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AlertStar: Path-Aware Alert Prediction on Hyper-Relational Knowledge Graphs
Zahra Makki Nayeri, Mohsen Rezvani
Intelligence
Status: succeeded | Model: google/gemini-3.1-flash-lite-preview | Prompt: intel-v1 | Confidence: 94%
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Summary
The paper introduces AlertStar and a suite of hyper-relational knowledge graph completion (HR-KGC) models to improve network intrusion detection. By representing network alerts as qualified statements (h, r, t, Q) rather than binary triples, the authors enable path-aware reasoning over attacker-victim interactions. The proposed models include HR-NBFNet (path-based propagation), AlertStar (embedding-based fusion), and their multi-task variants, which are evaluated on the Warden and UNSW-NB15 benchmarks, demonstrating superior performance and efficiency in predicting future attack targets.
Entities (5)
Relation Signals (3)
AlertStar → evaluatedon → Warden
confidence 95% · Evaluated inductively on the Warden and UNSW-NB15 benchmarks
HR-NBFNet → solves → HR-KGC
confidence 95% · HR-NBFNet extends Neural Bellman-Ford Networks to the hyper-relational setting
AlertStar → outperforms → HR-NBFNet
confidence 90% · AlertStar achieves up to 50x per-epoch speedup over HR-NBFNet
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Abstract
Abstract:Cyber-attacks continue to grow in scale and sophistication, yet existing network intrusion detection approaches lack the semantic depth required for path reasoning over attacker-victim interactions. We address this by first modelling network alerts as a knowledge graph, then formulating hyper-relational alert prediction as a hyper-relational knowledge graph completion (HR-KGC) problem, representing each network alert as a qualified statement (h, r, t, Q), where h and t are source and destination IPs, r denotes the attack type, and Q encodes flow-level metadata such as timestamps, ports, protocols, and attack intensity, going beyond standard KGC binary triples (h, r, t) that would discard this contextual richness. We introduce five models across three contributions: first, Hyper-relational Neural Bellman-Ford (HR-NBFNet) extends Neural Bellman-Ford Networks to the hyper-relational setting with qualifier-aware multi-hop path reasoning, while its multi-task variant MT-HR-NBFNet jointly predicts tail, relation, and qualifier-value within a single traversal pass; second, AlertStar fuses qualifier context and structural path information entirely in embedding space via cross-attention and learned path composition, and its multi-task extension MT-AlertStar eliminates the overhead of full knowledge graph propagation; third, HR-NBFNet-CQ extends qualifier-aware representations to answer complex first-order logic queries, including one-hop, two-hop chain, two-anchor intersection, and union, enabling multi-condition threat reasoning over the alert knowledge graph. Evaluated inductively on the Warden and UNSW-NB15 benchmarks across three qualifier-density regimes, AlertStar and MT-AlertStar achieve superior MR, MRR, and Hits@k, demonstrating that local qualifier fusion is both sufficient and more efficient than global path propagation for hyper-relational alert prediction.
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- Source: https://arxiv.org/abs/2604.03104v1
- Canonical: https://arxiv.org/abs/2604.03104v1
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AlertStar: Path-Aware Alert Prediction on Hyper-Relational Knowledge Graphs Zahra Makki Nayeri, Mohsen Rezvani Zahra Makki Nayeri, Faculty of Computer Engineering at Shahrood University of Technology, Shahrood, Iran. E-mail: zmakki@shahroodut.ac.ir Mohsen Rezvani, Faculty of Computer Engineering at Shahrood University of Technology, Shahrood, Iran. E-mail: mrezvani@shahroodut.ac.ir Abstract Cyber-attacks continue to grow in scale and sophistication, yet existing network intrusion detection approaches lack the semantic depth required for path reasoning over attacker–victim interactions. We address this by first modelling network alerts as a knowledge graph, then formulating hyper-relational alert prediction as a hyper-relational knowledge graph completion (HR-KGC) problem, representing each network alert as a qualified statement (h,r,t,)(h,r,t,Q), where h and t are source and destination IPs, r denotes the attack type, and Q encodes flow-level metadata such as timestamps, ports, protocols, and attack intensity, going beyond standard KGC binary triples (h,r,t)(h,r,t) that would discard this contextual richness. We introduce five models across three contributions: first, Hyper-relational Neural Bellman–Ford (HR-NBFNet) extends Neural Bellman–Ford Networks to the hyper-relational setting with qualifier-aware multi-hop path reasoning, while its multi-task variant MT-HR-NBFNet jointly predicts tail, relation, and qualifier-value within a single traversal pass; second, AlertStar fuses qualifier context and structural path information entirely in embedding space via cross-attention and learned path composition, and its multi-task extension MT-AlertStar eliminates the overhead of full knowledge graph propagation; third, HR-NBFNet-CQ extends qualifier-aware representations to answer complex first-order logic queries, including one-hop, two-hop chain, two-anchor intersection, and union, enabling multi-condition threat reasoning over the alert knowledge graph. Evaluated inductively on the Warden and UNSW-NB15 benchmarks across three qualifier-density regimes, AlertStar and MT-AlertStar achieve superior MR, MRR, and Hits@k, demonstrating that local qualifier fusion is both sufficient and more efficient than global path propagation for hyper-relational alert prediction. Index Terms: Hyper-relational knowledge graphs, knowledge graph completion, qualifier prediction, Neural Bellman-Ford, alert prediction, network intrusion detection. 1 Introduction Cyber-attacks continue to grow in scale and sophistication, rendering purely reactive defences insufficient for modern network security. Operational intrusion detection systems (IDSs) are highly sensitive by design, routinely generating tens of thousands of alerts per day [5], yet the most dangerous threats — advanced persistent threat (APT) campaigns — are rarely revealed by any single alert. APT actors deliberately decompose their campaigns into individually innocuous steps: reconnaissance, initial compromise, lateral movement, and data exfiltration, each generating only a low-priority alert that may be dismissed in isolation [5]. The central challenge facing Security Operations Centers (SOCs) and Computer Emergency Response Teams (CERTs) is reasoning across sequences of alerts distributed over time and across different host pairs to reconstruct the underlying campaign [6]. Consider a representative scenario: a denial-of-service alert at 02:14, a reconnaissance alert on a different source IP at 03:47, and a data exfiltration alert on a third IP pair at 06:22 — each assessed as medium priority in isolation, yet collectively the signature of a coordinated lateral-movement campaign. A CERT team that could have predicted the second and third alerts from the first could have intervened before exfiltration. This is the problem of alert prediction: given the observed alert history, infer which IP will be targeted next, under