic control over a blockchain network [42]. This represents an important move toward measurable and multi-dimensional treatment. However, the measure remains explicitly blockchain-centric and models decentralization through subsystem control based on node counts alone. It therefore quantifies concentration within a specific architecture rather than defining decentralization as a transferable structural property. Similar limitations appear in approaches that model decentralization through effective power distributions among protocol participants, where quantification remains tightly coupled to blockchain-specific consensus mechanisms and resource assumptions [25]. Related works has also adapt measures from adjacent disciplines. Examples include the Gini coefficient, used to capture inequality in resource distribution, and Shannon entropy, used to quantify uncertainty or dispersion among participants [28]. While these approaches introduce useful analytical expressiveness, they inherit the same foundational limitation: they measure how resources or participation are distributed among nodes but do not formally define decentralization itself. They abstract away the relational structure between entities, and as a result, decentralization is fundamentally expressed through relationships and dependencies between participants rather than through participant attributes in isolation [37]. I-D Systematization A large leap in the discussion around decentralization can be observed with works that attempt to systematize and quantify decentralization in certain contexts beyond just single instantiations [46, 41]. The authors of [46] have informally drawn an epistemological distinction between distributed and decentralized systems, further grounding decentralization as a relational property where distribution is also a direct prerequisite of decentralization. However, the authors from the beginning limit their scope to privacy and put emphasis on trust among authorities. As much as that is considered to be sufficient in security literature, it creates tension for systems that are considered decentralized yet some of it’s elements are not distributed (e.g., Cryptocurrency nodes physically clustered on the same physical host, whether it would be a data center or a cloud region [13]) as well as any discussions of decentralization outside of security. The discussions around decentralization are en bloc, with greater focus on how it affects privacy rather than what decentralization is. On the other side of the spectrum lies [41], where the authors focus was the quantification of decentralization. An analytical approach is presented limited to blockchain networks (often operating on unstructured peer-to-peer architectures). Interestingly, the distinction between distribution and decentralization is not specified by the authors, and as such is not a contrasted concept. Meanwhile decentralization is defined as a form of expression within blockchain, rather than a formal concept. Once again, authors focus lies with the properties of decentralization rather than its definition, providing yet another motivation to pursue this research question. I-E Summary Taken together, the literature demonstrates substantial progress in modeling resilience, consensus, trust assumptions, and measurable concentration across distributed systems. Yet these contributions largely characterize properties associated with decentralized systems rather than formalizing decentralization itself. Existing approaches remain either descriptive or context-dependent, and it is this gap that motivates the ontology introduced in this work. IV Preliminaries IV-A Adjectival Dimensionality of Decentralization The root of the problem becomes much clearer once we view decentralization through the lens of formal semantics, as decentralization is a multi-dimensional gradable adjective, where what it attributes cannot be fully confined in a single dimension. A rudimentary work in this direction by Sassoon [40] shows that multidimensional adjectives require a dimension parameter, that is, when a speaker uses an adjective such as "healthy", the relevant dimensions are often left to context, and some binding operation determines how those dimensions jointly produce a single interpretation. This is directly relevant to decentralization, given that in much of the literature, a system is described as decentralized only with respect to some implicitly selected set of dimensions. However, the operation by which these dimensions are bound together is often left unspecified. As a result, different works may appear to disagree about decentralization while in fact relying on different contextually selected dimensions, or on different ways of combining those dimensions. Following Sassoon’s work, multidimensional adjectives may be interpreted conjunctively, disjunctively, or in a mixed manner: (a) an entity may need to satisfy all relevant dimensions, (b) one relevant dimensions, or (c) a pragmatically determined subset of them in order to fall under the adjective [40]. Decentralized appears to behave as a mixed adjective in this sense. Consider a system whose communication is decentralized, but in which aggregation authority or data remain centralized. We then begin to ask ourselves a question if that system should or shouldn’t be described as decentralized and why. Conversely, consider a system with decentralized data, but whose infrastructure depends on a small number of coordinating entities. Whether that system is decentralized or not depends on both the degrees reached along each dimension, but also on which dimensions are taken to be relevant and how they are bound into a single judgment. IV-B Distribution vs. Decentralization The academic community generally agrees that decentralization and distribution are distinct concepts. Distribution commonly refers to the placement or dispersion of system elements across multiple locations, whereas decentralization concerns the structure, organization, and governance of relationships between entities within a system [37]. If we combine this with previously discussed formal semantics, it helps explain why decentralization is difficult to compare across contexts. Both "decentralized" and "not decentralized" may be asserted with respect to some dimension of the same system, just as "centralized" and "not centralized" may also hold relative to different dimensions. Distribution, by contrast, is more commonly used disjunctively in technical discourse, where a system may be called distributed if at least one relevant class of components or resources is placed across multiple locations or entities [37]. Decentralization does not usually permit the same looseness without further clarification. Its dimensions must be specified, and the standard for satisfying them must be made explicit. This is precisely where an ontological framework becomes useful, it provides a cross-contextual structure for identifying the relevant dimensions of decentralization and for making explicit the operation by which those dimensions are combined into a coherent assessment. IV-C Formal vs. Informal Reasoning in Decentralization Broadly speaking, reasoning is defined as the a cognitive process of drawing inferences, evaluating arguments, and forming conclusions from premises. The main distinction between formal and informal reasoning is thus drawn by what governs them both. Formal reasoning requires explicit rule-bound systems (in our context provided by mathematical logic) where validity of arguments is entirely independent from the context, prioritizing form over contents. Requiring closed systems that are precisely defined any by their very nature incomplete [14]. Informal reasoning however, operates within the domain of natural language where formal logical validity alone is deemed as insufficient (or inapplicable [45]). Within this framing, the term decentralization, as it is used in computer communication systems, exposes the limitations of a strict formal/informal divide. Formally, decentralization can be specified in terms of graph-theoretic properties of distributed networks, protocol-level constraints, and the absence of a singular coordinating authority, thereby lending itself to representation within rule-governed systems of specification and verification. However, its practical invocation in discourse about system design frequently extends beyond these formal properties to include informal evaluative claims concerning robustness, transparency, and governance structure. This dual usage showcases how formal technical concepts are often interpreted through informal reasoning which incorporates contextual judgment. On that basis decentralization serves as a great example of how reasoning in computer science oscillates between formally tractable descriptions and informally grounded assessments, thereby reinforcing the view that the boundary between formal and informal reasoning is not absolute but functionally dependent in applied contexts. Although this distinction might appear as self-evident from the contemporary standpoint in the field of computer science, it is crucial to highlight for our argument. Our arguments that link decentralization, ontologies, and formal logic, will be inspired by the claims of Finocchiaro [12, 11]; specifically, those which establish that this divide is a spectrum rather than a binary divide (as it is often perceived). V The Decentralization Problem V-A Informal Reasoning as a Basis In practice, the informal component of reasoning often becomes the de facto basis upon which “decentralization” is defined and assessed [24, 34, 10, 51, 43, 31] where discourse frequently defaults to intuitively accessible but weak notions such as “absence of centralization” [47], or resorts to their own interpretations of the term [51]. The term acquires a degree of flexibility that allows it to function as a normative label as much as a technical descriptor. This presents a structural problem as evaluative judgments about decentralized systems may be formed on the basis of rhetorically important but analytically weak criteria. In this sense, “decentralization” becomes a case study in how informal reasoning can silently dominate conceptual formation. V-B Pragmatic Multi-Dimensionality of Decentralization Previous works that focus on questions around defining decentralization in different contexts have converged on a shared intuition that decentralization is not a binary property, but multi-dimensional. Notably, [37] provides an extensive treatment of this problem, drawing lessons directly from blockchain networks to present a framework that breaks down decentralization across several distinct dimensions, each of which can be independently more or less centralized. While the scope of that work remains largely confined to blockchain infrastructures and the framework itself is conceptual rather than formal, it nonetheless establishes a precedent that rigorous attempts to describe decentralization must account for multiple axes of evaluation. Across broader literature, there is a growing acknowledgment that decentralization resists a single, unified measure. As noted in [1], there exists no exact mathematical definition of decentralization; rather, quantitative approaches such as the Gini coefficient are employed to approximate the degree to which a system tends toward one end of a spectrum or the other. As such, decentralization is treated not as an absolute state, but as a scalar property. The problem associated with lack of formalization is widely acknowledged yet consistently left out of scope, leaving the field without a principled basis for comparing or reasoning about decentralization across different system types. The approach, instead, is to first informally assert what do the authors mean by decentralization, and only then follow with their analysis or design. Building on the above, we contend that a sufficiently general definition must satisfy two requirements. First, it must be multi-dimensional from the perspective of formal semantics, which in essence account for distinct aspects of a system such as data locality, model locality, and network topology, each of which contributes independently to an overall characterization. Second, it must treat each dimension as a spectrum rather than a binary truth-value, allowing for gradability rather than collapsing to an informal centralized-or-not distinction. V-C Abstraction Level for Approaches The works surveyed in Section I, taken together, reveal a recurring and systematic limitation. Namely, the level of abstraction at which decentralization has been approached has remained insufficient, constrained by the specific contexts from which each contribution emerged and by the informal reasoning that has served as its conceptual basis. Rather than producing general formal definitions, the existing literature has largely produced taxonomies and context-specific measures, a tendency we can observe from most recent attempts to formalise decentralization doing so specifically for the purpose of evaluating a particular system or setting the authors wish to examine [42, 25, 28]. The result is a body of work in which the definition follows from the application, rather than the application following from the definition. It is precisely this inversion that the present work seeks to correct by approaching decentralization at a level of abstraction sufficient to accommodate a formal treatment. V-D Culminating the Decentralization Problem Despite its prevalent use in formal settings across multiple domains of computer science, decentralization remains a concept grounded in informal reasoning, defined contextually and inconsistently such that it does not transfer reliably between applications or system classes. The methodological issues identified in the more recent literature compound this further: while later approaches acknowledge the multi-dimensional nature of decentralization as an improvement over earlier binary characterisations, they do so while simultaneously disregarding the relational aspect of communication systems, reducing their analyses to the properties of individual entities and omitting the connections between them. This tendency is closely tied to the significant influence of blockchain-based systems on the literature, which following their introduction effectively became the implicit conceptual standard against which decentralization is defined and measured. As we have already demonstrated, this is an insufficient basis for a general treatment of the concept, given the significant structural omissions it entails. It is the combination of these factors, namely the absence of more formal and transferable definition, the methodological inconsistencies in quantification, and the over-reliance on a single application context as a conceptual baseline, that altogether constitute what we refer to in this work as the Decentralization Problem. Table I consolidates our analysis by comparing representative approaches against the requirements derived from the above. Specifically, the comparison considers whether each approach: (P1) defines decentralization itself rather than an associated property; (P2) explicitly distinguishes decentralization from distribution; (P3) represents decentralization through