which attack type, and via which propagation path. Knowledge graphs (KGs) offer a principled solution by representing attacker–victim interactions as relational triples (h,r,t)(h,r,t), enabling heterogeneous, semantically rich reasoning over unseen entities — a critical capability for inductive alert prediction. However, standard KGC models are limited to binary triples, which cannot encode the auxiliary flow-level metadata that distinguishes network alerts of the same type. Two alerts sharing the same source IP, attack type, and destination IP may represent fundamentally different threat scenarios depending on their timestamp, port, protocol, and flow intensity. This motivates hyper-relational knowledge graphs, where each alert is represented as a qualified statement (h,r,t,)(h,r,t,Q) with Q encoding auxiliary key–value pairs beyond the core triple. Existing hyper-relational KGC models (StarE [4], StarQE [1], HyNT [3]) are predominantly transductive, limiting their applicability when new IP addresses unseen during training must be reasoned about at inference time. Neural Bellman–Ford Networks (NBFNet) [12] address inductive link prediction via path-based message passing, but operate on binary triples with no mechanism for qualifier context. In this work, we model network alerts as a hyper-relational knowledge graph and formulate alert prediction as an HR-KGC problem. We introduce five models across three paradigms: qualifier-aware path propagation, embedding-based fusion, and complex query answering over alert graphs. The main contributions are: • HR-KGC alert benchmarks. We formulate alert prediction as HR-KGC and construct inductive benchmarks from Warden and UNSW-NB15 across three qualifier-density regimes (Q33, Q66, Q100), retaining 33%, 66%, and 100% of qualifier pairs respectively, to study robustness to metadata availability. • HR-NBFNet and MT-HR-NBFNet. HR-NBFNet extends NBFNet to the hyper-relational setting by injecting qualifier context at every message-passing step. MT-HR-NBFNet augments this backbone with joint prediction of tail, relation, and qualifier-value objectives within a single Bellman–Ford pass at no additional graph-traversal cost. • AlertStar and MT-AlertStar. AlertStar fuses qualifier-aware multi-head cross-attention with a learned path-composition branch via a trainable scalar gate, operating entirely in embedding space without graph traversal, yielding more stable optimisation on sparse, inductive alert graphs than HR-NBFNet. MT-AlertStar extends this with joint prediction from a shared Transformer encoder, completing a symmetric multi-task design across paradigms with stronger qualifier supervision than graph-propagation alternatives. • HR-NBFNet-CQ. HR-NBFNet-CQ extends qualifier-aware representations to answer four first-order logic query types — one-hop (1p), two-hop chain (2p), two-anchor intersection (2i), and two-anchor union (2u) — enabling complex multi-condition threat investigations directly over the alert knowledge graph. • Comprehensive evaluation. We evaluate all five models against state-of-the-art HR-KGC baselines under inductive settings, with complexity analysis showing AlertStar achieves up to 50×50× per-epoch speedup over HR-NBFNet, and four ablation studies (A1–A4) dissecting component contributions and qualifier-density sensitivity. The remainder of this paper is organised as follows. Section 2 introduces background on hyper-relational KGs and the threat model. Section 3 formalises the problem and presents the five models. Section 4 describes experiments and evaluation. Section 5 reviews related work. Section 6 analyses results. Section 7 concludes and outlines future work. 2 Preliminaries and Threat Model This section introduces the formal background required for the remainder of the paper, and defines the operational threat model and assumptions under which the proposed models are evaluated. 2.1 Knowledge Graphs Definition 2.1 (Knowledge Graph). A knowledge graph is a directed, labeled multigraph =(ℰ,ℛ,)G=(E,R,T), where ℰE is a finite set of entities, ℛR is a finite set of relation types, and ⊆ℰ×ℛ×ℰT ×R×E is a set of facts, each represented as a triple (h,r,t)(h,r,t) with head entity h∈ℰh , relation r∈ℛr , and tail entity t∈ℰt . In the network alert setting, ℰE is the set of observed IP addresses, ℛR is the set of attack categories (e.g., Recon, DoS, DDoS, Anomaly), and each triple (h,r,t)∈(h,r,t) records that source IP h launched an attack of type r against destination IP t. Definition 2.2 (Knowledge Graph Completion). Given a knowledge graph G, the knowledge graph completion (KGC) task is to score and rank candidate tail entities t′t for an incomplete query (h,r,?)(h,r,?), or equivalently candidate head entities h′h for a query (?,r,t)(?,r,t), using a scoring function f:ℰ×ℛ×ℰ→ℝf:E×R×E . Applied to alert prediction, the KGC task answers: given that source IP h is conducting an attack of type r, which destination IP t′t is the most likely next target. 2.2 Hyper-Relational Knowledge Graphs Standard KGs represent each alert as a binary triple (h,r,t)(h,r,t), discarding the flow-level metadata that accompanies every alert record. Hyper-relational knowledge graphs (HR-KGs) extend this representation to incorporate predicate-scoped contextual attributes known as qualifiers [4]. Definition 2.3 (Hyper-Relational Knowledge Graph). A hyper-relational knowledge graph is a set of qualified statements +=(h,r,t,)G^+=\(h,r,t,Q)\, where (h,r,t)∈ℰ×ℛ×ℰ(h,r,t) ×R×E is the main triple and =(qki,qvi)i=1nQ=\(q_k^i,q_v^i)\_i=1^n is a set of qualifier pairs with qki∈ℛq_k^i (qualifier relations) and qvi∈ℰq_v^i (qualifier entities). StarE StarE [4] is a message-passing GNN designed for representation learning over HR-KGs. It augments each main relation embedding with a qualifier vector computed by composing and aggregating all qualifier pairs of a given fact: q=q∑(qk,qv)∈ϕq(qk,qv),h_q=W_q _(q_k,\,q_v) _q(h_q_k,\,h_q_v), (1) where qW_q is a learned projection matrix and ϕq _q is a composition function such as DistMult or RotatE [9]. The qualifier vector is then merged with the main relation embedding rh_r via: γ(r,q)=α⊙r+(1−α)⊙q,γ(h_r,h_q)=α _r+(1-α) _q, (2) where α∈[0,1]α∈[0,1] is a hyperparameter controlling the flow of qualifier information into the relation representation. Node embeddings are then updated by message passing: v=f(∑(u,r)∈(v)λ(r)ϕr(u,γ(r,q)vu)),h_v=f\! ( _(u,r) (v)W_λ(r)\, _r\! (h_u,\,γ(h_r,h_q)_vu ) ), (3) where ϕr _r is a relation composition function, λ(r)W_λ(r) is a direction-specific weight matrix for incoming, outgoing, and self-loop edges, and (v)N(v) denotes the set of incoming edges to v. 2.3 Neural Bellman-Ford Networks Path-based knowledge graph completion methods reason over multi-hop paths between a query source entity and candidate answer entities. Neural Bellman-Ford Networks (NBFNet) [12] provide a principled and general framework for this by casting link prediction as a pair representation learning problem. Definition 2.4 (Pair Representation). For a query (u,q,?)