relationships and dependencies between entities; and (P4) provides quantification transferable beyond its original application domain (these are based on earlier established research problems). We additionally report whether an approach explicitly supports multiple dimensions of decentralization and whether it provides an ontological foundation. TABLE I: Comparison of representative approaches against the requirements derived from the Decentralization Problem. Work Primary focus and scope Formal definition (P1) Decentralization/ distribution distinction (P2) Relational structure (P3) Transferable quantification (P4) Multi- dimensional Baran [5] Structural network typology and resilience under node and link failures. – △ ✓ – – Lamport et al. [26] Consensus feasibility and Byzantine fault tolerance under adversarial participation. – – △ – – Brewer [7] Trade-offs among consistency, availability, and partition tolerance in distributed systems. – – △ – – Nakamoto [35] Peer-to-peer electronic cash and removal of a trusted third party. – – △ – – Srinivasan and Lee; Kwon et al. [42, 25] Blockchain-specific concentration, subsystem control, and effective protocol power. – – – – △ Lin et al. [28] Blockchain decentralization proxies based on resource inequality and participation dispersion. – – – – △ Rossi et al. [37] Conceptual and multidimensional systematization of decentralization, primarily in blockchain infrastructures. – ✓ ✓ – ✓ Troncoso et al. [46] Privacy- and trust-oriented distinction between distributed and decentralized architectures. – ✓ ✓ – – Shahsavari et al. [41] Analytical quantification of decentralization in blockchain peer-to-peer networks. – – – – △ Our work Domain-independent, graph-based ontology with subject-specific classification and analytical metrics. ✓ ✓ ✓ ✓ ✓ Legend: ✓ indicates that the requirement is explicitly addressed; △ indicates that it is addressed partially, descriptively, or only within a restricted context; “–” indicates that the requirement is not addressed as part of the work’s treatment of decentralization. P1–P4 correspond respectively to the research problems: definitional ambiguity, confusion with distribution, absence of structural formalism, and absence of transferable quantification. The table evaluates each work only with respect to its treatment of decentralization; it does not assess the broader significance of its original technical contribution. VI Ontology as the Foundation Defining decentralization and addressing the Decentralization Problem requires a framework capable of operating simultaneously within the domain of formal reasoning and across the breadth of contextual variation that the concept exhibits in practice. Ontologies, as formalized conceptualizations of knowledge, present themselves as a natural candidate for this purpose. Meanwhile as a tool, they provide a rigorous approach within which concepts and their relationships can be defined such that logical statements about them can be decidable and quantifiable. Crucially, however, ontologies are not restricted to purely logical primitives; they are equally capable of encapsulating non-logical concepts, that is, concepts whose meaning is not fully reducible to formal symbols alone but which can nonetheless be given precise structural descriptions within an ontological framework. More importantly, ontologies allow for context-dependence while still being regarded as formal. These properties operate with the rigor demanded by formal reasoning while remaining expressive enough to capture concepts that originate from informal discourse, making ontologies a particularly well-suited tool for addressing the methodological gaps identified in our earlier discussion. The application of ontologies to decentralized settings is not without precedent. Existing work has employed ontological frameworks across a range of domains in which decentralization plays a central role, including decentralized energy systems [50, 23], eco-industrial information management [53], decentralized production control in cloud manufacturing [22], and self-organising multi-agent systems [29]. Broader cross-disciplinary treatments have also sought to map the conceptual landscape of decentralization across domains [18], and the relationship between decentralization and the organisation of knowledge itself has been explored at a philosophical level [17]. However, the existing use of ontologies in this space suffers from a limitation that closely mirrors the one identified in the preceding discussion of decentralization definitions, as the level of abstraction remains insufficient to function as a definition. In each case, the ontological framework is developed to formalize conceptual knowledge within a particular application domain, whether energy trading, manufacturing, or collective decision-making, rather than to capture decentralization as a general and transferable concept. The ontologies produced are therefore domain ontologies in the strict sense, encoding the semantics of a specific context rather than providing a foundation from which a general formal treatment of decentralization could be derived. As a result, the same problem of context-dependence and limited transferability that characterizes the definitional literature on decentralization is reproduced at the ontological level. What is absent from the existing literature is an ontological treatment of decentralization that operates at a level of abstraction sufficient to serve as a general conceptual foundation, independent of any particular application context. VII Our Definition To make our approach easier to follow, we organize our ontology-based definition into four layers: (a) Ontological Layer, where the core ontology will be defined with conceptual commitments, competency questions, signature, and taxonomy, (b) Logical Layer, where formal semantics, axiomitization, and reasoning properties will be specified, (c) Modeling Layer where we bind the graph-based topological representation, and (d) Analytical Layer where we introduce graph-based metrics as measures for decentralization and evaluate them on previous instantiations to showcase the following computational capabilities when using our definition. We do this also to make sure that our approach is aligned with important works for ontology development such as [16, 36], while using standard notation for ontology development and expressing logic [4, 3]. Similarly to those works, we also note that ontology development methodologies are idiosyncratic, as there is no standardized or universally accepted approach to their engineering. In our case, these layers are necessary to function as a definition. Provided that in formal reasoning direct definitions are bounded to the underlying context in order to make precise logical inferences. For us to have the capacity to produce truth-value expressions across many we are required to construct an ontological foundation, around which we will be able to formalize axioms governing inferences. Simultaneously, if we want to elevate this foundation onto practical systems and introduce quantifiable metrics based on the ontology, we need to specify how do we model our ontology for later analytical evaluation. Figure 1 visualises how these layers are connected altogether. Figure 1: Ontology Roadmap. VIII Ontological Layer This section aims to first formalize the core ontology behind decentralized and distributed systems (without using "decentralization" and "distribution" as primitive concepts), from which we can derive a formal definition for decentralization. VIII-A Core Ontology Ontological Commitments. Conceptually, the first and most fundamental commitment is to a graph-theoretic realism about systems. A system is taken to exist as a configuration of computational entities and their interconnections, formally representable as a non-empty graph G=(V,E)G=(V,E). This commits us to the position that systems are not merely conceptual abstractions but structured objects. The ontology does not address systems that cannot be represented as a graph, thereby implicitly constraining the domain to computer systems in which nodes are computationally capable entities (which can both physical and non-physical). A second commitment concerns temporality. Our ontology evaluates systems at a fixed point in time. The framework treats distinct operational phases as distinct topologies, where each can be assessed independently. This commitment entails that dynamic behaviour is handled through static "snapshots". The practical consequence is that a system undergoing phase-dependent structural change (such as a federated learning protocol alternating between a cryptographic setup phase and an inference phase [27]) is not treated as a single topology that evolves, but as a family of topologies corresponding to distinct points in time, each governed by its own constraint set. Lastly, the ontology adopts a multi-dimensionality for subjects of decentralization. Following from what has been extensively discussed in this work, the framework recognizes that any distributed component of a system (whether data localization, role assignment, delegation authority, or model training etc.) constitutes a legitimate subject of decentralization, provided it admits a graph-realistic representation. The cardinality of the set of such subjects for a given system defines its dimensionality of decentralization, with centralized systems constituting the degenerate case of zero-dimensional decentralization. Crucially, no cardinality constraints are imposed on this set, reflecting the empirical reality that decentralized systems vary enormously in their architectural complexity and that the ontology should remain general across instantiations. Now formally, we follow the approach presented in [16] for consistency with established ontology literature, as such, given a formal language ℒL, we construct our ontological commitment as follows: =<,>K=\ ,\,J (1) Where C is our formal conceptualization expressed as: =<Δ,,ℛ>C=\ ,\,W,\,R (2) Where Δ is our domain (containing entities relevant to the computational systems under consideration), W is a set of possible worlds (maximal states of affairs for our domain [16], in our case, arrangement of these entities and their subjects of decentralization in the system), and ℛR represents a set of conceptual relations for <Δ,> ,\,W . Meanwhile J denotes a conceptual interpretation function that maps elements from our signature Σ (vocabulary, which will be specified in the following subsections) to their intended meanings within the conceptualization C. A model (or a model structure) for the language ℒL can be expressed as <ℳ,I> ,\,I such that ℳM denotes a world-relative relational structure, while II provides the interpretation mapping over that structure. ℳM can be expressed as: ℳ=<Δ,R>M= ,\,R (3) Where RR is a set of extensions relative to the world specific structure that models ℒL such that R=p(w)|p∈ℛ,w∈R=\p(w)\ |\ p ,\,w \. Note that ℛR denotes conceptual relations at the level of conceptualization, whereas RR denotes their world-relative extensions. With our commitment we construct an ontology O for the language ℒL that commits to conceptualization C: =<Δ,,ℛ,,Σ,Φ>O= ,\,W,\,R,\,J,\, ,\, (4) A model <ℳ,I> ,\,I satisfies our ontology O as long as it satisfies our axioms φ , this follows the standard textbook notation [4]. <ℳ,I>⊨⇔∀φ∈Φ,<ℳ,I>⊨φ ,\,I ∀ ∈ ,\; ,\,I (5) Ontology Signature. In principle, we aim for our ontology to be as minimal as it is possible while sustaining the necessary expressiveness to define decentralized systems. Reason being is to avoid over-axiomatization with overly large number of primitive concepts. With that said, we define our ontology singature in the following manner. Σ=(C,R) =(C,R) (6) Where C denotes our primitive concepts and R primitive relations. Our primitive concepts consist of the following: C=System,Topology,SubjectC=\System,\,Topology,\,Subject\ (7) These terms are defined as follows: SystemSystem is a set of computational entities that admits a structural representation, TopologyTopology is the said graph-realistic relational structure representation, and SubjectSubject is a context-relative projection over the structure representation. Similarly, our primitive relations contain: R= R=\ hasTopology,hasSubject,hasProjection, hasTopology,\,hasSubject,\,hasProjection, (8) ofSubject,inTopology,hasRealization, ofSubject,\,inTopology,\,hasRealization,\, realizedAt realizedAt\,\ Where hasTopologyhasTopology indicates a relation between a given graph-based structure (which can be expressed with graph-theoretic methods; e.g., adjacency matrices) and a system instance, while hasSubjecthasSubject describes a relation between a system instance and a context-relative subject. The relation hasProjectionhasProjection connects a system to a projection particular p; ofSubjectofSubject and inTopologyinTopology identify, respectively, the unique subject and topology associated with p; hasRealizationhasRealization relates p to its distinct realization particulars; and realizedAtrealizedAt links each realization particular to the vertex at which it is realized. Projection and realization particulars are not introduced as additional primitive concepts as they are individuals in Δ identified through these relation patterns. Ontology Taxonomy and Schema. Next, we establish a taxonomy as its proper part [3] where a hierarchy is required as it follows a graph-theoretic structure with a single root node. However, we are interested in addressing the decentralization problem through a clear separation of representational units that altogether constitute a system (specifically, those that can characterize a system particular as canonically decentralized or not). This must be done without constructing our taxonomy backwards from the terms of decentralization or distribution, as that would risk cyclic definitions. This way we also avoid binding ourselves to a specific context in which decentralization could be used in. Adding to that, providing the context-dependent expressions of SubjectSubject and TopologyTopology, it isn’t possible to construct an is_a hierarchy between them (as well as the SystemSystem) as they are described via other relations specified in R. Instead, each concept in C has a respective taxonomic backbone. In order to sustain perspectivalism and realism of our ontology, both TopologyTopology and SubjectSubject taxonomies can be expanded upon and established with specific contexts during the design and/or evaluation of a system. Figure 2: System-class taxonomic backbone. Meanwhile, the set of representational units subsumed by SystemSystem concept is finite (Figure 2), each describing the extent of decentralization a system has that follows from our system-centric ontological schema (Figure 3). The schema shows the mapping of non-taxonomical relations from R across all concepts in C, along with additional relations describing how particulars are instantiated (for TopologyTopology). Figure 3: System-centric ontology schema. Individuals p, r, and v denote representative instances (particulars) used to illustrate object property assertions between ontology classes. Dashed arrows indicate informal conceptual associations only; v is obtained from the topology representation and r represents a realization of the subject, included solely to aid interpretation. Solid arrows denote ontology roles. More on Projection Particulars and Multiplicity. The last caveat that needs addressing is the intention and interpretation of projection particulars. Our ontology identifies a projection particular p through its relational structure. Under an interpretation II, the set of projection particulars is derived as: ΠI=p∈ΔI| ^I= \p∈ ^I\ | ∃s,u,T(hasProjection(s,p) ∃ s,u,T\, (hasProjection(s,p) ∧ofSubject(p,u)∧inTopology(p,T)) ofSubject(p,u) inTopology(p,T) ) \ For a subject u belonging to a system s with topology T, we write pup_u for the unique projection particular satisfying hasProjection(s,pu)hasProjection(s,p_u), ofSubject(pu,u)ofSubject(p_u,u), and inTopology(pu,T)inTopology(p_u,T). The multiplicity of the subject in that context is then derived from the distinct realization particulars related to pup_u. Intuitively, p can be thought of as a projection linking the system, topology, and subject together. This distinction is necessary because decentralization and distribution are not equivalent. We later derive a function δ(pu)δ(p_u) which counts distinct realizations of the subject, whereas the derived function λ(pu)λ(p_u) counts the distinct vertices supporting one or more of those realizations. Multiple realization particulars may therefore be located at the same vertex, allowing δ(pu)>1δ(p_u)>1 while λ(pu)=1λ(p_u)=1. Centralization and decentralization depend on δ, while distribution depends on λ, preserving the intended separation between multiplicity and spatial dispersion. Notice how this interpretation is substantially different from works like [46] which focus on trust and direct control, but often limit the discussion on what can and cannot be controlled (e.g., an entity holds data which belongs to multiple clients is cryptographically bidden, that same entity can erase data completely from itself but cannot modify it without clients noticing) as well as particular contexts (blockchain-based privacy). This is partly the reason why our focus shifts directly onto subjects themselves, as we want to avoid restricting ourselves to particular elements that are associated with decentralization in some contexts. Competency Questions. The questions this ontology answers address the existence and extent of decentralization across different dimensions in reference to the graph-realistic structure any system can be described as. It allows to identify the exact components that are to be decentralized (subjects of decentralization) in accordance with the intended function of the system (contextual operationalism), as well as the topological relations defining how those components are structurally positioned and interconnected (structural relationalism). Take note that, in principle, competency questions themselves are not a part of our ontology, but are here to present external constraints for validation. With that said, to construct our competency questions we follow the methodology from [15], where we first phrase them informally to then develop their formal instances in later sections: • CQ 1. When is a component considered to be centralized or decentralized? • CQ 2. When is a component considered to be distributed? • CQ 3. How many dimensions of decentralization does a system have? • CQ 4. How do we derive a notion for centralized system from decentralization? • CQ 5. Can we state that a system is fully decentralized? • CQ 6. Can a system be decentralized in one component but centralized in another? Altogether, above competency questions make up for external constraints on acceptable models of the ontology. IX Logical Layer Semantics. We approach semantics for our ontology using competency questions and formalize their semantic expressions. Take note that we carry over the symbol notation throughout our semantics due to space limits. CQ 1. A singular component which is considered to be a subject is described as centralized if and only if the multiplicity of its unique projection particular is equal to 1, implying that the system contains one realization of that subject on the associated topology: δ(pu)=1 δ(p_u)=1 ⇔Centralized(u), Centralized(u), where hasProjection(s,pu) hasProjection(s,p_u) ∧ofSubject(pu,u)∧inTopology(pu,T). ofSubject(p_u,u) inTopology(p_u,T). Conversely, that same subject is considered to be decentralized if and only if its projection multiplicity is greater than 1: δ(pu)>1⇔Decentralized(u)δ(p_u)>1 Decentralized(u) CQ 2. The notion of distribution in our ontology is defined by the subject realizations over vertices. We say that a subject of decentralization is distributed if and only if the realizations associated with its projection particular are located over more than one vertex. λ(pu)>1⇔Distributed(u)λ(p_u)>1 Distributed(u) Here, λ(pu)=|Vpu|λ(p_u)=|V_p_u|, where Vpu⊆V_p_u V is the derived set of vertices supporting at least one realization associated with pup_u and the realization of T is a graph G(V,E)G(V,E). This separation allows us to semantically differentiate between decentralization and distribution. CQ 3. Dimensionality of decentralization in a given model is defined by the count of subjects under which it is decentralized: dim(s) dim(s) =|Ud(s)|, =|U_d(s)|, Ud(s) U_d(s) =u|hasSubject(s,u) = \u\ |\ hasSubject(s,u) ∧∃p(hasProjection(s,p) ∃ p\, (hasProjection(s,p) ∧ofSubject(p,u)∧δ(p)>1) ofSubject(p,u) δ(p)>1 ) \ CQ 4. Centralization of a whole model is implied directly by the absence of decentralized subjects, meaning that the dimension of decentralization for a centralized system equals 0: Centralized(s)⇔ Centralized(s) System(s) System(s) ∧∀p(hasProjection(s,p) ∀ p\, (hasProjection(s,p) →δ(p)=1), 68.2866pt→δ(p)=1 ), Centralized(s)⟹ Centralized(s) dim(s)=0 dim(s)=0 CQ 5. Now we semantically introduce a formal distinction between a model that has some decentralization and one that is fully decentralized. Full decentralization requires at least one subject projection and requires every projection particular associated with the system to have multiplicity greater than 1: System(s)∧∃phasProjection(s,p) System(s) ∃ p\,hasProjection(s,p) ∧∀p(hasProjection(s,p)→δ(p)>1) ∀ p\, (hasProjection(s,p)→δ(p)>1 ) ⟹Fully Decentralized(s) 28.45274pt Fully Decentralized(s) Similarly, a model is partially decentralized when it contains at least one centralized projection particular and at least one decentralized projection particular: System(s)∧∃pc,pd( System(s) ∃ p_c,p_d\, ( hasProjection(s,pc)∧δ(pc)=1 hasProjection(s,p_c) δ(p_c)=1 ∧hasProjection(s,pd)∧δ(pd)>1) hasProjection(s,p_d) δ(p_d)>1 ) ⟹Partially Decentralized(s). 28.45274pt Partially Decentralized(s). CQ 6. Here we are essentially asking ourselves if different subjects of decentralization within the same system can be classified differently in terms of decentralization, which directly relates to the notion of decentralization targeting unique subjects. It is satisfiable to answer this if one can demonstrate that the same system has one subject that is centralized and another that is decentralized. Axiomitization. The axioms presented here serve as a specification by encoding formal constraints over the ontology’s intended models in accordance with [15]. Where the semantics specify what it means for a subject to be centralized or for a system to be fully decentralized, and competency questions clarify the purpose and exact questions our ontology is meant to answer, here we specify what must hold in any model satisfying O, ruling out interpretations that are formally consistent with Σ but ontologically unintended. We use description-logic TBox statements for role typing and cardinality restrictions, supported by first-order constraints where coherence between several role paths must be expressed. We follow the standard treatment of [4] for TBox construction. Domain and Range. We define every relation in R through formal domain, range, and relational-structure constraints: ⊤⊑∀hasTopology−.System⊤⊑∀hasTopology.Topology⊤⊑∀hasSubject−.System⊤⊑∀hasSubject.Subject⊤⊑∀hasProjection−.System⊤⊑∀ofSubject.Subject⊤⊑∀inTopology.Topology∃ofSubject.⊤⊑∃hasProjection−.System∃inTopology.⊤⊑∃hasProjection−.System∃hasRealization.⊤⊑∃hasProjection−.System∃realizedAt.⊤⊑∃hasRealization−⊤ array[]l ∀ hasTopology^-.System\\ ∀ hasTopology.Topology\\ \\[-2.0pt] ∀ hasSubject^-.System\\ ∀ hasSubject.Subject\\ \\[-2.0pt] ∀ hasProjection^-.System\\ ∀ ofSubject.Subject\\ ∀ inTopology.Topology\\ \\[-2.0pt] ∃ ofSubject. ∃ hasProjection^-.System\\ ∃ inTopology. ∃ hasProjection^-.System\\ ∃ hasRealization. ∃ hasProjection^-.System\\ ∃ realizedAt. ∃ hasRealization^- array Since projection and realization particulars are not primitive concepts, their admissible extensions are derived from the relation patterns above rather than named as additional classes. Existence and Cardinality. Existence of decentralizable subjects is not assumed by our ontology as a model with no subjects remains admissible and has dimensionality 0. Consequently, no positive lower-bound axiom is imposed on hasSubjecthasSubject. Every system must, however, have exactly one topology: System⊑(=1hasTopology.Topology)System (=1\,hasTopology.Topology) Because subjects are context-relative particulars, every subject belongs to exactly one system: Subject⊑(=1hasSubject−.System)Subject (=1\,hasSubject^-.System) Every projection particular belongs to exactly one system and, within that system, has exactly one subject, exactly one topology, and at least one realization particular: ∃hasProjection−.⊤ ∃ hasProjection^-. ⊑(=1hasProjection−.System), (=1\,hasProjection^-.System), System System ⊑∀hasProjection.( ∀ hasProjection. ( (=1ofSubject.Subject) (=1\,ofSubject.Subject) ⊓(=1inTopology.Topology) (=1\,inTopology.Topology) ⊓(≥1hasRealization.⊤)) (≥ 1\,hasRealization. ) ) Every realization particular belongs to exactly one projection particular and is located at exactly one vertex: ∃hasRealization−.⊤⊑ ∃ hasRealization^-. (=1hasRealization−.⊤) (=1\,hasRealization^-. ) ⊓(=1realizedAt.⊤) (=1\,realizedAt. ) The following first-order constraints ensure that the role fillers refer to the same system context and that every subject admitted by a system has exactly one projection particular: ∀s,u( ∀ s,u\, ( hasSubject(s,u) hasSubject(s,u) →∃!p(hasProjection(s,p) →∃!p\, (hasProjection(s,p) ∧ofSubject(p,u))) 88.20354pt ofSubject(p,u) ) ) ∀s,p( ∀ s,p\, ( hasProjection(s,p) hasProjection(s,p) →∃!u(hasSubject(s,u)∧ofSubject(p,u)) →∃!u\, (hasSubject(s,u) ofSubject(p,u) ) ∧∃!T(hasTopology(s,T)∧inTopology(p,T))) → ∃!T\, (hasTopology(s,T) inTopology(p,T) ) ) Finally, realization locations (vertices) must belong to the graph that realizes the topology associated with the same projection: ∀p,T,r,v( ∀ p,T,r,v\, ( inTopology(p,T)∧hasRealization(p,r) inTopology(p,T) hasRealization(p,r) ∧realizedAt(r,v) realizedAt(r,v) →v∈VT) → v∈ V_T ) Where VTV_T denotes the vertex set of the graph realizing T. Relation to Multiplicity. The core requirement for the ontology to function is the ability to quantify the projection count for each subject. We derive the realization set of each projection particular p∈ΠIp∈ ^I as: ℛp=r∈ΔI| _p= \r∈ ^I\ | (p,r) (p,r) ∈hasRealizationI ∈ hasRealization^I \ Admissible models require ℛpR_p to be finite and non-empty. Projection multiplicity is then the positive-integer-valued function: δ:ΠI δ: ^I ⟶ℕ>0, _>0, δ(p) δ(p) =|ℛp| =|R_p| Here, p is a derived projection particular and r ranges over the distinct realization particulars associated with it. ℛpR_p is the set collected by the relation hasRealizationhasRealization. To distinguish multiplicity from distribution, we separately derive the vertex set: Vp=v∈VT| V_p= \v∈ V_T\ | ∃r∈ℛp ∃ r _p\, ((r,v)∈realizedAtI), ((r,v)∈ realizedAt^I ) \, and the vertex cardinality function: λ:ΠI λ: ^I ⟶ℕ>0, _>0, λ(p) λ(p) =|Vp| =|V_p| Because every realization particular has exactly one location, while several realizations may share that location, every admissible projection satisfies: 1≤λ(p)≤δ(p)1≤λ(p)≤δ(p) Consequently, δ(p)>1δ(p)>1 and λ(p)=1λ(p)=1 is satisfiable, while δ(p)=1δ(p)=1 entails λ(p)=1λ(p)=1. Refined Concepts. Our ontology derives concepts such as Centralized(s) Centralized(s) and Decentralized(s) Decentralized(s) from the primitive signature. Starting from centralization and decentralization of individual subjects: Centralized(u)⇔ Centralized(u) Subject(u)∧∃p( Subject(u) ∃ p\, ( ofSubject(p,u)∧δ(p)=1), ofSubject(p,u) δ(p)=1 ), Decentralized(u)⇔ Decentralized(u) Subject(u)∧∃p( Subject(u) ∃ p\, ( ofSubject(p,u)∧δ(p)>1) ofSubject(p,u) δ(p)>1 ) A centralized SystemSystem is axiomatically defined by every projection particular associated with it having multiplicity equal to one. A system with no subjects satisfies this condition vacuously and therefore constitutes the zero-dimensional case. Centralized(s)⇔ Centralized(s) System(s) System(s) ∧∀p(hasProjection(s,p) ∀ p\, (hasProjection(s,p) →δ(p)=1). 68.2866pt→δ(p)=1 ). The notion of distribution is derived from the vertex cardinality of the unique projection particular associated with a subject: Distributed(u)⇔ Distributed(u) Subject(u) Subject(u) ∧∃p(ofSubject(p,u)∧λ(p)>1). ∃ p\, (ofSubject(p,u) λ(p)>1 ). If this expression is not satisfied by the subject, we classify it as Undistributed, or simply, ¬Distributed Distributed. Partial and Full system decentralization are established with respect to the earlier notion of decentralization of a subject: Fully Decentralized(s)⇔ Fully Decentralized(s) System(s)∧∃phasProjection(s,p) -85.31874ptSystem(s) ∃ p\,hasProjection(s,p) ∧∀p(hasProjection(s,p)→δ(p)>1), -85.31874pt ∀ p\, (hasProjection(s,p)→δ(p)>1 ), Partially Decentralized(s)⇔ Partially Decentralized(s) System(s)∧∃pc,pd( -85.31874ptSystem(s) ∃ p_c,p_d\, ( hasProjection(s,pc)∧δ(pc)=1 -85.31874pt hasProjection(s,p_c) δ(p_c)=1 ∧hasProjection(s,pd)∧δ(pd)>1) -85.31874pt hasProjection(s,p_d) δ(p_d)>1 ) It is important to note that we can only say as far as that a system is Fully Decentralized with respect to established subjects of decentralization. As we account for perspectivalism required from ontologies (there are multiple correct descriptions of reality). This, although at first might seem like a downside, doesn’t void applicability as even these different descriptions of the same system will still be comparable. The positive integer codomain of δ yields an exhaustive and mutually exclusive classification of every subject projection: ∀u( ∀ u\, ( Subject(u) Subject(u) →(Centralized(u)⊻Decentralized(u))), → ( Centralized(u) Decentralized(u) ) ), Centralized(u)∧Decentralized(u) Centralized(u) Decentralized(u) →⊥. 