(u,q,?), the pair representation q(u,v)h_q(u,v) is defined as the generalised sum of all path representations between u and v, where each path representation is the generalised product of the edge representations along the path: q(u,v)=⨁P∈uvq(P),h_q(u,v)= _P _uvh_q(P), (4) q(P=(e1,…,e|P|))=⨂i=1|P|q(ei).h_q(P=(e_1,…,e_|P|))= _i=1^|P|w_q(e_i). (5) where ⊕ and ⊗ denote a commutative summation operator and a (not necessarily commutative) multiplication operator, respectively, and q(e)w_q(e) is the representation of edge e under query relation q. This formulation subsumes several classical link-prediction heuristics as special cases: Katz index uses ⊕=+ =+ and ⊗=× =×; graph distance uses ⊕=min = and ⊗=+ =+; widest path uses ⊕=max = and ⊗=min = [12]. Generalised Bellman-Ford Algorithm. Directly computing Equations (4)–(5) is intractable, because the number of paths grows exponentially with path length. When ⟨⊕,⊗⟩ , satisfy a semiring [12], the pair representation can instead be computed efficiently via the generalised Bellman-Ford algorithm. Letting q1_q and q0_q denote the multiplication and summation identities of the semiring, the algorithm initialises q(0)(u,v)←q(u=v),h_q^(0)(u,v) 1_q(u=v), (6) and iterates q(t)(u,v)←(⨁(x,r,v)∈ℰ(v)q(t−1)(u,x)⊗q(x,r,v))⊕q(0)(u,v), splith_q^(t)(u,v)&← ( _(x,r,v) (v)h_q^(t-1)(u,x) _q(x,r,v) )\\ & \,h_q^(0)(u,v), split (7) where the residual connection to q(0)(u,v)h_q^(0)(u,v) preserves the boundary condition at every layer. The algorithm computes q(u,v)h_q(u,v) for a fixed source u, a fixed query relation q, and all v∈v in parallel, exploiting the distributive property of ⊗ over ⊕ to avoid explicit path enumeration. Setting ⊕=min = and ⊗=+ =+ recovers the original Bellman-Ford shortest-path algorithm [2]. Neural Bellman-Ford Networks NBFNet relaxes the semiring assumption and replaces the hand-crafted operators with three learned neural components: 1. INDICATOR replaces the boundary condition q(u=v)1_q(u=v). In practice it is instantiated as (u=v)⋅q1(u=v)·e_q, where q∈ℝde_q ^d is a learned query-relation embedding. 2. MESSAGE replaces the binary multiplication operator ⊗ . It is instantiated using relational operators from knowledge-graph embeddings [9], e.g. translation (+qh+w_q) or element-wise multiplication (⊙qh _q). 3. AGGREGATE replaces the n-ary summation operator ⨁ . It is a permutation-invariant set aggregation function, e.g. sum, mean, max, or PNA, followed by a linear transformation and a non-linear activation. Substituting these functions into Equations (6)–(7) gives the NBFNet update, where (x,r,v)∈ℰ(v)(x,r,v) (v) denotes the set of incoming edges to node v: v(0) _v^(0) ←Indicator(u,v,q), ← Indicator(u,v,q), (8) v(t) _v^(t) ←Aggregate(t)(Message(t)(x(t−1),q(x,r,v)) ← Aggregate^(t)\! ( \ Message^(t)\! (h_x^(t-1),\,w_q(x,r,v) ) \ . ∪v(0)). .∪\, \h_v^(0) \ ). After L propagation steps the pair representation v(L)h_v^(L) is scored by a shallow MLP: f(u,q,v)=σ(MLP(v(L))).f(u,q,v)=σ\! ( MLP\! (h_v^(L) ) ). (9) 2.4 Threat Model and Assumptions We adopt a bounded adversarial model tailored to the properties of the evaluated datasets. The following assumptions define the operational threat context. A1 — External and Internal Adversaries. The adversary may originate from outside or within the network perimeter. Both attacker positions are handled uniformly: source and destination IPs are entities in ℰE regardless of their network location. A2 — Limited-Knowledge Adversary. The adversary has no knowledge of the alert prediction model, its parameters, or its outputs, ensuring that observed patterns reflect genuine attacker behavior [6]. A3 — Unidirectional Attack Interactions. Each alert represents a unidirectional flow from a source (attacker) IP to a destination (victim) IP. A victim may subsequently launch a counter-attack toward the original attacker, but such interactions are captured as separate alert events in a different time window, rather than modeled as a single simultaneous bidirectional exchange [11]. A4 — Role Disjointness per Interaction. Within a single alert, an IP plays exactly one role: attacker (h) or victim (t). An IP may switch roles across alerts but not within the same event. A5 — Closed-World Attack Taxonomy. Predicted attack categories are limited to those annotated in the training data. While our inductive approach generalizes to unseen entities at test time, it can only predict whether an interaction exists between previously unseen IPs — it cannot infer the attack type for categories not observed during training. Zero-day or novel attack taxonomies therefore remain outside the current scope. 3 Methodology Figure 1: Three representational levels: (A) triple-based facts, (B) hyper-relational facts, (C) path-based hyper-relational reasoning. Figure 1 illustrates the three representational levels motivating this work. Triple-based facts (A) cannot distinguish two attacks sharing the same (h,r,t)(h,r,t) structure. Hyper-relational facts (B) solve this ambiguity by attaching flow-level qualifiers, protocol, port, flow count, and detect time, to each alert. Path-based hyper-relational reasoning (C) further captures multi-hop attacker pivoting that qualifier-enriched node representations alone cannot detect. These three levels motivate a three-paradigm investigation. Paradigm I: Graph Propagation. HR-NBFNet extends NBFNet [12] to the hyper-relational setting by injecting qualifier context at every message-passing step via per-edge DistMult embeddings. MT-HR-NBFNet augments this backbone with three auxiliary tasks, tail, attack type, and qualifier-value prediction, sharing a single Bellman–Ford pass at no additional graph-traversal cost. However, Bellman–Ford propagation has a known scalability limitation: (||Qmaxd)O(|T|\,Q_ \,d) per query, with path counts growing exponentially with graph diameter. In dense alert graphs where a single attacker IP interacts with hundreds of victims, this overhead becomes prohibitive and redundant traversals beyond moderate path lengths carry negligible signal [12]. Paradigm I: Embedding-Based. AlertStar fuses qualifier context and structural path information entirely in embedding space via qualifier-aware multi-head cross-attention and a feed-forward path-composition branch, balanced by a trainable scalar gate, no graph traversal is performed. MT-AlertStar extends this with a Transformer encoder and the same three auxiliary tasks, completing a symmetric multi-task design across paradigms. The empirical comparison between HR-NBFNet and AlertStar directly answers whether global path reasoning is worth its computational cost for alert prediction. Paradigm I: HR-NBFNet-CQ. Answering complex first-order logic queries, one-hop, two-hop chain, two-anchor intersection, and two-anchor union, over the hyper-relational alert graph, enabling multi-condition threat investigations by SOC and CERT analysts. 