68.2866pt→ . Where ⊻ denotes exclusive disjunction. Reasoning Properties. We validate the reasoning properties of our ontology using standard approaches by showing: consistency, satisfiability, disjointness, and entailment. Definition 1. (Consistency). Let O be an ontology with a set of axioms Φ . The ontology O is said to be consistent if and only if there exists a structure ℳM and an interpretation II such that ℳM under II satisfies every axiom in Φ . Formally: Consistent() (O) ⇔∃<ℳ,I> such that <ℳ,I>⊨ ∃ ,\,I such that ,\,I ⇔∃<ℳ,I> such that ∀φ∈Φ, ∃ ,\,I such that ∀ ∈ , <ℳ,I>⊨φ 62.59605pt ,\,I Proof. To prove consistency we are required to show that there exists an interpretation that models our ontology: ∃<ℳ,I>⊨∃ ,\,I To do that, we show that the interpretation satisfies all axioms in Φ as stated by Equation 5. Construct an interpretation I=<ΔI,⋅I>I= ^I,·^I and a finite witness structure =s,t,U,P,Q,V W=\s,t,U,P,Q,V\, where s∈SystemIs∈ System^I, t∈TopologyIt∈ Topology^I is realized by a non-empty finite graph G(V,E)G(V,E), U=u1,…,un⊆SubjectIU=\u_1,…,u_n\ Subject^I, P=p1,…,pn⊆ΠIP=\p_1,…,p_n\ ^I, and Q=⋃iℛpiQ= _iR_p_i. For every uiu_i, let pip_i be its unique projection particular, let ℛpiR_p_i be a finite non-empty set, and let ℓ:Q→V :Q→ V assign each realization particular to exactly one vertex. Role interpretations are: hasTopologyI=(s,t),hasSubjectI=(s,ui):ui∈U,hasProjectionI=(s,pi):pi∈P,ofSubjectI=(pi,ui):1≤i≤n,inTopologyI=(pi,t):pi∈P,hasRealizationI=(pi,r):pi∈P,r∈ℛpi,realizedAtI=(r,ℓ(r)):r∈Q array[]lhasTopology^I=\(s,t)\,\\ hasSubject^I=\(s,u_i):u_i∈ U\,\\ hasProjection^I=\(s,p_i):p_i∈ P\,\\ ofSubject^I=\(p_i,u_i):1≤ i≤ n\,\\ inTopology^I=\(p_i,t):p_i∈ P\,\\ hasRealization^I=\(p_i,r):p_i∈ P,\ r _p_i\,\\ realizedAt^I=\(r, (r)):r∈ Q\ array Under this interpretation, every projection has one system, subject, and topology, at least one finite realization, and every realization has one topological location. All axioms in Φ are therefore satisfied: <,I>⊨ W,\,I Hence, the ontology is consistent. □ Definition 2. (Disjointness) Let O be an ontology, and let C and D be concepts in the signature of O. The concepts C and D are said to be disjoint with respect to O if and only if their interpretations have no common instances in every model of O. Formally: Disjoint(C,D) _O(C,D) ⇔⊨C⊓D⊑⊥ C D ⇔∀<ℳ,I>, ∀ ,\,I , <ℳ,I>⊨⟹CI∩DI=∅. 28.45274pt ,\,I C^I∩ D^I= . Proof. We show disjointness of concepts in our ontology by expanding the derived definitions. For every subject u, its unique projection particular pup_u satisfies exactly one of δ(pu)=1δ(p_u)=1 and δ(pu)>1δ(p_u)>1, because δ(pu)∈ℕ>0δ(p_u) _>0. Therefore: Centralized(u)∧Decentralized(u)→⊥ Centralized(u) Decentralized(u)→ It follows immediately that Fully Decentralized(s) Fully Decentralized(s) and Centralized(s) Centralized(s) are disjoint. The same logic applies to the disjointness of Partially Decentralized(s) Partially Decentralized(s) from both Fully Decentralized(s) Fully Decentralized(s) and Centralized(s) Centralized(s), since the system-level definitions require mutually exclusive combinations of projection multiplicities. The notion of distribution is defined over vertex count rather than realization multiplicity. A subject is distributed if and only if λ(pu)>1λ(p_u)>1, whereas centralization and decentralization are determined by δ(pu)=1δ(p_u)=1 and δ(pu)>1δ(p_u)>1, respectively. Since every admissible projection satisfies 1≤λ(pu)≤δ(pu)1≤λ(p_u)≤δ(p_u), assuming that a subject is both Distributed(u) Distributed(u) and Centralized(u) Centralized(u) would require λ(pu)>1λ(p_u)>1 and δ(pu)=1δ(p_u)=1, contradicting λ(pu)≤δ(pu)λ(p_u)≤δ(p_u). Hence: Distributed(u)∧Centralized(u)→⊥ Distributed(u) Centralized(u)→ By contrast, Distributed(u)∧Decentralized(u) Distributed(u) Decentralized(u) is satisfiable whenever δ(pu)>1δ(p_u)>1 and λ(pu)>1λ(p_u)>1. A subject may also be both Undistributed and Decentralized whenever δ(pu)>1δ(p_u)>1 but λ(pu)=1λ(p_u)=1, because multiple realization particulars may be located at the same vertex. This reinforces the claim that distribution and decentralization are distinct properties. Hence, our ontology satisfies disjointness. □ Definition 3. (Satisfiability) Let O be an ontology, and let C be a concept in the signature of O. The ontology O is said to be satisfiable if and only if it admits at least one model. The concept C is said to be satisfiable with respect to O if and only if its interpretation is non-empty in at least one model of O. Formally: Satisfiable() (O) ⇔∃<ℳ,I>, ∃ ,\,I , <ℳ,I>⊨, 28.45274pt ,\,I , Satisfiable(C) _O(C) ⇔⊭C⊑⊥ C ⇔∃<ℳ,I>, ∃ ,\,I , <ℳ,I>⊨andCI≠∅ 28.45274pt ,\,I C^I≠ Proof. The first expression is equivalent to consistency and therefore holds for our ontology. For concept satisfiability, the domain and range axioms are satisfied by the witness model W. Every hasTopologyhasTopology pair has a source in SystemISystem^I and a target in TopologyITopology^I; every hasSubjecthasSubject pair has a source in SystemISystem^I and a target in SubjectISubject^I; and every projection and realization particular satisfies the relation pattern constraints imposed by hasProjectionhasProjection, ofSubjectofSubject, inTopologyinTopology, hasRealizationhasRealization, and realizedAtrealizedAt. The witness model satisfies the existence and cardinality axioms, as s has exactly one topology t; each ui∈Uu_i∈ U belongs to s and has exactly one associated projection particular pip_i; each pip_i has exactly one subject and topology and a finite non-empty realization set ℛpiR_p_i; and each r∈ℛpir _p_i has exactly one location in V. Consequently, δ(pi)≥1δ(p_i)≥ 1 and λ(pi)≥1λ(p_i)≥ 1 for every admitted projection particular. Refined concepts follow as every positive integer satisfies exactly one of =1=1 and >1>1. Hence each subject is classified as exactly one of Centralized and Decentralized, and each system is classified according to the multiplicities of its associated projection particulars. A system with no subjects remains centralized through the vacuous universal condition and has dim(s)=0dim(s)=0. The distributed predicate is satisfiable by choosing at least one projection pkp_k whose realization particulars occupy at least two distinct vertices, so that λ(pk)>1λ(p_k)>1. Formally: ⟨,I⟩⊨ W,I ∃u∈SubjectI∃p( ∃ u∈ Subject^I\,∃ p\, ( ofSubject(p,u)∧λ(p)>1 ofSubject(p,u) λ(p)>1 ∧Distributed(u)) Distributed(u) ) Therefore, all concepts are satisfiable within O. □ Definition 4. (Entailment) Let O be an ontology, and let α be an axiom or assertion expressed in the signature of O. The ontology O is said to entail α if and only if every model of O is also a model of α. Formally: ⊨α α ⇔∀<ℳ,I>, ∀ ,\,I , <ℳ,I>⊨⟹<ℳ,I>⊨α. 28.45274pt ,\,I ,\,I α. Proof. To validate entailment, we show that every model satisfying the ontology also satisfies the derived classifications. Let 1=s,t,U,P,Q,V W_1=\s,t,U,P,Q,V\ with U=u1,u2U=\u_1,u_2\. If every p∈Pp∈ P associated with s satisfies δ(p)>1δ(p)>1, then s∈Fully Decentralized(s)s∈ Fully Decentralized(s). If one associated projection satisfies δ(p)=1δ(p)=1 and another satisfies δ(p)>1δ(p)>1, then s∈Partially Decentralized(s)s∈ Partially Decentralized(s). Otherwise, all associated projections satisfy δ(p)=1δ(p)=1, and therefore s∈Centralized(s)s∈ Centralized(s). Dimensionality follows from the cardinality of the set of associated subjects whose projection multiplicity is greater than one. Entailment for distribution follows directly from the λ function. For any model satisfying O, if the unique projection particular pup_u associated with subject u satisfies λ(pu)>1λ(p_u)>1, then Distributed(u) Distributed(u) holds by definition: ⊨Distributed(u)⇔λ(pu)>1.O Distributed(u) λ(p_u)>1. □ Evaluation. We approach evaluation of our ontology twofold: (a) we evaluate our competency questions so that we can show how our ontology can actually provide answers to them, and (b) we provide instantiations of our ontology using architectural models such as Block-chain, and Federated Learning. Competency Questions Evaluation. CQ 1 is answered by the Centralized(u) Centralized(u) and Decentralized(u) Decentralized(u) definitions. Subject u is centralized if and only if the multiplicity of its unique projection particular satisfies δ(pu)=1δ(p_u)=1, and decentralized if and only if δ(pu)>1δ(p_u)>1. These are necessary and sufficient conditions derived directly from the axioms, with mutual exclusivity guaranteed by the positive-integer codomain of δ. CQ 2 is answered by the Distributed(u) Distributed(u) predicate. Subject is distributed if and only if λ(pu)>1λ(p_u)>1. The separate functions δ and λ allow distribution to be distinguished from projection multiplicity. CQ 3 is answered by dim(s)=|Ud|dim(s)=|U_d|, the cardinality of the set of decentralized subjects. Any model satisfying O admits a count of its decentralized subjects, yielding the dimensionality. CQ 4 follows directly: a system with dim(s)=0dim(s)=0 satisfies Centralized(s) Centralized(s), recovering the classical notion of a centralized system as the degenerate zero-dimensional case. CQ 5 is answered by the Fully Decentralized(s) Fully Decentralized(s) and Partially Decentralized(s) Partially Decentralized(s) definitions. Full decentralization holds when the system has at least one projection particular and every such projection has multiplicity greater than one; partial decentralization holds when at least one projection has multiplicity one and at least one has multiplicity greater than one. Satisfiability and entailment together guarantee that a given system model is classified under exactly one of the three system-level concepts. CQ 6 is answered by the combination of CQ 3 and CQ 5: since each subject is independently assessed through its own projection particular, it is entirely possible and consistent for Decentralized(ui) Decentralized(u_i) and Centralized(uj) Centralized(u_j) to hold simultaneously for distinct subjects ui,uju_i,u_j within the same system s. This yields Partially Decentralized(s) Partially Decentralized(s) for that system. Intantiations. We instantiate examples to be first evaluated ontologically to demonstrate inference properties across different contexts. Later we evaluate the same instances analytically. Instantiation 1: Vanilla Federated Learning. We first consider a star-topology federated learning system where v1v_1 is the aggregation server and v2v_2-v5v_5 are clients that hold private data and perform local model training. We identify three subjects of decentralization: • udatau_data: data storage, projected over the client nodes in 4 instances v2,v3,v4,v5v_2,v_3,v_4,v_5, δ(pdata)=4⟹Decentralized(udata)δ(p_data)=4 Decentralized(u_data). • utrainu_train: local model training, also projected in the same fashion over v2,v3,v4,v5v_2,v_3,v_4,v_5, δ(ptrain)=4⟹Decentralized(utrain)δ(p_train)=4 Decentralized(u_train). • uAggAuthu_AggAuth: aggregation authority, one instance projected only over the aggregation server v1v_1, δ(pAggAuth)=1⟹Centralized(uAggAuth)δ(p_AggAuth)=1 Centralized(u_AggAuth). Under this instantiation, dim(sFL)=|udata,utrain|=2dim(s_FL)=|\u_data,u_train\|=2. Since data storage and local model training are decentralized while aggregation authority remains centralized, we infer using our ontology that this system is Partially Decentralized(sFL) Partially Decentralized(s_FL). Corresponding to the conventional cross-silo federated learning architecture in which data and local computation are decentralized across clients, while global aggregation remains under the authority of a single server [21, 8]. Moreover, λ(pdata)=λ(ptrain)=4λ(p_data)=λ(p_train)=4. Thus, data storage and local model training are Distributed, while aggregation authority is not. Consequently, the system is both partially decentralized and distributed. Instantiation 2: Decentralized Federated Learning. We next consider a federated learning system consisting of five client entities, each of which holds private data and performs local model training. Aggregation authority is jointly realized by v1v_1, v2v_2, and v3v_3. We again identify three subjects: • udatau_data: data storage, projected over all nodes v1,v2,v3,v4,v5v_1,v_2,v_3,v_4,v_5, δ(pdata)=5⟹Decentralized(udata)δ(p_data)=5 Decentralized(u_data). • utrainu_train: local model training, also projected over all five nodes, δ(ptrain)=5⟹Decentralized(utrain)δ(p_train)=5 Decentralized(u_train). • uAggAuthu_AggAuth: aggregation authority, jointly projected over v1,v2,v3v_1,v_2,v_3, δ(pAggAuth)=3⟹Decentralized(uAggAuth)δ(p_AggAuth)=3 Decentralized(u_AggAuth). Here, dim(sFL)=|udata,utrain,uAggAuth|=3dim(s_FL)=|\u_data,u_train,u_AggAuth\|=3. Since every identified subject is decentralized, the system satisfies Fully Decentralized(sFL) Fully Decentralized(s_FL). Furthermore, λ(pdata)=λ(ptrain)=5λ(p_data)=λ(p_train)=5 and λ(pAggAuth)=3λ(p_AggAuth)=3. All three subjects are therefore Distributed, yielding a system that is both fully decentralized and distributed. Compared with the first federated learning instantiation, the decentralization dimension increases from 22 to 33 because aggregation authority is no longer realized by a single entity. Instantiation 3: Blockchain 1. We consider a permissioned blockchain over seven nodes. Node v2v_2 acts as the sole block-ordering authority, the four core nodes v2,v3,v4,v5\v_2,v_3,v_4,v_5\ replicate the ledger, and the endpoint nodes v1,v6,v7\v_1,v_6,v_7\ submit transactions. We identify three subjects: • uconsu_cons: consensus authority, projected only over v2\v_2\, δ(pcons)=1⟹Centralized(ucons)δ(p_cons)=1 Centralized(u_cons). • uledgeru_ledger: ledger replication, projected over the four core nodes v2,v3,v4,v5v_2,v_3,v_4,v_5, δ(pledger)=4⟹Decentralized(uledger)δ(p_ledger)=4 Decentralized(u_ledger). • utxu_tx: transaction submission, projected over v1,v6,v7v_1,v_6,v_7, δ(ptx)=3⟹Decentralized(utx)δ(p_tx)=3 Decentralized(u_tx). Under this instantiation, dim(sBC)=|uledger,utx|=2dim(s_BC)=|\u_ledger,u_tx\|=2. Since ledger replication and transaction submission are decentralized while consensus authority remains centralized, the system satisfies Partially Decentralized(sBC) Partially Decentralized(s_BC). The corresponding realization cardinalities are λ(pcons)=1λ(p_cons)=1, λ(pledger)=4λ(p_ledger)=4, and λ(ptx)=3λ(p_tx)=3. Accordingly, ledger replication and transaction submission are Distributed. The system is therefore both partially decentralized and distributed. Instantiation 4: Blockchain 2. We retain the same seven-node blockchain system but assign its functions differently. Consensus authority is jointly realized by the four core nodes v2,v3,v4,v5\v_2,v_3,v_4,v_5\, every node replicates the ledger, and transaction submission is exposed through the two gateway nodes v3,v4\v_3,v_4\. The three subjects are instantiated as follows: • uconsu_cons: consensus authority, projected over v2,v3,v4,v5\v_2,v_3,v_4,v_5\, δ(pcons)=4>1⟹Decentralized(ucons)δ(p_cons)=4>1 Decentralized(u_cons). • uledgeru_ledger: ledger replication, instances projected over all seven nodes v1,…,v7v_1,…,v_7, δ(pledger)=7>1⟹Decentralized(uledger)δ(p_ledger)=7>1 Decentralized(u_ledger). • utxu_tx: transaction submission, two instances projected over the gateway nodes v3,v4\v_3,v_4\, δ(ptx)=2>1⟹Decentralized(utx)δ(p_tx)=2>1 Decentralized(u_tx). Under this instantiation, dim(sBC-D)=|ucons,uledger,utx|=3dim(s_BC-D)=|u_cons,u_ledger,u_tx|=3. Since all three subjects are decentralized, the system satisfies Fully Decentralized(sBC) Fully Decentralized(s_BC). Moreover, λ(pcons)=4λ(p_cons)=4, λ(pledger)=7λ(p_ledger)=7, and λ(ptx)=2λ(p_tx)=2. Every subject is therefore Distributed, yielding a system that is both fully decentralized and distributed. Compared with the preceding blockchain instantiation, the decentralization dimension increases from 22 to 33 because consensus authority is jointly realized by multiple entities. These paired instantiations demonstrate that decentralization is evaluated relative to individual subjects rather than assigned as an undifferentiated property of the entire system. Two