3.1 Alert Graph Representation Table I illustrates the three representational levels applied to alert data. Each alert is first modeled as a binary triple (h,r,t)(h,r,t) — source IP, attack category, and destination IP — forming =(ℰ,ℛ,)G=(E,R,T). This captures who attacked whom, but treats all alerts of the same category between the same IP pair as equivalent, discarding flow-level context. To resolve this ambiguity, we extend to a hyper-relational KG +G^+ by attaching qualifier pairs Q encoding detect time, flow count, port, and protocol to each triple — for example, distinguishing the two DDoS alerts in Figure 1 by port and flow count. The StarE encoder [4] incorporates this context into message passing via Eqs. (1)–(3). However, qualifier-enriched node representations cannot detect how an attacker pivots across multiple victims via intermediate nodes (Figure 1(C)). NBFNet [12] addresses this by computing a pair representation q(u,v)h_q(u,v) for each source–target pair via multi-hop path propagation (Eqs. (6)–(9)), enabling detection of coordinated multi-stage attacks and lateral movement patterns beyond what local neighbourhood aggregation can capture. TABLE I: Example alert records and their qualified-statement representation (h,r,t,)(h,r,t,Q). The qualifier set Q encodes detect time, flow count, port, and protocol. Detect Time Flow Count Source IP (h)(h) Target IP (t)(t) Port Protocol Category (r)(r) 2019-03-11 00:05 17,094 185.192.59.136 142.252.135.136 22 TCP Recon Scan 2019-03-12 00:45 5,113 78.234.46.141 142.252.32.63 443 TCP Availability DoS 2019-03-14 00:25 15 185.192.59.136 142.252.32.63 81 TCP Availability DDoS 2019-03-14 00:25 39 78.234.46.141 142.252.32.63 22 UDP Anomaly Traffic Qualified-statement mapping for row 1: h = 185.192.59.136, r = Recon Scan, t = 142.252.135.136 Q = detectTime: 2019-03-11 00:05, flowCount: 17094, port: 22, protocol: TCP 3.2 Paradigm I: Hyper-Relational NBFNet HR-NBFNet extends NBFNet [12] to the hyper-relational setting by injecting qualifier context at every propagation step, in the query-conditioned initialisation and in each per-edge message. Notation and base formulation follow Section 2. Figure 2: HR-NBFNet combines StarE qualifier encoding [4] with NBFNet Bellman-Ford propagation [12]. Qualifier pairs are composed via ϕq _q, aggregated, projected by qW_q, and merged with rh_r via γ, injecting flow-level context into the pair representation uvqq′(t)h_uvqq ^(t) at every layer. Inverse edges (t,r−1,h)(t,r^-1,h) enable bidirectional propagation for both tail and head prediction. Qualifier vector. For each edge (u,r,v)(u,r,v), the qualifier vector is: q=q∑(qk,qv)∈rvuqk⊙qv.h_q=W_q _(q_k,q_v) _r_vuh_q_k _q_v. (10) Relation–qualifier merging. γ(r,q)=α⊙r+(1−α)⊙q,γ(h_r,h_q)=α _r+(1-α) _q, (11) where α∈[0,1]α∈[0,1] is a learned scalar. Qualifier-conditioned propagation. The hidden state uvqq′(t)h_uvqq ^(t) encodes all qualifier-conditioned paths from source u to node v at layer t, where q′q denotes the query relation and q denotes the qualifier context q=(qki,qvi)i=1nQ_q=\(q_k^i,q_v^i)\_i=1^n associated with the query. It is initialised at the source node as: uvqq′(0)=q′qry+proj∑(qk,qv)∈qqkqry⊙qvqryv=u,otherwise,h_uvqq ^(0)= casese_q ^qry+W_proj _(q_k,q_v) _qe_q_k^qry _q_v^qry&v=u,\\ 0&otherwise, cases (12) where q′qrye_q ^qry is the query relation embedding and qkqry,qvqrye_q_k^qry,e_q_v^qry are dedicated query-side qualifier embeddings. proj∈ℝd×dW_proj ^d× d projects the qualifier context into relation embedding space. This extends NBFNet [12] at two levels: (i) the Indicator boundary condition (Eq. 12) is conditioned on both the query relation q′q and the query-level qualifier context q; and (i) the aggregation at each layer (Eq. 13) incorporates the edge-level qualifier vector qxrvh_q^xrv via γ(r,qxrv)γ(h_r,h_q^xrv), injecting flow-level metadata into every Bellman-Ford iteration. v(t) _v^(t) =∑(x,r,v)∈ℰ(v)uxqq′(t−1)+γ(r,qxrv), = _(x,r,v) (v)h_uxqq ^(t-1)+γ(h_r,h_q^xrv), (13) uvqq′(t) _uvqq ^(t) =Dropout(ReLU(LN((t)[uvqq′(t−1)∥v(t)])))+uvqq′(0), =Dropout(ReLU(LN(W^(t)[h_uvqq ^(t-1)\|a_v^(t)])))+h_uvqq ^(0), (14) where the residual to uvqq′(0)h_uvqq ^(0) preserves the query qualifier signal across all layers. Scoring and training. After L layers, candidate tails are scored by: fHR(h,q′,t,)=MLPHR([htqq′(L)∥q′qry]),f_HR(h,q ,t,Q)=MLP_HR\! ([h_htqq ^(L)\|e_q ^qry] ), (15) trained with margin ranking loss. Since each (h,q′)(h,q ) query requires one full Bellman–Ford pass, training groups triples by (h,q′)(h,q ) and samples kmaxk_ tails per group. The complete procedure is given in Algorithm 1. Algorithm 1 HR-NBFNet: Inference 1:source node h, query relation q′q , query qualifiers q=q=(qki,qvi)i=1nq=Q_q=\(q_k^i,q_v^i)\_i=1^n, graph +G^+, #layers L 2:score vector ∈ℝNs ^N where [v]=fHR(h,q′,v,q)s[v]=f_HR(h,q ,v,Q_q) for all v∈v 3:⊳ Boundary condition 4:(0)←N×dH^(0) 0^N× d 5:ϕq←∑(qki,qvi)∈qqkiqry⊙qviqry φ_q← _(q_k^i,q_v^i)∈ qe_q_k^i^qry _q_v^i^qry ⊳ encode q 6:(0)[h]←qry[q′]+projϕqH^(0)[h] ^qry[q ]+W_proj φ_q (12) ⊳ inject q′q and q 7:res0←(0).clone()H^0_res ^(0).clone() 8:⊳ Bellman-Ford iteration 9:for t=1t=1 to L do 10: Compute qh_q per edge via Eq. (10) ⊳ edge qualifier 11: Compute γ via Eq. (11) ⊳ relation–qualifier merge 12: (t)←scatter_add((t−1)[]+,)A^(t) \_add(H^(t-1)[u]+ γ,\,v) (13) 13: (t)←Dropout(ReLU(LN((t)[(t−1)∥(t)])))+res0H^(t) (ReLU(LN(W^(t)[H^(t-1)\|A^(t)])))+H^0_res (14) 14:end for 15:⊳ Scoring 16:return ←MLPHR([(L)∥qry[q′].expand(N,d)])s _HR([H^(L)\|e^qry[q ].expand(N,d)]) (15) Time complexity. O(L||Qmaxd+|ℰ|d2)O(L\,|T|\,Q_ \,d+|E|\,d^2) per query. 3.2.1 Multi-Task Hyper-NBFNet MT-HR-NBFNet augments HR-NBFNet with two auxiliary tasks, relation and qualifier-value prediction, alongside tail prediction. All three tasks share a single Bellman-Ford pass, so multi-task supervision adds only three lightweight MLP heads with no extra graph-traversal cost. Three prediction heads. Using (L)H^(L) from Eqs. 12–14: ftail(t) f_tail(t) =MLPt([(L)[t]∥q′qry]), =MLP_t\! ([H^(L)[t]\|e_q ^qry] ), (16) ^q′ y_q =MLPq′((L)[h])∈ℝ|ℛ|, =MLP_q \! (H^(L)[h] ) ^|R|, (17) ^qv y_q_v =MLPqv(σ(g[(L)[h]∥Eqkhead[qk∗]])⊙(L)[h]), =MLP_q_v\! (σ\! (W_g[H^(L)[h]\|E_q_k^head[q_k^*]] ) ^(L)[h] ), (18) where EqkheadE_q_k^head is a dedicated qualifier-key embedding table and qk∗q_k^* is the target qualifier key. The joint loss is: ℒMT-HR=λtℒt+λq′ℒq′+λqvℒqv,L_MT -HR= _t\,L_t+ _q \,L_q + _q_v\,L_q_v, (19) where ℒtL_t is margin ranking, and ℒq′,ℒqvL_q ,L_q_v are cross-entropy. λq′<λt _q < _t since propagated representations are already partially discriminative for relation type. Algorithm 2 MT-HR-NBFNet: Training Step 1:group (h,q′)(h,q ), positives ti+i=1B\t_i^+\_i=1^B, qualifiers i\Q_i\, graph +G^+, weights λt,λq′,λqv _t, _q , _q_v, margin δ 2:updated Θ 3:⊳ Single shared BF pass 4:(L)←HR-Infer(h,q′,rep,+)H^(L) -Infer(h,q ,Q_rep,G^+) Alg. 1 5:⊳ Task 1: tail (margin ranking) 6:ℒt←1B∑imax(0,δ−ftail(ti+)+ftail(ti−))L_t← 1B _i (0,\,δ-f_tail(t_i^+)+f_tail(t_i^-)) (16) 7:⊳ Task 2: relation (cross-entropy) 8:ℒq′←CE(MLPq′((L)[h]),q′)L_q (MLP_q (H^(L)[h]),\,q ) (17) 9:⊳ Task 3: qualifier-value (cross-entropy) 10:ℒqv←CE(^qv,qv∗)L_q_v ( y_q_v,\,q_v^*) if rep≠∅Q_rep≠ , else 0 (18) 11:⊳ Combined loss and update 12:Θ←Adam(∇Θ(λtℒt+λq′ℒq′+λqvℒqv),clip=1.0) ( _ ( _tL_t+ _q L_q + _q_vL_q_v),\,clip=1.0) 13:return Θ Time complexity. One step runs a single BF pass shared across all three tasks: O(L||Qmaxd+|ℰ|d2)O(L\,|T|\,Q_ \,d+|E|\,d^2), asymptotically identical to HR-NBFNet. 