systems within the same application context may therefore have different decentralization dimensions depending on the multiplicities of their subject projections. The distinction between decentralization and distribution is also preserved as decentralization concerns the number of distinct realizations associated with a subject projection, whereas distribution concerns whether those realizations occupy multiple system vertices. What follows from the instantiations is the reason why the number of instances of a subject must be distinguished from the number of vertices over which those instances are realized. As we reiterate for clarity, δ(pu)δ(p_u) denotes the projection multiplicity of u, meaning the number of distinct realization particulars associated with its unique projection particular. By contrast, λ(pu)λ(p_u) is the number of distinct vertices that support one or more of those realizations. Hence, several distinct realizations of the same subject may be located at a single vertex. For example, it is possible that δ(pu)=4δ(p_u)=4 while λ(pu)=1λ(p_u)=1. Under the ontology, such a subject is Decentralized, because it has multiple realizations, but Undistributed, because all of those realizations occur at one topological location. This way we can capture deployment arrangements that would otherwise be obscured by counting nodes alone. A cryptocurrency network may contain multiple logically distinct protocol nodes while those nodes remain physically located on the same host, or concentrated within a single data centre [13]. When the topology is instantiated at the level of physical deployment locations, these nodes contribute multiple realizations to δ(pu)δ(p_u) but may contribute only one vertex, or a small number of vertices, to λ(pu)λ(p_u). The system may therefore exhibit decentralization with respect to protocol participation while remaining undistributed with respect to its physical infrastructure. Intuitively, this is consequence of decentralization and distribution describing different ontological properties rather than a contradiction. The difference also explains why decentralization admits partially and fully decentralized system classes, whereas distribution is not assigned a partially distributed class. Following Sassoon’s account of multidimensional adjectives and formal semantics, an adjective may depend on several contextually relevant dimensions whose contributions are bound conjunctively, disjunctively, or through a mixed interpretation [40]. In this framework, decentralized behaves as a multidimensional gradable adjective: a system may be decentralized with respect to data, model training, or ledger replication, while remaining centralized with respect to aggregation or consensus authority. Its overall characterization is therefore not an absolute true/false judgment detached from context. Instead, the ontology records the relevant subjects explicitly, with dim(s)dim(s) expressing how many of those dimensions are decentralized. Distribution is treated differently; once the topology and its vertices have been fixed, distribution expresses the extensional placement of subject realizations and is evaluated as a bivalent condition: a subject either spans more than one vertex, satisfying λ(pu)>1λ(p_u)>1, or it does not. We do not compare the dimensions for distribution themselves due to pragmatics and what the term is meant to entail when designing or analyzing a system. At the system level, the subjects taken together either span multiple vertices or remain confined to one, yielding Distributed or Undistributed, respectively. There is no distribution analogue of dim(s)dim(s) and therefore no intermediate Partially Distributed system class (which strongly reflects the approach seen in literature when referring to distribution [35, 19, 48], especially when referring to "distributed computing"). A system may contain individual subjects that are not distributed and still be distributed overall, provided that its combined subject realizations extend across more than one vertex. Decentralization is consequently dimensional and gradable at the system level, while distribution is an absolute property of physical or topological placement. X Modeling Layer Topological Representation. The ontological commitment to graph-theoretic realism entails that any system admitted by O must have a topology T whose realization is a graph G=(V,E)G=(V,E). This is strictly to link the previous layers to analytical layer, such that there exists a mapping from our logical formalizations to computational representations that can be evaluated analytically. Crucially, our ontology imposes no constraint on the choice of representational formalism, provided that the chosen structure faithfully encodes the graph G. An adjacency matrix A∈0,1|V|×|V|A∈\0,1\^|V|×|V|, an edge list E⊆V×VE V× V, or any equivalent structure each constitute admissible realizations of T, as they all preserve the relational structure required for subject projection. The modeling layer therefore acts as a binding between the abstract topology T defined at the ontological level and its concrete computational instantiation. Subject Projection as Vertex Selection. At the ontological level, a subject u and topology T are connected through their unique projection particular pup_u, identified by ofSubject(pu,u)ofSubject(p_u,u) and inTopology(pu,T)inTopology(p_u,T). At the modeling layer, the relation realizedAtrealizedAt derives the non-empty vertex-support set Vpu⊆V_p_u V, containing precisely those vertices of G that support at least one realization associated with pup_u. Its cardinality is λ(pu)=|Vpu|λ(p_u)=|V_p_u|, while the distinct realization count remains δ(pu)=|ℛpu|δ(p_u)=|R_p_u|. For compatibility with the graph-specific notation used below, we write VuV_u as shorthand for VpuV_p_u whenever the associated projection particular is unambiguous. Computationally, the support of each subject can therefore be represented as a vertex set, or equivalently as a binary indicator vector with a value of 1 at position i indicating that vertex vi∈Vpuv_i∈ V_p_u. This provides a direct computational handle on subject support while preserving the distinction between realization multiplicity and vertex placement. Shared Vertices and Subjects. Two structural observations follow naturally from this modeling. First, a single vertex vi∈Vv_i∈ V may belong to the support sets of multiple distinct subject projections simultaneously, i.e., vi∈Vpuj∩Vpulv_i∈ V_p_u_j∩ V_p_u_l for uj≠ulu_j≠ u_l. This reflects the practical reality that a single computational entity can simultaneously perform storage, computation, and coordination roles, each constituting a distinct subject of decentralization. Second, a subject may have several realization particulars whose locations span multiple vertices, which is precisely the condition λ(pu)>1λ(p_u)>1 for Distributed(u) Distributed(u). Decentralization is instead determined by δ(pu)>1δ(p_u)>1; consequently, multiple realizations may remain co-located at one vertex. These observations are jointly unrestricted within O beyond the derived bounds 1≤λ(pu)≤δ(pu)1≤λ(p_u)≤δ(p_u). XI Analytical Layer Graph-based Metrics. Building on the modeling layer, we introduce two derived analytical operators that quantify the extent of decentralization for a given subject over a system topology. Both metrics are subject-specific, they are evaluated exclusively over the subgraph induced by the vertices in VuV_u for a subject u, and result in values in [0,1][0,1]. The first metric, Void Tolerance TLT_L, measures vertex-focused decentralization by assessing the resilience of the subject-induced subgraph to vertex removal. Concretely, it asks how many vertices exist whose deletion would isolate a portion of the subgraph, and how large that isolated portion could be. The second operator, Imperviousness ILI_L, measures edge-focused decentralization by assessing the minimum number of edge deletions required to compromise the connectivity of any subject-specific vertex, relative to the scale of the topology. Proposition 1 Given a system topology GtG_t, Void Tolerance is defined as: TL T_L =0ifδ(pu)=1,1if|Gs|=0,|Gt|>1,λ(pu)>1,e−rv2|Gs|−ϵ|Gt|otherwise = (9) where|Gt|>|Gs|,ϵ>0 |G_t|>|G_s|, ε>0 Where |Gt||G_t| is the order of the system topology, |Gs||G_s| is the sum order of the isolated subgraphs, λ(pu)λ(p_u) is the number of distinct subject-supporting vertices, rvr_v is the number of vertices whose removal can isolate a subgraph, δ(pu)δ(p_u) is the realization multiplicity of the subject projection, and ϵε is a subject weight factor. When deciding on the size of the subgraph that is removed (that is, which side is the “removed” one), priority is given to the distribution of subject-specific vertices. Following the removal of a vertex, the connected component containing the greatest number of subject-specific vertices is selected as the reference component. All remaining connected components are collectively treated as the isolated subgraph. Thus, if the removed vertex connects three or more components, the component containing the greatest number of subject-specific vertices serves as the reference point, while the vertices of all other components are combined when determining the order of the isolated subgraph. If two or more components contain the same maximal number of subject-specific vertices, the component of greatest order is selected as the reference component. This convention reflects the subject-oriented nature of the metric as the reference component represents the portion of the topology that remains accessible to the greatest concentration of subject realizations. Proposition 2 Given a system topology GtG_t, Imperviousness is defined as: IL I_L =0if|E(Gt)|=0orδ(pu)=1,1ifre=|Gt|−1andλ(pu)>1,e1−(λ(pu)|Gt|)−ϵ2reotherwise = (10) where|Gt|≥λ(pu),ϵ>0 |G_t|≥λ(p_u), ε>0 Where λ(pu)λ(p_u) is the number of distinct vertices supporting the subject projection, rer_e is the minimum number of edges whose removal compromises subject functionality (where no path from the subject vertex to the rest of the system where subject-specific vertices are located exists), and ϵε is the subject weight factor. Both metrics assign 0 to complete centralization and 11 to complete decentralization. The centralized edge case (TL=IL=0T_L=I_L=0) arises when δ(pu)=1δ(p_u)=1, reflecting the zero-dimensional degenerate case established in the ontological layer when every subject projection has multiplicity of one. The fully decentralized bound of 11 is recovered via limits rather than an explicit construction: limrv→0(e−rv2|Gs|−ϵ|Gt|)=1,limre→+∞(e1−(λ(pu)|Gt|)−ϵ2re)=1 _r_v→ 0 (e^- r_v^2|G_s|^-ε|G_t| )=1, _r_e→+∞ (e^1- ( λ(p_u)|G_t| )^- ε^2r_e )=1 (11) This corresponds respectively to the absence of any isolating vertices and a fully connected subject-specific subgraph. In practice, both limits are bounded by the topology: rvr_v cannot fall below 0 and rer_e is bounded above by |Gt|−1|G_t|-1, which provides sufficient justification for replacing the maxima of both formulations with 11 for any concrete system topology. Together, both operators can be expressed as a decentralization vector per subject: d→u=[TLIL]whereTL,IL∈[0,1] d_u= bmatrixT_L\\ I_L bmatrix T_L,I_L∈[0,1] (12) This vector representation enables direct comparison of decentralization across subjects and across systems, and is consistent with the earlier notions of full and partial decentralization established at the logical layer. Subject Weight Factor. Prior to evaluation, we briefly address the subject weight factor ϵε, which appears in both TLT_L and ILI_L. Formally, ϵε is a strictly positive sensitivity parameter that encodes the relative importance of decentralizing a given subject within the system. It is meant to be chosen arbitrarily in accordance with the intended function and design priorities of the system under consideration: a subject deemed critical to decentralization (such as data storage in a privacy-sensitive system) would receive a higher ϵε, amplifying the penalty incurred by structural or functional weaknesses, whereas a subject of lesser importance would receive a lower value. No universal assignment procedure is prescribed, which is meant to reflect the contextual operationalism of the ontology: the same system topology may warrant different weight assignments depending on the application context. For the purposes of the analytical evaluation that follows, we fix ϵ=1ε=1 uniformly across all subjects, treating all subjects as equally weighted. Multi-dimensional Aggregate. Both propositions and their resulting vectors from all subjects can be aggregated through simple averaging to describe the systems average degree of decentralization. In turn, this can support claims that one system is more or less decentralized than another with respect to the subjects of decentralization represented by the dimensions of the vector. This idea naturally develops the multidimensional account for gradable adjectives. On Sassoon’s account [40], a multidimensional adjective is associated with a contextually selected set of dimensions, each represented by a degree function. Later works also propose that these dimensions may be aggregated through universal quantification, existential quantification, or dimension counting, depending on the adjective. Authors of [9] preserve this dimension-based representation but generalize the aggregation stage by treating an aggregation function as a function that takes a profile of dimension-specific value functions and returns an overall ordering of the objects under comparison [9, p. 259-262, 269-270]. Simple element-wise averaging instantiates this general framework as an equally weighted aggregation function, since the aggregate value of subject vectors in U is A()=1n∑i=1nui→A(U)= 1n _i=1^n u_i. It therefore makes the equal treatment of the included dimensions explicit, rather than deriving the overall comparison solely from threshold satisfaction or from counting the dimensions on which a threshold is met. To compare two dimensions or aggregates of different systems, it is sufficient to compare the lengths of these vectors. We note that the use of subject weights within individual propositions does not necessarily conflict with this account; those weights affect the value assigned to a subject when the proposition’s vector is calculated, and thus influence the coordinates of the resulting vector. They do not alter how the completed vector is subsequently treated en bloc by the aggregation function. The averaging operation therefore remains equally weighted at the level of vector aggregation, even where the values within the vector already reflect proposition-specific differences in the relevance or contribution of particular subjects. Intuitively, subject weight factor affects how fast the values reach 1 or 0 for both Void Tolerance and Imperviousness and not how much the resulting vector is treated among other vectors. Simple averaging is appropriate where the dimensional values are represented on scales and where equal weighting is a justified modeling assumption. Under these conditions, averaging preserves the intuition that if one system is at least as decentralized as another on every included dimension and more decentralized on at least one dimension, then its aggregate score