3.3 Paradigm I: AlertStar As illustrated in Figure 3, AlertStar fuses qualifier context and structural path information entirely in embedding space for a single triple (h,r,)(h,r,Q), no graph traversal is performed. It is orders of magnitude faster than HR-NBFNet while retaining full qualifier awareness. Figure 3: Architecture of AlertStar. Qualifier pairs are aggregated into U_Q and used as key/value in MHA with re_r as query, producing ~r e_r. Two branches — cross-attention and path-composition — are fused via a trainable gate α=σ(g)α=σ(g) into z, scored against candidate tails via dot product. Qualifier enrichment. Each qualifier pair is combined as i=qki+qviu^i=e_q_k^i+e_q_v^i, forming ∈ℝn×dU_Q ^n× d. The relation is enriched via Multi-head cross-attention (MHA): ~r=MHA(r,,)≠∅,rotherwise. e_r= casesMHA(e_r,U_Q,U_Q)&Q≠ ,\\ e_r&otherwise. cases (20) Cross-attention branch. Fuses he_h with ~r e_r via residual connection, followed by Layer Normalisation (LN), preserving the attacker IP signal: ~rattn=LN(h+~r). e_r^attn=LN(e_h+ e_r). (21) Path-composition branch. Models implicit compositional head–relation structure without graph traversal: =LN(h+FFNpath([h∥~rattn])),p=LN\! (e_h+FFN_path\! ([e_h\| e_r^attn] ) ), (22) where FFNpathFFN_path is a two-layer feed-forward network consisting of Linear(2d→d)Linear(2d→d), Layer Normalisation, ReLU activation, Dropout, and Linear(d→d)Linear(d→d). Gated fusion and scoring. A trainable scalar gate g∈ℝg (initialised to 0.50.5) balances both branches via sigmoid σ: α=σ(g),=α~rattn+(1−α)∈ℝd,α=σ(g), =α\, e_r^attn+(1-α)\,p ^d, (23) where z is the fused alert representation. Candidate tails are scored via dot product against the entity embedding matrix E∈ℝN×dE ^N× d — note that t is not an input but a scoring target: fAS(h,r,t,)=Dropout()⊤t,=E⊤∈ℝN,f_AS(h,r,t,Q)=Dropout(z) e_t, =z\,E ^N, (24) where t∈ℝde_t ^d is the embedding of candidate tail t and N=|ℰ|N=|E| is the number of entities. Algorithm 3 AlertStar: Forward Pass 1:head h, relation r, qualifiers Q, optional tail t+t^+ (training only) 2:score f∈ℝf or ∈ℝNs ^N 3:⊳ Qualifier enrichment 4:←[Eqk[qki]+Eqv[qvi]]i=1nU_Q←[E_q_k[q_k^i]+E_q_v[q_v^i]]_i=1^n 5:~r←MHA(r,,) e_r (e_r,U_Q,U_Q) if n>0n>0, else re_r (20) 6:⊳ Cross-attention branch 7:~rattn←LN(h+~r) e_r^attn (e_h+ e_r) (21) 8:⊳ Path-composition branch 9:←LN(h+FFNpath([h∥~rattn]))p (e_h+FFN_path([e_h\| e_r^attn])) (22) 10:⊳ Gated fusion 11:←σ(g)~rattn+(1−σ(g))z←σ(g)\, e_r^attn+(1-σ(g))\,p (23) 12:⊳ Scoring 13:if training then 14: return ⊤t+z e_t^+ 15:else 16: return E⊤z\,E 17:end if(24) Time complexity. O(Mnd+d2+Nd)O(Mnd+d^2+Nd) per triple — no graph traversal. Typically 10310^3–104×10^4× faster per sample than HR-NBFNet on dense alert graphs. 3.3.1 MT-AlertStar: Multi-Task AlertStar As illustrated in Figure 4, MT-AlertStar extends AlertStar with a Transformer encoder [10], and three simultaneous prediction objectives, tail, relation, and qualifier-value, mirroring MT-HR-NBFNet within the embedding-based paradigm. Figure 4: Architecture of MT-AlertStar. The masked token sequence X is encoded by a Transformer, whose relation-token output r=[1]h_r=H[1] serves as shared context for three MLP heads predicting tail, relation, and qualifier value jointly. Input and masking. For a qualified triple (h,r,t,)(h,r,t,Q) with n=||n=|Q| qualifier pairs, all components are embedded into ℝdR^d and arranged into a token sequence ∈ℝ(3+2n)×dX ^(3+2n)× d: =[h,r,t,qk1,qv1,…,qkn,qvn]∈ℝ(3+2n)×d,X=[e_h,\,e_r,\,e_t,\,e_q_k^1,\,e_q_v^1,\,…,\,e_q_k^n,\,e_q_v^n] ^(3+2n)× d, (25) where qki,qvi∈ℝde_q_k^i,e_q_v^i ^d are learned embeddings for the i-th qualifier key and value respectively. Task-specific masking prevents target leakage: t←e_t 0 for tail prediction; r←e_r 0 for relation prediction; qualifier pair j is removed for qualifier-value prediction. Transformer encoder. The masked sequence is passed through a standard Transformer encoder [10]: =TransformerEncoder()∈ℝ(3+2n)×dH=TransformerEncoder(X) ^(3+2n)× d, where each layer applies multi-head self-attention, position-wise FFN, LayerNorm, and residual connections. The relation-token output r=[1]∈ℝdh_r=H[1] ^d — position 1 in the sequence — serves as shared context for all three prediction heads, as it attends over all other tokens including the head entity, qualifier pairs, and masked target position. Three prediction heads. All heads share the same architecture: two linear layers with LayerNorm, ReLU, and Dropout interleaved, mapping ℝd→ℝdoutR^d ^d_out. Using the shared context rh_r, each head produces a score vector over its prediction space: ^t y_t =MLPt(r)∈ℝN, =MLP_t(h_r) ^N, (26) ^q′ y_q =MLPq′(r)∈ℝ|ℛ|, =MLP_q (h_r) ^|R|, (27) ^qv y_q_v =MLPqv(r)∈ℝ|V|, =MLP_q_v(h_r) ^|Q_V|, (28) where ^t y_t, ^q′ y_q , and ^qv y_q_v are the predicted score vectors over N entities, |ℛ||R| attack categories, and |V||Q_V| qualifier values respectively. The three tasks are trained jointly: ℒMT-AS=λtℒt+λq′ℒq′+λqvℒqv,L_MT -AS= _t\,L_t+ _q \,L_q + _q_v\,L_q_v, (29) where ℒtL_t, ℒq′L_q , and ℒqvL_q_v are cross-entropy losses for tail, relation, and qualifier-value prediction respectively, and λt,λq′,λqv∈ℝ+ _t, _q , _q_v ^+ are task-specific loss weights. Algorithm 4 MT-AlertStar: Training Step 1:triple (h,r,t,)(h,r,t,Q), task τ∈tail,rel,qvalτ∈\ tail, rel, qval\ 2:updated Θ 3:⊳ Build masked token sequence 4:h,r,t←E[h],Er[r],E[t]x_h,x_r,x_t← E[h],E_r[r],E[t] 5:Mask: t←x_t←0 if tail; r←x_r←0 if rel; remove pair j if qval 6:←[h;r;t;qk1;qv1;…]X←[x_h;x_r;x_t;\,e_q_k^1;e_q_v^1;…] (25) 7:⊳ Transformer encoder 8:r←TransformerEncoder()[1]h_r (X)[1] 9:⊳ Task-specific head and loss 10:^←MLPτ(r) y _τ(h_r) (26)–(28) 11:ℒτ←λτ⋅CE(^,y∗)L_τ← _τ·CE( y,\,y^*) 12:⊳ Update 13:Θ←Adam(∇Θℒτ,clip=1.0) ( _ L_τ,\,clip=1.0) 14:return Θ Time complexity. O(n2d+ndF+Nd)O(n^2d+ndF+Nd) per sample, no graph traversal. 3.4 Paradigm I: Complex Hyper-Relational As illustrated in Figure 5, HR-NBFNet-CQ answers complex first-order logic queries over the hyper-relational alert graph, enabling richer threat intelligence beyond single-hop link prediction. Figure 5: Hyper-relational query templates. Qualifier pairs on each edge may vary from 0 to n. Query types. Four templates are supported, each grounded in a concrete threat scenario: 1p (?t.r(h,t,1))(?t.\;r(h,t,Q_1)) — direct prediction. “Which IP was targeted by 185.192.59.136 via DDoS with flow count 17,094 over TCP on port 22?” 2p (?t.∃y:r1(h,y,1)∧r2(y,t,2))(?t.\;∃ y:\;r_1(h,y,Q_1) r_2(y,t,Q_2)) — lateral movement: attacker first compromises an intermediate host y, which then attacks t. y is inferred implicitly during propagation. “185.192.59.136 performed a Recon Scan on y; y then launched a DoS — who was the final target?” 2i (?t.r1(h1,t,1)∧r2(h2,t,2))(?t.\;r_1(h_1,t,Q_1) r_2(h_2,t,Q_2)) — coordinated attack: finds a shared victim targeted by two source IPs simultaneously. “Which IP was hit by both 185.192.59.136 via DDoS and 78.234.46.141 via Recon Scan?” 2u (?t.r1(h1,t,1)∨r2(h2,t,2))(?t.\;r_1(h_1,t,Q_1) r_2(h_2,t,Q_2)) — campaign victim set: retrieves all IPs targeted by either attacker. Unlike StarQE [1], which restricts to conjunctive templates, we support this disjunctive query motivated by campaign-level co-targeting in Warden, where coordinated attacks against overlapping victim sets are operationally meaningful for CERT analysts. Composition operator. All query types share the residual path-composition operator ϕNBF _NBF, which models one hop entirely in embedding space. Given a current node embedding ∈ℝdx ^d, a relation r, qualifier set Q, and an anchor embedding 0∈ℝdx_0 ^d (the source entity, preserved via residual): ϕNBF(,r,,0)=0+FFNϕ([∥~r]), _NBF(x,r,Q,x_0)=x_0+FFN_φ([x\| e_r]), (30) where [⋅∥⋅][·\|·] denotes concatenation, ~r e_r is the qualifier-enriched relation embedding from Eq. 20, and FFNϕ:ℝ2d→ℝdFFN_φ:\,R^2d ^d comprises two linear layers with LayerNorm, ReLU, and Dropout. Query formulations. Each query type produces a query vector ∈ℝdq ^d by composing ϕNBF _NBF in different structural patterns. For 2p, r1r_1 and r2r_2 are two sequential relations and ∅ denotes an empty qualifier set on the second hop. For 2i and 