will be equal or higher. Averaging also produces a complete and transitive ordering, which permits comparative claims about whether one system is more decentralized than another while retaining the underlying vector that identifies the respects in which the comparison is made. Equal weighting should nevertheless be treated as an explicit aggregation choice rather than as a context-independent feature of decentralization. Authors of [9] emphasize that dimensions may differ in weight and that several aggregation functions may remain admissible in a given context. The aggregate value should therefore be reported together with the dimensions included in the analysis and the assumption that those dimensions receive equal weight. Analytical Evaluation. We evaluate both metrics on the instantiations from the logical layer, using an arbitrary topology GtG_t with |Gt|=5|G_t|=5, and fix ϵ=1ε=1 throughout. We instantiate our examples within two contexts to show cross-contextual applicability: Federated Learning, and Blockchain. With each context we also showcase two instances to demonstrate comparability. Federated Learning. Encode a star-topology resembling a vanilla Federated Learning, containing 11 aggregator, 44 clients, and |E(Gt)|=4|E(G_t)|=4 (Figure 4). Account for 3 subjects: Data (udatau_data), Aggregation Authority (uAggAuthu_AggAuth), and Model Training (utrainu_train). For uAggAuthu_AggAuth, there is only one aggregator so we let Vu(AggAuth)=v1V_u(AggAuth)=\v_1\ which also corresponds to a single Aggregation Authority (implying λ(pAggAuth)=1λ(p_AggAuth)=1), in accordance with both of our propositions, TL(uAggAuth)T_L(u_AggAuth) and IL(uAggAuth)I_L(u_AggAuth) both equate to 0. From which we can infer that the subject is Centralized both logically and analytically. Figure 4: Star-topology centralized federated learning setup with one aggregator and four clients. For both udatau_data and utrainu_train, the subject specific vertices are Vu=v2,v3,v4,v5V_u=\v_2,v_3,v_4,v_5\ (clients perform local training on their respective private datasets). Connectivity between subject specific vertices for both udatau_data and utrainu_train passes entirely through v1v_1, as such, no vertex can be removed from Vu(data)V_u(data) or Vu(train)V_u(train) that would isolate a subgraph making the subjects Void Tolerance TL=1T_L=1. For imperviousness, however, provided that each client is held up by only one connection to the aggregator makes re=1r_e=1, and there being 44 clients in a 55 node system results with λ(pu)/Gt=4/5λ(p_u)/G_t=4/5. We then compute: IL(udata)=IL(utrain)=e1−(0.8)−1≈0.779I_L(u_data)=I_L(u_train)=e^1-(0.8)^-1≈ 0.779 From the above, the resulting vectors are: u→data=u→train=[10.779],u→AggAuth=[00] u_data= u_train= bmatrix1\\ 0.779 bmatrix, u_AggAuth= bmatrix0\\ 0 bmatrix As well as the aggregate: A(FL)=[0.670.519]A(U_FL)= bmatrix0.67\\ 0.519 bmatrix For the second example, encode a system topology for Federated Learning setup, assume that all entities are clients who hold private data, with 33 aggregation authorities that collaboratively aggregate the model, and |E(Gt)|=6|E(G_t)|=6 (Figure 5). Similarly to the other example, we account for 3 subjects: Data (udatau_data), Aggregation Authority (uAggAuthu_AggAuth), and Model Training (utrainu_train). For uAggAuthu_AggAuth, we let Vu(AggAuth)=v1,v2,v3V_u(AggAuth)=\v_1,v_2,v_3\ with realization λ(pAggAuth)=3λ(p_AggAuth)=3. Vertex v1v_1 serves as the only vertex that can isolate any subgraph (maximum order of 22 in this case), given that v1v_1 exists in Vu(AggAuth)V_u(AggAuth), we compute Void Tolerance: TL(uAggAuth)=e−122−1(5)=e−15/2≈0.67T_L(u_AggAuth)=e^- 1^22^-1(5)=e^- 15/2≈ 0.67 For uAggAuthu_AggAuth Imperviousness, one needs to remove at minimum two edges from any vertex in Vu(AggAuth)V_u(AggAuth) to disconnect one from the rest of the system (re=2r_e=2). Given |Vu(AggAuth)|=3|V_u(AggAuth)|=3, we compute: IL(uAggAuth)=e1−(35)−122=e1−53≈0.748I_L(u_AggAuth)=e^1-( 35)^- 1^22=e^1- 53≈ 0.748 For both udatau_data and utrainu_train, all nodes hold data and train their own local models, including the aggregation authorities. Similarly to uAggAuthu_AggAuth, only one vertex can isolate a subgraph of the same size, resulting with the same TLT_L for all subjects. Figure 5: Decentralized federated learning setup with three aggregation authorities, each also acting as data holders and local trainers. For imperviousness, however, both subjects have the same cardinality of |Vu||V_u| (realization of both subjects cover the entire topology). As such, we compute: IL(udata)=IL(utrain)=e1−(55)−122=1I_L(u_data)=I_L(u_train)=e^1-( 55)^- 1^22=1 Since udatau_data and utrainu_train are all realized across the entire topology, their subject vectors are identical. As such, all subject vectors for this example are as follows: u→data=u→train=[0.671],u→AggAuth=[0.670.748] u_data= u_train= bmatrix0.67\\ 1 bmatrix, u_AggAuth= bmatrix0.67\\ 0.748 bmatrix Along with the aggregate: A(FL)=[0.670.916]A(U_FL)= bmatrix0.67\\ 0.916 bmatrix Now, comparing the two examples based on the results inferred from our analytical layer, we notice how the overall extent of decentralization in the second example is greater than that of a first, despite both being commonly known as instances of federated learning. Void Tolerance of data and training nodes is maximum in the first instance as no disappearing nodes from their sets would result in separating a chunk of a system, meanwhile the same subjects in that example have lower, but relatively high, Imperviousness given that all of them are all help up by only one edge, but there is relatively many of them with respect to the total size of the system. This differs from the second example as sets of realized vertices for both data and training now contain nodes that can isolate a subgraph, meaning that the impact of their absence can have greater effects on the system; indicating that, unlike in the first example, both data and train nodes contain a point of failure. The Imperviousness of data and train subjects in the second example is significantly higher than that of first, provided that they are realized across the entire topology, indicating that it is much harder to isolate those subjects away from the rest of the system with respect to how are they connected across the system, showcasing both pros and cons of both examples across different designs for data and training. When looking closer at the aggregation authority subject, we reveal more concrete differences between these two systems. First example has centralized aggregation authority resulting in a [0,0][0,0] vector, meanwhile second is said to be decentralized albeit with similar issues in points of failure to both data and train subjects. It is not realized across the entire topology but only 33 vertices, which then results with much greater but not complete imperviousness. Using our ontology we infer that the first example is only Partially Decentralized meanwhile second is Fully Decentralized, given that only some subjects of the first example are decentralized and not all like in the second, which is also shown by our analytical methods. Combining these vectors altogether, we see how the second example can be described analytically as "more" decentralized and not just logically. Not only that, we can clearly point out which subjects have what flaws and why across different designs for that context. The above discussion is also reflected in comparison of the aggregates of both systems during evaluation. In our second example, the length of A(FL)A(U_FL) is greater than that of the first. Hence, it can be described as a more decentralized federated learning system with respect to data, training, and aggregation authority. Blockchain. We now instantiate both examples using the same blockchain topology GtG_t from Figure 6, where |Gt|=7|G_t|=7 and |E(Gt)|=9|E(G_t)|=9. The topology contains a dense core formed by v2,v3,v4,v5v_2,v_3,v_4,v_5, one endpoint v1v_1 connected through v2v_2, and a two-hop endpoint branch v6,v7v_6,v_7 connected through v5v_5. We fix ϵ=1ε=1 throughout. Figure 6: Shared blockchain topology used for both blockchain examples. For the first example, encode a simple permissioned blockchain where one node acts as the block-ordering authority, four nodes store the ledger, and three nodes act as transaction-submission clients. We account for three subjects: Consensus Authority (uconsu_cons), Ledger Replication (uledgeru_ledger), and Transaction Submission (utxu_tx). For uconsu_cons, suppose that only v2v_2 orders blocks. Hence, Vu(cons)=v2V_u(cons)=\v_2\ and λ(pcons)=1λ(p_cons)=1. By the centralized edge case in both propositions, we obtain: TL(ucons)=0,IL(ucons)=0T_L(u_cons)=0, I_L(u_cons)=0 Thus, the consensus subject is analytically Centralized. For uledgeru_ledger, suppose that the ledger is replicated only by the four core full nodes Vu(ledger)=v2,v3,v4,v5V_u(ledger)=\v_2,v_3,v_4,v_5\. Within this subject set, v2v_2 and v5v_5 are vertices whose removal can isolate a subgraph. Removing v2v_2 isolates v1v_1, while removing v5v_5 isolates the branch v6,v7v_6,v_7. Hence, rv2r_v2 and the maximum isolated subgraph has order |Gs|=2|G_s|=2. We compute: TL(uledger)=e−222−1(7)=e−87≈0.319T_L(u_ledger)=e^- 2^22^-1(7)=e^- 87≈ 0.319 For Imperviousness, the weakest ledger node in the core requires removing three edges to isolate it from the rest of the topology, so re=3r_e=3. Since λ(pledger)=4λ(p_ledger)=4, we compute: IL(uledger)=e1−(47)−13≈0.815I_L(u_ledger)=e^1- ( 47 )^- 13≈ 0.815 For utxu_tx, suppose that transactions are submitted only by client-facing nodes Vu(tx)=v1,v6,v7V_u(tx)=\v_1,v_6,v_7\. Only v6v_6 can isolate a subgraph when removed, since removing v6v_6 separates v7v_7 from the rest of the system. Hence, rv=1r_v=1 and |Gs|=1|G_s|=1. We compute: TL(utx)=e−121−1(7)=e−17≈0.867T_L(u_tx)=e^- 1^21^-1(7)=e^- 17≈ 0.867 For Imperviousness, either v1v_1 or v7v_7 can be isolated by removing one edge, so re=1r_e=1. Since λ(ptx)=3λ(p_tx)=3, we compute: IL(utx)=e1−(37)−1≈0.264I_L(u_tx)=e^1- ( 37 )^-1≈ 0.264 The subject vectors for the first blockchain example are therefore: u→cons=[00],u→ledger=[0.3190.815],u→tx=[0.8670.264] u_cons= bmatrix0\\ 0 bmatrix, u_ledger= bmatrix0.319\\ 0.815 bmatrix, u_tx= bmatrix0.867\\ 0.264 bmatrix With an aggregate: A(BC)=[0.3950.36]A(U_BC)= bmatrix0.395\\ 0.36 bmatrix For the second example, keep the exact same system topology, but assign blockchain functionality differently. Encode a blockchain where consensus is performed by the four core nodes, the ledger is replicated by all nodes, and transaction submission is exposed only through two core gateway nodes. We again account for the same three subjects: Consensus Authority (uconsu_cons), Ledger Replication (uledgeru_ledger), and Transaction Submission (utxu_tx). For uconsu_cons, suppose that the four core nodes jointly validate and order blocks Vu(cons)=v2,v3,v4,v5V_u(cons)=\v_2,v_3,v_4,v_5\. This is the same realization used for ledger replication in the first example. Hence, rv=2r_v=2, |Gs|=2|G_s|=2, re=3r_e=3, and λ(pcons)=4λ(p_cons)=4. We compute: TL(ucons)=e−222−1(7)=e−87≈0.319T_L(u_cons)=e^- 2^22^-1(7)=e^- 87≈ 0.319 and IL(ucons)=e1−(47)−13≈0.815I_L(u_cons)=e^1- ( 47 )^- 13≈ 0.815 For uledgeru_ledger, suppose that every node stores a copy of the ledger Vu(ledger)=v1,…,v7V_u(ledger)=\v_1,…,v_7\. Now the subject is realized across the whole topology. However, v2v_2, v5v_5, and v6v_6 can each isolate some part of the topology when removed. The largest isolated subgraph is still the branch v6,v7\v_6,v_7\, so rv=3r_v=3 and |Gs|=2|G_s|=2. We compute: TL(uledger)=e−322−1(7)=e−187≈0.076T_L(u_ledger)=e^- 3^22^-1(7)=e^- 187≈ 0.076 For Imperviousness, since the ledger subject is realized across the entire topology, λ(pledger)/|Gt|=1λ(p_ledger)/|G_t|=1. Therefore: IL(uledger)=e1−(77)−1re=1I_L(u_ledger)=e^1- ( 77 )^- 1r_e=1 For utxu_tx, suppose that transaction submission is exposed only through two non-articulation gateway nodes in the core Vu(tx)=v3,v4V_u(tx)=\v_3,v_4\. Neither v3v_3 nor v4v_4 can isolate a subgraph when removed. Hence, |Gs|=0|G_s|=0, |Gt|>1|G_t|>1, and λ(ptx)>1λ(p_tx)>1, so: TL(utx)=1T_L(u_tx)=1 For Imperviousness, isolating either v3v_3 or v4v_4 requires removing three edges, so re=3r_e=3. Since λ(ptx)=2λ(p_tx)=2, we compute: IL(utx)=e1−(27)−13≈0.596I_L(u_tx)=e^1- ( 27 )^- 13≈ 0.596 The subject vectors for the second blockchain example are therefore: u→cons=[0.3190.815],u→ledger=[0.0761],u→tx=[10.596] u_cons= bmatrix0.319\\ 0.815 bmatrix, u_ledger= bmatrix0.076\\ 1 bmatrix, u_tx= bmatrix1\\ 0.596 bmatrix Alongside the aggregate: A(BC)=[0.4650.804]A(U_BC)= bmatrix0.465\\ 0.804 bmatrix Now, comparing both blockchain examples over the same topology, we see that the analytical layer distinguishes decentralization by subject rather than by graph structure alone. In the first example, consensus is centralized because only one node performs block ordering. Ledger replication is more decentralized than consensus because it is realized over the dense core, although it still has lower Void Tolerance because two ledger nodes, v2v_2 and v5v_5, are also structurally important for maintaining connectivity. Transaction submission has high Void Tolerance because most submission nodes are not structurally critical, but it has low Imperviousness because endpoint clients such as v1v_1 and v7v_7 can be isolated by removing only one edge. In the second example, the same topology supports a different decentralization profile. Consensus becomes more decentralized because validation is moved from one ordering node to the four-node core. Ledger replication becomes maximally impervious because every node stores the ledger, but its Void Tolerance decreases because the subject now includes all articulation vertices of the topology. Transaction submission, meanwhile, has maximum Void Tolerance because it is assigned only to non-articulation core nodes, with high but not maximum Imperviousness because only two nodes realize the subject that are still difficult to isolate. Therefore, even under a fixed topology, the resulting subject vectors differ according to how context functions are realized over that topology. When comparing the aggregates, however, the second blockchain example is substantially more decentralized than the first. Visibly influenced by the centralized consensus in the first example and greater decentralization across the transactions dimension in the second. XII Implementation The framework is implemented as a standalone browser application, readers may find the working implementation in [2]. A system topology is represented as a finite simple undirected graph: Gt=(V,E),n=|V|,m=|E|G_t=(V,E), n=|V|, m=|E| (13) Stored internally as an adjacency list. The graph canvas maintains a vertex identifier and a pair of square-plane coordinates for each vertex, together with unordered endpoint pairs for the edges. Coordinates affect only the layout and evaluation depends exclusively on (V,E)(V,E). Graphs may be constructed by vertex and edge insertion/deletion or imported from JSON. For each subject u, the implementation stores a positive realization count cu(v)c_u(v) at every vertex. The projection multiplicity and support cardinality are therefore computed as Vu=v∈V:cu(v)>0,λ(pu)=|Vu|,δ(pu)=∑v∈Vucu(v) array[]cV_u=\v∈ V:c_u(v)>0\, λ(p_u)=|V_u|,\\[5.0pt] δ(p_u)= _v∈ V_uc_u(v) array (14) Before evaluation, the implementation requires unique non-empty vertex and subject identifiers, valid edge endpoints, no self-loops or duplicate edges, cu(v)∈ℕ>0c_u(v) _>0, δ(pu)∈ℕ>0δ(p_u) _>0, ∑vcu(v)=δ(pu) _vc_u(v)=δ(p_u), Vu≠∅V_u≠ , and ϵu>0 _u>0. Void Tolerance An iterative Tarjan traversal [44] computes discovery indices, low-link values, connected components (subgraphs), and articulation vertices. Let vC_v be the connected components of Gt−vG_t-v, and let the reference component be selected lexicographically by subject support and then component order: Cv⋆∈argmaxC∈vlex(|C∩(Vu∖v)|,|C|)C_v ∈ *arg\,max_C _v^lex (|C∩(V_u \v\)|,|C| ) (15) For every subject-supporting articulation vertex (vertex that manages to isolate a subgraph after removal), the isolated order is: gu(v)=∑C∈v∖Cv⋆|C|=n−1−|Cv⋆|g_u(v)= _C _v \C_v \|C|=n-1-|C_v | (16) Therefore: Au=v∈Vu:v is an articulation vertex of GtA_u=\v∈ V_u:v is an articulation vertex of G_t\ (17) the implemented quantities are rv(u)=|Au|,|Gs(u)|=0,Au=∅,maxv∈Augu(v),otherwiser_v(u)=|A_u|, |G_s(u)|= cases0,&A_u= ,\\ _v∈ A_ug_u(v),&otherwise cases (18) Void Tolerance is then evaluated as per our first Proposition. Imperviousness The ontology interprets “compromise” as isolating a subject-specific vertex from the topology. In a simple undirected graph, this requires deleting all edges incident to that vertex; hence: re(u)=minv∈VudegGt(v)r_e(u)= _v∈ V_u _G_t(v) (19) Imperviousness is then evaluated in accordance with our second Proposition. Vectors, aggregation, and classification For every subject: d→u=[TL(u)IL(u)],‖d→u‖2=TL(u)2+IL(u)2 d_u= bmatrixT_L(u)\\ I_L(u) bmatrix, \| d_u\|_2= T_L(u)^2+I_L(u)^2 (20) For a declared subject set =u1,…,ukU=\u_1,…,u_k\, the equal-weight aggregate is A()=1k∑i=1kd→ui,‖A()‖2=AT()2+AI()2A(U)= 1k _i=1^k d_u_i, \|A(U)\|_2= A_T(U)^2+A_I(U)^2 (21) Aggregate comparisons are admitted only for identical declared subject sets. Subject u is classified as Centralized iff δ(pu)=1δ(p_u)=1, otherwise as Decentralized. It is Distributed iff λ(pu)>1λ(p_u)>1. A non-empty system is Fully Decentralized when every subject is decentralized, Partially Decentralized when both centralized and decentralized subjects occur, and Centralized otherwise. Simulation Simulation is deterministic under a user-supplied seed. For simulation index i, a seeded pseudorandom generator samples the graph order, topology family, generator parameters, perturbations, and subject projections. Supplied with various topology families during generation. The order nin_i is sampled from one of the following distributions: fixed, uniform, triangular, truncated-normal, or log-uniform. Generator parameters may be swept or independently jittered, after which edges may be deleted or added at configured rates and disconnected components optionally joined. Support vertices are sampled uniformly or by degree, connected clustering, articulation status, or periphery, with optional overlap, separation, or nesting relative to a preceding subject. Per-vertex counts are sampled from uniform, fixed, Poisson, or geometric distributions and give δi(pu)=∑v∈Vu,icu,i(v) _i(p_u)= _v∈ V_u,ic_u,i(v) (22) The implementation permits up to 250,000250,000 vertices per system and 100,000100,000 systems per batch, subject to explicit vertex- and edge-work budgets. Topology preprocessing is O(n+m)O(n+m). Reusing traversal, each subject requires O(n+λ(pu)+au)O(n+λ(p_u)+a_u) time and O(n)O(n) temporary storage, where aua_u is the number of relevant articulation-tree incidences. The metric implementation is regression-tested against the paper examples and large synthetic graphs. XIII Discussion Having established the ontology, its formal properties, and analytical capabilities, we now reflect on what the proposed definition contributes relative to the existing literature and the broader implications that follow from it. XIII-A Distribution and Decentralization as Distinct Properties. A central contribution of the proposed framework is the formal separation of distribution and decentralization, a distinction that, while acknowledged in principle across the literature [37], has rarely been enforced at the formal level [30]. In our ontology, distribution is defined as a property of subject support (focusing more on locality). Subject is distributed if and only if λ(pu)>1λ(p_u)>1, that is, if the realizations associated with its projection particular occupy more than one vertex. Decentralization, by contrast, is defined by the realization-multiplicity function δ(pu)δ(p_u), with δ(pu)>1δ(p_u)>1 indicating more than one distinct realization (focusing more on relationalism). These predicates are formally distinct as several realizations may be located at one vertex, yielding δ(pu)>1δ(p_u)>1 and λ(pu)=1λ(p_u)=1, while the structural axioms ensure that δ(pu)=1δ(p_u)=1 entails λ(pu)=1λ(p_u)=1. This distinction captures deployment arrangements that node-counting approaches obscure and makes the conflation of realization multiplicity with topological placement formally inadmissible. XIII-B Epistemological Distinction between Distribution and Decentralization. Distribution is typically observable from the physical arrangement of a system, whereas we have shown that decentralization is an inferred property of the relations that hold between entities, subjects, and their projections. The fact that a function is implemented across several nodes does not by itself establish what those nodes are capable of affecting or representing within the topology. Therefore, our ontology prevents decentralization from being treated as a surface-level architectural descriptor and instead requires it to be justified in the relational structure of the system itself. This epistemological importance is highlighted by many existing examples that move too quickly from multiplicity to decentralization, allowing an empirical fact about deployment to stand in for a stronger claim about organizational structure. Our definition prevents this inference, forcing decentralization to be demonstrated rather than presumed. XIII-C Relational Character of Subject Multiplicity. Although δ(pu)δ(p_u) returns a cardinal value, it is not a node-counting statistic detached from the ontology’s relational structure. Its extension is determined by the unique projection particular pup_u, which connects a system, a topology, and a contextually identified subject through hasProjection, ofSubject, and inTopology. The realization set RpuR_p_u subsequently contains only those particulars connected to that projection through hasRealization. Consequently, δ(pu)=|Rpu|δ(p_u)=|R_p_u| counts the distinct ways in which a particular subject is realized within a particular system topology. Its multiplicity is therefore inherently subject-specific and derives its meaning from the relations that identify what is being realized, where it is being considered, and to which system it belongs. In this sense, the output of δ is cardinal, but the property from which that cardinality is derived is relational [16, 37]. Graph realism supports our account through the relation realizedAt, which maps each realization particular to a vertex of the topology. From this mapping, λ(pu)λ(p_u) derives the number of distinct vertices supporting the realizations counted by δ(pu)δ(p_u). The two functions therefore describe different aspects of the same subject projection: δ captures plurality of realization, whereas λ captures the topological separation, or lack of separation, among those realizations. Multiple realizations may coincide at one vertex, producing δ(pu)>1δ(p_u)>1 and λ(pu)=1λ(p_u)=1, or may span several vertices, producing both decentralization and distribution. The graph’s edge structure subsequently determines the resilience of that placement through the analytical layer. Thus, relationality at the logical layer should not be confused with edge sensitivity, given that the former arises from the subject-projection-realization structure, while the latter evaluates how the resulting realization support is connected within G. The federated-learning and blockchain instantiations illustrate why this distinction is substantive. In the star-topology federated-learning example, data storage and model training have several realizations, while aggregation authority has only one, despite all three subjects being evaluated over the same graph. Likewise, the blockchain examples retain the same topology while changing the realizations of consensus authority, ledger replication, and transaction submission, producing different subject classifications and different system-level dimensionalities. A node-counting approach cannot express these differences because the number of vertices may remain unchanged. The ontology instead determines decentralization from which subject is instantiated by which realizations and then separately determines whether those realizations are distributed across the graph. This subject-relative relational structure is what permits decentralization to be formally distinguished from mere distribution while remaining grounded in a graph-realistic representation. XIII-D Collective Amnesia Regarding the Relational Aspect of Decentralization. The broader literature on decentralization exhibits what might be described as a collective amnesia with respect to the relational aspect of the concept. Early foundational treatments were notably attentive to this aspect, the seminal network topology work of [5] explicitly separated node-level and link-level analysis, and the Byzantine Generals problem [26] grounded decentralization in the structure of communication relationships between participants. The CAP theorem [7] similarly situated decentralization within a framework of trade-offs that are inherently relational in character. However, following the introduction of Bitcoin [35] and the subsequent dominance of blockchain-based systems as the implicit reference point for decentralization [10], a significant portion of academic literature effectively reverted to node-centric analysis. The Nakamoto coefficient [42], Gini-based measures [28], and Shannon entropy approaches all reduce decentralization to a distributional property over participants, enumerating entities and their resource holdings while omitting the edges that connect them. As we argue in the problem statement, it is precisely the structure of relationships between participants that distinguishes decentralization from distribution. The star topology instantiations in this work illustrate this directly: both the blockchain and federated learning examples are distributed under any node-counting measure, yet both exhibit significant structural vulnerability when the relational dimension is examined through Void Tolerance and Imperviousness, neither of which any node-centric metric can detect. XIII-E Analytical Depth Beyond Existing Metrics. The graph-based metrics introduced in the analytical layer provide a strictly richer characterization of decentralization than existing quantitative approaches. Gini coefficients and Shannon entropy summarize the distribution of a single scalar quantity across a population of nodes, yielding a single scalar output. The Nakamoto coefficient reduces this further to a single threshold value. None of these measures are sensitive to the topological structure of the system, the identity of which nodes hold which subjects, or the relational fragility of subject-induced subgraphs. Void Tolerance and Imperviousness, by contrast, are explicitly subject-specific and edge-sensitive. They assess, respectively, whether any vertex removal can isolate a portion of the subject-induced subgraph, and how many edge deletions are required to compromise subject functionality. The resulting decentralization vector d→u=[TL,IL] d_u=[T_L,I_L] is two-dimensional per subject, and the full characterization of a system with n subjects yields n such vectors, each exposing a distinct axis of structural and functional vulnerability. This granularity is not achievable by scalar-only measures, and it is precisely what is needed to reason about partially decentralized systems where different subjects exhibit different degrees of robustness. XIII-F Visualization and Cross-Contextual Comparison. The vector representation of decentralization enables a visualization capability that is absent from existing approaches. By placing d→u d_u in the unit square [0,1]2[0,1]^2, subjects from different systems and different contexts can be directly compared along the same two axes. A subject approaching [1,1][1,1] is both structurally robust and functionally resilient; a subject near [0,0][0,0] is centralized; and subjects occupying intermediate positions reveal the specific nature of their weakness, whether structural (low TLT_L, adequate ILI_L) or functional (adequate TLT_L, low ILI_L). This visualization can further support cross-contextual analysis, as a practitioner comparing two systems can immediately identify that their decentralized subjects share the same vulnerability profile whether operating in entirely different application domains or not, one of the conclusions that no existing metric could support. This generalization across contexts is a direct consequence of grounding the analysis in graph-theoretic structure and formal semantics behind decentralization rather than application-specific semantics. XIII-G Bridging Informal and Formal Reasoning. As established in the introductory sections, decentralization has perviously oscillated between formal technical descriptions and informally grounded assessments, with the informal component frequently dominating conceptually in practice. By adopting an ontological approach rather than a purely axiomatic or measure-theoretic one, the framework retains the capacity to encode context-dependent concepts, such as the identity of subjects of decentralization, which are determined by the intended function of the system rather than by the formalism itself, while simultaneously grounding all reasoning in a formally specified conceptualization with decidable semantics. This is precisely the capability that Guarino [16] identifies as the distinguishing feature of ontologies relative to other formal specification approaches: the ability to commit to a conceptualization in a way that bridges informal intuition and formal reasoning. The contextual operationalism of the ontology, where subjects are identified by the analyst in accordance with the system’s design intent, ensures that the framework does not impose an artificially narrow reading of decentralization, while the formal semantics, axiomitization, and reasoning properties ensure that the resulting classifications are not arbitrary. In this sense, we also demonstrate that ontologies are a uniquely well-suited tool for addressing conceptual problems of this kind, where the informal basis of a concept must be preserved rather than eliminated in order for the definition to remain applicable across contexts. XIV Limitations Although the proposed ontology establishes a formally grounded and cross-contextual account of decentralization, it remains subject to several limitations that arise from the assumptions necessary for its construction. These limitations should be understood as