2u, h1h_1 and h2h_2 are two anchor source IPs with relations r1r_1 and r2r_2: 1p ^1p =ϕNBF(h,r,,h), = _NBF(e_h,r,Q,e_h), (31) 2p ^2p =ϕNBF(ϕNBF(h,r1,,h),r2,∅,h), = _NBF( _NBF(e_h,r_1,Q,e_h),r_2, ,e_h), (32) 2i ^2i =Wi[ϕNBF(h1,r1,,h1)∥ϕNBF(h2,r2,∅,h2)], =W_i[ _NBF(e_h_1,r_1,Q,e_h_1)\| _NBF(e_h_2,r_2, ,e_h_2)], (33) 2u ^2u =12(ϕNBF(h1,r1,,h1)+ϕNBF(h2,r2,∅,h2)), = 12( _NBF(e_h_1,r_1,Q,e_h_1)+ _NBF(e_h_2,r_2, ,e_h_2)), (34) where Wi∈ℝd×2dW_i ^d× 2d is a learned projection for asymmetric intersection. All query types are scored by dot product against candidate entity embedding e∈ℝde_e ^d: f=Dropout()⊤ef=Dropout(q) e_e, with inference =E⊤∈ℝNs=qE ^N over all N entities costing O(Nd)O(Nd). Algorithm 5 HR-NBFNet-CQ: Training Step 1:triple (h,r,t+,)(h,r,t^+,Q), neighbour index ℐoutI_out, anchor index ℐinI_in, margin δ 2:updated Θ 3:⊳ Qualifier enrichment 4:~r←QualEnrich(Er[r],) e_r (E_r[r],Q) (20) 5:t−←Uniform(ℰ)t^- (E); ←∅A← ; ℒ←0L← 0 6:⊳ 1p (always) 7:1p←ϕNBF(E[h],r,,E[h])q^1p← _NBF(E[h],r,Q,E[h]) (31) 8:ℒ+=max(0,δ−1p⊤E[t+]+1p⊤E[t−])L += (0,δ-q^1p E[t^+]+q^1p E[t^-]); +=1pA +=\1p\ 9:⊳ 2p (if chain exists) 10:if ℐout[t+]≠∅I_out[t^+]≠ then 11: Sample (r2,t2)(r_2,t_2); 2p←ϕNBF(1p,r2,∅,E[h])q^2p← _NBF(q^1p,r_2, ,E[h]) (32) 12: ℒ+=max(0,δ−2p⊤E[t2]+2p⊤E[t−])L += (0,δ-q^2p E[t_2]+q^2p E[t^-]); +=2pA +=\2p\ 13:end if 14:⊳ 2i (if second anchor exists) 15:if ∃(h2,r2)∈ℐin[t+],h2≠h∃(h_2,r_2) _in[t^+],\,h_2≠ h then 16: 2i←Wi[1p∥ϕNBF(E[h2],r2,∅,E[h2])]q^2i← W_i[q^1p\| _NBF(E[h_2],r_2, ,E[h_2])] (33) 17: ℒ+=max(0,δ−2i⊤E[t+]+2i⊤E[t−])L += (0,δ-q^2i E[t^+]+q^2i E[t^-]); +=2iA +=\2i\ 18:end if 19:⊳ Average and update 20:Θ←Adam(∇Θ(ℒ/||),clip=1.0) ( _ (L/|A|),\,clip=1.0) 21:return Θ Time complexity. O(nd+d2+Nd)O(nd+d^2+Nd) per triple — same class as AlertStar, much cheaper than HR-NBFNet. 3.5 Shared Training Objective All five models are trained with Adam, gradient clipping to unit norm, and model selection by best validation MRR. The shared margin ranking loss is: ℒmargin=∑(h,r,t,)max(0,δ−f(h,r,t,)+f(h,r,t−,)),L_margin= _(h,r,t,Q) \! (0,\;δ-f(h,r,t,Q)+f(h,r,t^-,Q) ), (35) where t−∼Uniform(ℰ)t^- (E). MT-HR-NBFNet and MT-AlertStar use the joint objectives of Eqs. 19 and 29. HR-NBFNet-CQ averages the margin loss over active query types. Table I summarises per-sample inference complexity. TABLE I: Per-sample inference complexity. N=|ℰ|N=|E|, E=||E=|T|, L=L=layers, Q=QmaxQ=Q_ , n=||n=|Q|, S=3+2nS=3+2n, F=F=FFN width. Model Complexity Graph Traversal? AlertStar O(nd+d2+Nd)O(nd+d^2+Nd) No MT-AlertStar O(S2d+SdF+Nd)O(S^2d+SdF+Nd) No HR-NBFNet-CQ O(nd+d2+Nd)O(nd+d^2+Nd) No HR-NBFNet O(LEQd+Nd2)O(LEQd+Nd^2) Yes MT-HR-NBFNet O(LEQd+Nd2)O(LEQd+Nd^2) Yes TABLE I: KGC results on the Warden Alert dataset using statements under inductive and transductive settings. Inductive Transductive Model MR ↓ MRR ↑ H@1 ↑ H@3 ↑ H@10 ↑ MR ↓ MRR ↑ H@1 ↑ H@3 ↑ H@10 ↑ Paradigm I: Graph Propagation HR-NBFNet 4616.2 0.4666 0.4332 0.4791 0.5283 4182.4 0.4785 0.4408 0.4956 0.5465 MultiTask HR-NBFNet 5229.5 0.4729 0.4373 0.4849 0.5528 589.8 0.4134 0.3782 0.4171 0.4926 Paradigm I: Embedding-Based AlertStar 2645.2 0.4280 0.3400 0.4900 0.5620 3083.1 0.4620 0.3980 0.5040 0.5640 MultiTask AlertStar 3016.4 0.5618 0.5160 0.5860 0.6460 3341.5 0.5384 0.5040 0.5480 0.6140 TABLE IV: KGC results on the Warden Alert dataset using statements under the inductive and transductive settings. Inductive Transductive Model MR ↓ MRR ↑ H@1 ↑ H@3 ↑ H@10 ↑ MR ↓ MRR ↑ H@1 ↑ H@3 ↑ H@10 ↑ 33% ShrinkE 2724.0 0.4665 0.4380 0.4760 0.5220 3087.2 0.5012 0.4820 0.5060 0.5400 StarE 2858.4 0.4205 0.3640 0.4520 0.5300 3262.7 0.4451 0.3940 0.4680 0.4940 AlertStar 2765.3 0.4371 0.3520 0.5040 0.5840 3031.6 0.4627 0.3840 0.5280 0.5920 MultiTask AlertStar 2680.2 0.5860 0.5400 0.6100 0.6850 3174.2 0.5582 0.5200 0.5750 0.6350 66% ShrinkE 2920.0 0.4683 0.4440 0.4760 0.5100 2692.4 0.5026 0.4860 0.5040 0.5360 StarE 3187.6 0.3209 0.2100 0.4080 0.5140 3098.7 0.3799 0.2720 0.4540 0.5340 AlertStar 3111.1 0.4153 0.3420 0.4600 0.5420 3088.3 0.4666 0.3920 0.5260 0.5900 MultiTask AlertStar 2532.5 0.5968 0.5600 0.6100 0.6850 3329.0 0.5328 0.5000 0.5550 0.5800 100% ShrinkE 2784.8 0.4567 0.4380 0.4460 0.5220 2701.1 0.5104 0.4900 0.5180 0.5540 StarE 3522.9 0.4191 0.3360 0.4740 0.5480 3744.6 0.4227 0.3420 0.4840 0.5560 AlertStar 3289.9 0.4254 0.3260 0.5060 0.5860 3180.5 0.4797 0.4160 0.5220 0.5860 MultiTask AlertStar 2977.9 0.5838 0.5400 0.6050 0.6750 3307.0 0.5477 0.5150 0.5650 0.6100 TABLE V: KGC results on the UNSW dataset using statements under the inductive and transductive settings. Inductive Transductive Model MR ↓ MRR ↑ H@1 ↑ H@3 ↑ H@10 ↑ MR ↓ MRR ↑ H@1 ↑ H@3 ↑ H@10 ↑ 33% ShrinkE 4.2680 0.4220 0.2200 0.5000 1.000 4.2880 0.4083 0.2020 0.4800 1.000 StarE 4.1460 0.4367 0.2280 0.5480 0.996 4.1560 0.4267 0.2160 0.5080 1.000 AlertStar 3.9300 0.4497 0.2240 0.5640 1.000 4.0140 0.4394 0.2260 0.5300 1.000 MultiTask AlertStar 3.9350 0.4490 0.2350 0.5400 1.000 3.9250 0.4626 0.2600 0.5500 1.000 66% ShrinkE 4.1020 0.4483 0.2440 0.5500 1.000 4.1500 0.4241 0.2100 0.5100 1.000 StarE 4.2980 0.4296 0.2280 0.5280 0.996 4.0800 0.4404 0.2360 0.5060 1.000 AlertStar 4.0160 0.4575 0.2440 0.5740 1.000 4.0160 0.4575 0.2440 0.5740 1.000 MultiTask AlertStar 3.6800 0.5023 0.3200 0.5800 1.000 3.7900 0.4752 0.2750 0.5750 1.000 100% ShrinkE 4.0120 0.4449 0.2360 0.5680 1.000 4.0680 0.4477 0.2440 0.5500 1.000 StarE 4.0020 0.4595 0.2460 0.5760 0.996 4.0460 0.4236 0.2040 0.5320 1.000 AlertStar 3.8540 0.4707 0.2620 0.5900 1.000 3.8440 0.4647 0.2520 0.5820 1.000 MultiTask AlertStar 3.6450 0.4809 0.2700 0.6050 1.000 4.0000 0.4641 0.2650 0.5550 1.000 TABLE VI: Ablation A4: Model performance across qualifier density levels (inductive). Q33% Q66% Q100% Model MRR H@1 H@10 MRR H@1 H@10 MRR H@1 H@10 StarE 0.2783 0.0940 0.5140 0.3442 0.2180 0.5140 0.4051 0.3140 0.5340 AlertStar 0.4282 0.3380 0.5760 0.4167 0.3340 0.5440 0.4313 0.3420 0.5820 HyNT 0.5632 0.5140 0.6460 0.5726 0.5320 0.6440 0.5692 0.5280 0.6460 MultiTask AlertStar 0.5546 0.5040 0.6520 0.5663 0.5200 0.6540 0.5531 0.5060 0.6460 TABLE VII: KGC results on the Warden Alert dataset using triples under inductive and transductive settings. Inductive Transductive Model MR ↓ MRR ↑ H@1 ↑ H@3 ↑ H@10 ↑ MR ↓ MRR ↑ H@1 ↑ H@3 ↑ H@10 ↑ ShrinkE 2699.4 0.4580 0.4140 0.4840 0.5320 2889.4 0.4998 0.4680 0.5180 0.5500 StarE 2760.9 0.4098 0.3400 0.4680 0.5300 3150.9 0.4503 0.4000 0.4860 0.5460 NBFNet 3616.5 0.4853 0.4481 0.5060 0.5480 4524.8 0.4818 0.4451 0.5019 0.5511 AlertStar 2561.0 0.4565 0.3720 0.5140 0.5920 2746.2 0.4862 0.4320 0.5140 0.5840 MultiTask AlertStar (Tail+Rel) 2430.0 0.5860 0.5450 0.6000 0.6750 3138.6 0.5762 0.5450 0.5900 0.6400 TABLE VIII: KGC results on the UNSW dataset using triples under inductive and transductive settings. Inductive Transductive Model MR ↓ MRR ↑ H@1 ↑ H@3 ↑ H@10 ↑ MR ↓ MRR ↑ H@1 ↑ H@3 ↑ H@10 ↑ ShrinkE 4.1380 0.4405 0.2400 0.5320 1.000 4.1360 0.4305 0.2220 0.5140 1.000 StarE 4.1300 0.4445 0.2460 0.5240 0.996 4.1300 0.4163 0.1980 0.5120 1.000 NBFNet 4.2405 0.4288 0.2252 0.5120 1.000 4.2484 0.4279 0.2228 0.5142 1.000 AlertStar 4.0060 0.4469 0.2320 0.5480 1.000 4.0060 0.4355 0.2220 0.5320 1.000 MultiTask AlertStar (Tail+Rel) 3.9500 0.4409 0.2150 0.5850 1.000 3.9700 0.4519 0.2550 0.5200 1.000 TABLE IX: Complex query answering results on the Warden Alert dataset under the inductive and transductive settings. Inductive 33% 66% 100% Query MR ↓ MRR ↑ H@1 ↑ H@3 ↑ H@10 ↑ MR ↓ MRR ↑ H@1 ↑ H@3 ↑ H@10 ↑ MR ↓ MRR ↑ H@1 ↑ H@3 ↑ H@10 ↑ StarQE 1p 43.810 0.4875 0.460 0.475 0.555 79.305 0.4600 0.440 0.445 0.495 53.820 0.5103 0.4950 0.5050 0.5350 2p 100.66 0.0176 0.000 0.000 0.000 51.250 0.2611 0.250 0.250 0.250 34.545 0.6694 0.6363 0.7272 0.7272 2i 26.545 0.3584 0.005 0.685 0.715 31.965 0.3454 0.010 0.655 0.700 33.575 0.6835 0.6750 0.6750 0.6900 2u 50.260 0.4808 0.460 0.460 0.530 84.645 0.4503 0.435 0.440 0.465 60.715 0.4881 0.4600 0.4950 0.5300 HR-NBFNet-CQ 1p 32.080 0.4940 0.475 0.475 0.515 40.435 0.4461 0.415 0.430 0.500 33.115 0.4917 0.4600 0.4850 0.5450 2p 111.33 0.0232 0.000 0.000 0.000 139.250 0.0089 0.000 0.000 0.000 18.727 0.2565 0.0909 0.0909 0.7272 2i 17.850 0.7384 0.715 0.745 0.770 28.050 0.6399 0.620 0.635 0.670 25.985 0.6817 0.6450 0.6950 0.7300 2u 38.920 0.4818 0.455 0.475 0.515 48.165 0.4333 0.410 0.425 0.470 41.650 0.4782 0.4500 0.4700 0.5400 Transductive 33% 66% 100% Query MR ↓ MRR ↑ H@1 ↑ H@3 ↑ H@10 ↑ MR ↓ MRR ↑ H@1 ↑ H@3 ↑ H@10 ↑ MR ↓ MRR ↑ H@1 ↑ H@3 ↑ H@10 ↑ StarQE 1p 42.975 0.2782 0.02 0.500 0.555 42.255 0.5170 0.4950 0.5050 0.5750 55.500 0.4495 0.405 0.495 0.525 2p 6.750 0.3368 0.25 0.250 1.000 45.454 0.1977 0.0909 0.1818 0.4545 84.533 0.1092 0.000 0.267 0.400 2i 25.495 0.1219 0.00 0.015 0.770 22.145 0.4160 0.0850 0.7300 0.7500 27.720 0.6350 0.610 0.625 0.735 2u 51.210 0.2685 0.02 0.495 0.525 47.425 0.5122 0.4950 0.5000 0.5450 59.490 0.4572 0.405 0.485 0.520 HR-NBFNet-CQ 1p 29.955 0.5112 0.475 0.50 0.580 34.925 0.5658 0.540 0.565 0.605 36.960 0.4326 0.400 0.405 0.510 2p 12.250 0.1160 0.000 0.000 0.750 173.090 0.0068 0.000 0.000 0.000 59.400 0.0892 0.067 0.067 0.133 2i 19.390 0.6980 0.675 0.68 0.755 24.050 0.1637 0.015 0.015 0.730 18.725 0.6535 0.610 0.700 0.750 2u 37.405 0.4965 0.465 0.48 0.560 41.150 0.5573 0.535 0.560 0.590 44.315 0.4427 0.400 0.480 0.515 TABLE X: Complex query answering results on the UNSW dataset under the inductive and transductive settings. Inductive 33% 66% 100% Query MR ↓ MRR ↑ H@1 ↑ H@3 ↑ H@10 ↑ MR ↓ MRR ↑ H@1 ↑ H@3 ↑ H@10 ↑ MR ↓ MRR ↑ H@1 ↑ H@3 ↑ H@10 ↑ StarQE 1p 3.125 0.5368 0.335 0.665 1.000 3.340 0.5026 0.290 0.615 1.000 3.000 0.5228 0.290 0.685 1.000 2p 1.950 0.6633 0.435 0.950 1.000 1.715 0.7399 0.560 0.960 1.000 1.775 0.7469 0.580 0.940 1.000 2i 3.385 0.5149 0.310 0.645 1.000 3.730 0.4627 0.250 0.550 1.000 3.395 0.4995 0.275 0.630 0.995 2u 3.230 0.4973 0.265 0.660 1.000 3.300 0.5092 0.300 0.620 1.000 3.145 0.5126 0.280 0.660 1.000 HR-NBFNet-CQ 1p 3.485 0.5004 0.295 0.620 0.995 3.985 0.4207 0.205 0.495 1.000 3.510 0.4920 0.280 0.615 1.000 2p 1.780 0.7460 0.570 0.945 0.995 1.995 0.6555 0.415 0.940 1.000 1.930 0.6773 0.450 0.950 1.000 2i 3.840 0.4862 0.305 0.575 1.000 4.215 0.4183 0.215 0.495 0.995 3.675 0.4559 0.240 0.565 1.000 2u 3.575 0.5001 0.290 0.630 1.000 4.250 0.3906 0.180 0.445 1.000 3.740 0.4818 0.280 0.585 1.000 Transductive 33% 66% 100% Query MR ↓ MRR ↑ H@1 ↑ H@3 ↑ H@10 ↑ MR ↓ MRR ↑ H@1 ↑ H@3 ↑ H@10 ↑ MR ↓ MRR ↑ H@1 ↑ H@3 ↑ H@10 ↑ StarQE 1p 3.315 0.4962 0.280 0.610 1.000 3.390 0.4850 0.275 0.610 1.000 3.365 0.4721 0.240 0.620 0.995 2p 1.980 0.6982 0.525 0.840 1.000 2.245 0.6232 0.405 0.800 1.000 2.155 0.6210 0.395 0.870 1.000 2i 3.680 0.4448 0.205 0.595 1.000 3.570 0.4760 0.270 0.570 1.000 3.645 0.4740 0.285 0.555 1.000 2u 3.525 0.4947 0.300 0.590 0.990 3.535 0.4607 0.245 0.585 1.000 3.365 0.4774 0.235 0.625 1.000 HR-NBFNet-CQ 1p 3.395 0.4915 0.270 0.635 0.990 4.030 0.4135 0.195 0.505 1.000 3.925 0.4571 0.270 0.535 0.990 2p 2.125 0.6452 0.435 0.865 1.000 2.550 0.6143 0.400 0.805 0.980 2.135 0.6602 0.480 0.810 1.000 2i 3.830 0.4612 0.250 0.560 1.000 4.050 0.4195 0.210 0.480 1.000 3.730 0.4633 0.245 0.590 0.990 2u 3.540 0.4931 0.295 0.580 0.995 4.190 0.4061 0.200 0.485 0.995 4.135 0.4379 0.245 0.505 0.985 4 Experiment 4.1 Experiment Setup All models are implemented in PyTorch and trained on a single NVIDIA A100 GPU with 64 GB RAM. Shared hyperparameters: d=200d=200, dropout 0.20.2, Adam with lr 5×10−45×10^-4, 20 epochs, gradient clipping to unit norm, margin δ=1.0δ=1.0, uniform negative sampling. For HR-NBFNet and MT-HR-NBFNet: L=3L=3 layers, chunk size C=5,000C=5,000, kmax=8k_ =8 tails per (h,q′)(h,q ) group, Qmax=8Q_ =8 qualifier pairs per edge. For MT-AlertStar: 3 Transformer layers, 4 attention heads, FFN width F=800F=800. Batch sizes are 128128 (AlertStar), 6464 (MT-AlertStar), and 3232 (HR-NBFNet variants), reflecting the higher memory cost of graph traversal. The code is available 111https://gitfront.io/r/Zahra/pNhiE7GJ4P54/AlertStar/. 4.2 Main Results Paradigm comparison Table I compares graph propagation and embedding-based paradigms on Warden. Contrary to the assumption that explicit path reasoning yields superior performance, MT-AlertStar achieves the best overall results in both settings (MRR 0.5618 inductive, 0.5384 transductive), outperforming MT-HR-NBFNet by 19% and 30% respectively. Multi-task training strongly benefits the embedding-based paradigm (+0.1338 MRR inductively) but provides negligible gain or degrades graph propagation (−-0.0651 transductively), suggesting the Bellman-Ford backbone is less able to exploit auxiliary supervision. Although HR-NBFNet achieves lower MR transductively, AlertStar dominates on MRR and H@k — the operationally relevant metrics for CERT analysts inspecting top-ranked candidates. Hyper-relational baselines Table IV compares against StarE and ShrinkE across qualifier-coverage levels (33%, 66%, 100%). MT-AlertStar consistently outperforms ShrinkE by 25–28% relative MRR across all levels and both settings. StarE degrades with increasing qualifier density (MRR 0.4205→0.4191 inductively), while MT-AlertStar remains stable, demonstrating robustness to partial qualifier availability. AlertStar alone already outperforms StarE on H@3 and H@10 at all coverage levels. Similar trends hold on UNSW-NB15 (Table V), confirming generalisation across datasets. Effect of qualifier context On Warden (Table VII), triple-only models outperform hyper-relational variants, while on UNSW-NB15 (Table VIII) qualifiers consistently improve performance. We attribute this to Warden’s limited four-category attack taxonomy, where triples alone suffice, versus UNSW-NB15’s richer and more diverse taxonomy where qualifier attributes — port, protocol, flow count — are necessary to disambiguate identical (h,r,t)(h,r,t) structures. Qualifier context is beneficial when the graph is sufficiently complex, making UNSW-NB15 the more reliable benchmark for hyper-relational modelling. TABLE XI: MT-AlertStar on Warden. Inductive Transductive Metric 33% 66% 100% 33% 66% 100% Tail Prediction MR 2680.2 2532.5 2978.0 3174.2 3329.0 3307.0 MRR 0.5860 0.5968 0.5838 0.5582 0.5329 0.5477 H@1 0.5400 0.5600 0.5400 0.5200 0.5000 0.5150 H@3 0.6100 0.6100 0.6050 0.5750 0.5550 0.5650 H@10 0.6850 0.6850 0.6750 0.6350 0.5800 0.6100 Relation Prediction MR 1.0 1.0 1.0 1.0 1.1 1.0 MRR 1.0000 0.9975 1.0000 0.9950 0.9725 0.9975 H@1 1.0000 0.9950 1.0000 0.9900 0.9450 0.9950 H@3 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 H@10 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 Acc 1.0000 0.9950 1.0000 0.9900 0.9450 0.9950 Qualifier Value MR 1.0 1.0 1.0 1.0 1.0 1.0 MRR 0.9950 0.9900 0.9925 0.9975 0.9875 0.9875 H@1 0.9900 0.9800 0.9850 0.9950 0.9750 0.9750 H@3 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 H@10 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 