consequences of the intended scope of the framework rather than deficiencies of its formal foundations. XIV-A Classes of Decentralized Systems The present ontology is intentionally concerned with decentralization as a structural property of communication systems. Consequently, it does not attempt to capture behavioural, economic, or performance-related properties that are frequently associated with decentralized systems, such as latency, incentive compatibility, robustness, or fairness. While these characteristics undoubtedly influence practical system evaluation, they constitute distinct analytical dimensions whose formal treatment requires additional ontological commitments beyond those developed here. Likewise, the ontology evaluates communication systems as structural snapshots rather than temporally evolving entities. Dynamic processes, including topology evolution, adaptive coordination, or continuously changing authority relationships, must therefore be represented through successive realizations instead of explicit temporal reasoning. Although this abstraction is sufficient for establishing structural decentralization at a given point in time, it is not intended to provide a complete account of dynamic system behaviour. XIV-B Graph Realism as a Basis for System Representation Our framework adopts graph representations as ontologically meaningful abstractions of communication systems. This commitment follows from the ontological realist position developed throughout the paper, namely that relations between entities are treated as genuine features of the systems under investigation rather than merely convenient modelling artefacts. Nevertheless, this commitment necessarily limits the scope of the ontology to systems that admit faithful graph representations. Certain forms of interaction are more naturally expressed through richer mathematical structures, including higher-order or non-pairwise relationships that exceed the expressive capabilities of ordinary graphs. Furthermore, structural graph representations cannot always distinguish between relationships that are topologically identical but semantically different. Two systems may therefore possess identical graph structures while differing substantially in the functional significance of particular nodes or communication channels. These limitations do not undermine the ontological realist interpretation adopted here, but instead illustrate that ontological realism remains constrained by the expressive power of the modelling formalism chosen to represent reality. XIV-C Ontological Realism, Nominalism, and the Scope of Structural Decentralization A further limitation concerns the metaphysical status of decentralization and the extent to which its associated properties can be represented without altering the ontology’s foundational commitments. In the limited sense relevant here, a nominalist treatment would use “decentralized” primarily as a label for heterogeneous systems without committing to a common structure in reality that determines when the term applies. The present framework moves away from such a treatment by grounding decentralization in entities, relations, and realization particulars that admit explicit truth conditions within the ontology. Rather than introducing Decentralized as a primitive class whose extension is determined informally, the ontology derives it from the relational organization of a subject projection and, specifically, from its realization multiplicity δ(pu)δ(p_u). This follows the realist position that an ontology should constrain its models according to an intended conceptualization of an independently existing domain [16, 3]. This realist commitment is nevertheless relative to both the ontology’s declared domain and vocabulary. The system represented by the ontology is a computer communication system whose computational entities and interconnections admit a faithful graph representation. The ontology therefore commits to the existence of system entities, subject realizations, topological vertices, and the relations that connect them. It does not commit to every social, legal, economic, or organizational relation that may affect the operation of the represented system. Such relations may be genuine features of a broader socio-technological reality, but they are not thereby recoverable from the graph/communication topology adopted here. This limitation is particularly relevant to ownership and effective control. Consider two systems with the same topology G=(V,E)G=(V,E), the same subject projections, the same realization particulars, and the same mapping of those realizations to vertices. Suppose that, outside this representation, every realization in the first system is controlled by one organization, while the corresponding realizations in the second are controlled by mutually independent organizations. The two systems are indistinguishable under the ontology’s present signature. Consequently, every quantity derived from that signature, including δ, λ, Void Tolerance, and Imperviousness, must yield the same result for both systems. No predicate expressed in the existing language can distinguish them on the basis of information that the language neither represents nor ontologically commits to. It would be possible to extend the ontology with concepts such as Principal or Organization and relations such as ownedBy, controlledBy, or delegatesTo. Such an extension would not merely refine the existing graph representation, however. It would enlarge the universe of discourse, require identity and persistence criteria for controlling principals, and introduce additional assumptions concerning indirect ownership, shared credentials, collusion, and effective rather than nominal control. It would therefore constitute a different ontological commitment directed towards socio-technical or organizational decentralization. Moreover, if such information were introduced only through analyst-supplied labels without sufficiently general identity criteria and formally specified relations, it would risk reintroducing the nominal dependence that the present ontology seeks to avoid. Within the graph-realist perspective adopted here, the realization-multiplicity function δ therefore marks the furthest context-independent distinction that can be made at the graph-realistic, ontological layer without utilising properties from a different level of description. Independence of ownership, adversarial tolerance, and organizational control may remain important properties of a concrete system, but they are not treated as constitutive of structural decentralization under the present ontological commitment. Accordingly, a subject may be decentralized in realization while remaining undistributed in placement, structurally vulnerable, or centralized under an external control analysis, which from graph-realistic view only would be recoverable via δ. These descriptions concern different properties and are not logically contradictory. The classification produced by the ontology should therefore be interpreted as structural and subject-relative, as it establishes decentralization with respect to graph-representable subject realizations, rather than asserting independence across every possible economic, organizational, or governance dimension (although again, these could potentially be represented by reducing these properties to their individual subjects of decentralization, however we leave the verification of this direction for future work). Subject to this scope, δ continues to provide the promised domain-independent criterion for distinguishing centralization from decentralization, preserves the formal separation between decentralization and distribution, and supports the topology-dependent evaluation developed in the analytical layer. XIV-D Ad Hoc Metric Construction and Interpretative Scope A further limitation concerns the ad hoc construction of the analytical propositions introduced in this work. Void Tolerance and Imperviousness are purpose-built operators, and although their inputs and boundary conditions follow from the ontology, their particular functional forms are not uniquely entailed by its axioms. The use of exponential normalization, the sensitivity parameter ϵε, and the subsequent aggregation of subject vectors represent analytical design choices rather than logical consequences of the definition of decentralization (although largely inspired by formal semantics). Alternative functions could encode the same underlying structural quantities while producing different numerical scales or comparative orderings. Here, ad hoc should be understood in its methodological sense of being constructed for a particular analytical purpose, rather than as implying that the metrics are arbitrary. The ontology constrains what the propositions operate over and how their results may be interpreted. In particular, the propositions are subject-specific, preserve the centralized boundary established by δ(pu)=1δ(p_u)=1, operate over graph-realistic representations, and evaluate two explicitly identified structural phenomena. Void Tolerance evaluates the effect of subject-relevant vertex removal and the resulting isolation of portions of the topology, while Imperviousness evaluates the edge-removal effort associated with isolating a subject-supporting vertex. The formulas therefore assign numerical values to stated graph properties whose meanings remain independently inspectable. This distinction between a metrics mathematical meaning and its interpretation as decentralization is important because formally specified metrics do not, by themselves, provide a formal foundation for the concept they are used to evaluate. The Gini coefficient, for example, is formally a measure of inequality within a distribution without establishing that equality is identical to decentralization [28]. Indeed, a system containing a single participant will have a Gini coefficient of zero because no inequality exists within the singleton distribution, despite being centralized under almost any systems interpretation. Unlike when applying our ontology, where both are possible depending on the role and multiplicity of the subjects. Shannon entropy similarly measures uncertainty associated with a probability distribution, while the Herfindahl-Hirschman Index measures concentration. Their use as decentralization metrics requires an additional interpretative premise connecting equality, uncertainty, or concentration to decentralization. The underlying statistic may be objective relative to its inputs even though its interpretation as a measure of decentralization remains context-dependent. Based on the above, the formal precision of the statistic does not automatically transfer to the conceptual inference that the statistic measures decentralization itself. Different metrics may therefore produce equally valid descriptions of inequality, concentration, uncertainty, or attack thresholds while representing different and potentially incompatible interpretations of decentralization. Nevertheless, the purpose-built nature of Void Tolerance and Imperviousness remains a genuine methodological limitation. Their specific normalization, sensitivity to ϵε, and aggregation procedure require justification through sensitivity analysis, comparison with alternative functional forms, and evaluation against established graph and concentration measures. Numerical results should therefore be reported together with the declared subject set, topology, parameter assignments, and aggregation assumptions. The propositions provide useful and reproducible analytical interpretations under those conditions, but their scalar values should not be treated as context-independent natural constants of decentralization. As such, the analytical contribution of the framework does not rest on claiming that Void Tolerance and Imperviousness are uniquely correct measures. Its contribution lies in making the relationship between formal definition, graph property, metric construction, and subsequent interpretation explicit. This transparency reduces the conceptual circularity that arises when an existing statistic is first interpreted as decentralization and then used as evidence for the same interpretation. The proposed operators remain ad hoc in their functional construction, but they are ontologically constrained, semantically interpretable, and replaceable without undermining the foundational reasoning established by δ. XIV-E Formal and Informal Reasoning As argued throughout this work, following Finocchiaro [12, 11], the distinction between formal and informal reasoning is better understood as a continuum than as a binary divide. Consequently, our ontology should not be interpreted as eliminating informal judgement from decentralization analysis, but constraining such judgement by embedding it within a formally specified conceptual structure. In particular, the identification of subjects of decentralization and the abstraction of real-world systems into ontological representations remain dependent upon informed human judgement. Different analysts may legitimately select different subjects or construct different abstractions while remaining fully consistent with the formal ontology. The framework therefore reduces ambiguity by formalizing the reasoning process, but it does not eliminate the interpretative choices that necessarily accompany the modeling of complex communication systems. XV Conclusions This work addressed the Decentralization Problem by introducing a formal, graph-based ontology that defines decentralization independently of application domain. By integrating ontological commitments, formal semantics, graph-theoretic modeling, and subject-specific analytical metrics, the proposed framework establishes a transferable basis for reasoning about decentralization across computer communication systems. In doing so, it formally distinguishes decentralization from distribution and demonstrates that structurally identical topologies may exhibit different decentralization properties depending on the subject under consideration. The ontology was validated through two types of communication systems, showing that it provides consistent classifications while exposing structural characteristics that conventional scalar measures fail to capture. More broadly, this work establishes decentralization as a formally definable relational property rather than an application-dependent or informally interpreted concept, providing a common foundation for future theoretical and practical work on decentralized computer systems. 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A system with at least one decentarlized subject and one centralized subject. Fully Decentralized System = def. A system subjects of which are all decentralized. Undistributed Subject = def. A subject realization of which spans only one node on the associated system topology. Distributed Subject = def. A subject realization of which spans more than one node on the associated system topology. Undistributed System = def. A system where all subjects of which are, altogether, held on a singular node on the associated topology. Distributed System = def. A system all subjects of which are, altogether, projected across more than one node on the associated topology.