TABLE XII: MT-AlertStar on UNSW-NB15. Inductive Transductive Metric 33% 66% 100% 33% 66% 100% Tail Prediction MR 3.9 3.7 3.6 3.9 3.8 4.0 MRR 0.4491 0.5023 0.4810 0.4626 0.4753 0.4641 H@1 0.2350 0.3200 0.2700 0.2600 0.2750 0.2650 H@3 0.5400 0.5800 0.6050 0.5500 0.5750 0.5550 H@10 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 Relation Prediction MR 2.9 2.6 2.5 2.9 2.9 2.6 MRR 0.5912 0.6063 0.6366 0.5697 0.5780 0.6174 H@1 0.4150 0.4150 0.4650 0.3900 0.4000 0.4400 H@3 0.6900 0.7450 0.7550 0.6450 0.6700 0.7600 H@10 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 Acc 0.4150 0.4150 0.4650 0.3900 0.4000 0.4400 Qualifier Value MR 1.2 1.2 1.2 1.3 1.3 1.3 MRR 0.8992 0.8992 0.8958 0.8762 0.8896 0.8696 H@1 0.8050 0.8050 0.8000 0.7700 0.7950 0.7550 H@3 0.9900 1.0000 1.0000 0.9950 0.9950 0.9950 H@10 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 Multi-task auxiliary predictions. Tables XI and XII report MT-AlertStar’s three heads. On UNSW-NB15, tail prediction achieves H@10 of 1.000 across all settings while MRR (0.44–0.50) reflects the difficulty of precise top-1 ranking. Relation prediction improves with qualifier density (H@3: 0.690→0.755 inductively), confirming richer context aids attack category discrimination; accuracy (0.415–0.465) is lower than H@k as it measures exact top-1 match over a diverse taxonomy. Qualifier-value prediction achieves near-perfect performance (MRR >>0.87, H@3 >>0.99), confirming flow-level attributes are highly predictable. On Warden, all heads achieve near-perfect results due to its limited taxonomy, making UNSW-NB15 the more informative benchmark. Complex query answering. Tables IX and X evaluate HR-NBFNet-CQ against StarQE. StarQE leads on 1p and 2u across most settings. Two-hop queries (2p) are the hardest for both models, H@1 is near zero at low qualifier coverage, reflecting the sparsity of two-hop chains in alert graphs. HR-NBFNet-CQ outperforms StarQE on intersection queries (2i) on Warden (H@3: 0.745 vs 0.685 at 33%), the most operationally critical query type for coordinated attack detection. Qualifier density consistently improves complex queries: HR-NBFNet-CQ reaches H@10 of 0.727 on 2p at 100% vs 0.000 at 33%. On UNSW-NB15, StarQE leads across all query types including 2i, though both models achieve H@10 of 1.000 on most types, confirming answers always rank within the top 10 regardless of query complexity. 4.3 Ablation Study All ablations run on Warden unless stated otherwise. 4.3.1 A1: AlertStar Component Ablation Table XIII compares four variants: AS-NoQual (~r=r e_r=e_r, no MHA), AS-NoPath (α=1α=1, attention branch only), AS-NoGate (fixed α=0.5α=0.5), and AS-Full (complete model). Removing qualifier enrichment causes the largest single drop, confirming qualifier context as the primary discriminative signal. Removing the path-composition branch produces a consistent but smaller drop, demonstrating that structural path information complements rather than substitutes qualifier attention. Fixing the gate to α=0.5α=0.5 underperforms the learned gate, confirming that the optimal branch balance is query-dependent. 4.3.2 A2: Gate Value Trajectory Figure 6 tracks αe=σ(ge) _e=σ(g_e) across training epochs. Both inductive (0.611→0.8350.611→0.835) and transductive (0.614→0.8320.614→0.832) settings follow nearly identical trajectories, converging within 12–14 epochs. Two key findings emerge: (i) AlertStar consistently favours the cross-attention branch (α>0.5α>0.5), confirming that ~r e_r carries more discriminative signal than implicit path composition; (i) the near-identical trajectories across settings show branch preference is driven by qualifier structure rather than entity memorisation, supporting inductive generalisation. Figure 6: Gate value α=σ(g)α=σ(g) over 20 epochs on Warden. TABLE XIII: Ablation A1: AlertStar component ablation. Variant Qual Path Gate MR MRR H@1 H@3 H@10 AS-NoQual ✗ ✓ learned 2712.7 0.4423 0.352 0.518 0.584 AS-NoPath ✓ ✗ learned 2673.1 0.4454 0.354 0.508 0.590 AS-NoGate ✓ ✓ fixed=0.5 2809.9 0.4445 0.358 0.500 0.580 AS-Full ✓ ✓ learned 2615.2 0.4418 0.360 0.500 0.570 4.3.3 A3: MT-AlertStar Auxiliary Tasks Table XIV evaluates four MT-AlertStar variants: MT-Tail (ℒtL_t only), MT-Tail+Rel (+ℒq′+L_q ), MT-Tail+QV (+ℒqv+L_q_v), and MT-Full (all tasks, Eq. 29). All auxiliary tasks contribute positively when added individually. MT-Full outperforms all single-auxiliary variants, confirming the tasks are complementary rather than redundant. TABLE XIV: Ablation A3: MultiTask auxiliary. Variant Tasks MR MRR H@1 H@3 H@10 MT-TailOnly tail 3210.0 0.5607 0.512 0.594 0.650 MT-Tail+Rel tail, relation 2998.3 0.5604 0.514 0.586 0.646 MT-Tail+QualVal tail, qual_value 2881.2 0.5633 0.516 0.600 0.648 MT-Full tail, relation, qual_val 3100.6 0.5610 0.518 0.584 0.640 4.3.4 A4: Qualifier Density Sensitivity Table VI reports MRR degradation across all models. MT-AlertStar is the most robust, with AlertStar and MT-AlertStar exhibiting the smallest relative drop among all models. StarE, despite sharing the same MHA qualifier enrichment as AlertStar, degrades more sharply under sparse qualifiers — confirming that the path-composition branch and trainable gate provide robustness that qualifier attention alone cannot achieve. HR-NBFNet and MT-HR-NBFNet show the largest degradation, as missing qualifier noise accumulates across L=3L=3 Bellman–Ford layers. 5 Related Work Alert prediction and dynamic graph learning Graph-based approaches model attacker–victim interactions as evolving networks: Nayeri and Rezvani [7] introduce a dynamic graph deep learning framework using Temporal Graph Networks (TGN) for binary link prediction over time-evolving attack graphs, achieving 10–18% improvement over static baselines. Nayeri and Resvani [8] extend this paradigm with TGNE and TGNE-TA, replacing standard TGN message aggregation with a Transformer-based aggregator and a Dual-Level Temporal Encoding mechanism to capture both global and local temporal dependencies, achieving over 90% accuracy in multiclass attack type prediction. While these methods advance temporal reasoning over alert streams, they operate on binary or standard relational graphs and discard flow-level metadata, port, protocol, flow count, and timestamps, that is essential for disambiguating alerts of the same type. Our framework addresses this gap by modelling alerts as hyper-relational qualified statements (h,r,t,)(h,r,t,Q), enabling qualifier-conditioned prediction under both inductive and transductive settings. Hyper-relational knowledge graph completion Standard KGC models represent facts as binary triples and cannot encode auxiliary context. StarE [4] extends GNN-based KGC to hyper-relational graphs by composing qualifier pairs into relation embeddings via message passing, while ShrinkE [11] represents hyper-relational facts as geometric transformations in a shrinking entity space, achieving strong transductive performance. HyNT [3] encodes qualifier pairs through a nested Transformer architecture, attending jointly over the main triple and its qualifiers. For complex query answering, StarQE [1] extends GNN-based query embeddings to hyper-relational graphs, supporting conjunctive query templates over qualified facts. Despite their expressive power, these models are predominantly transductive — requiring all entities to be present at training time — limiting their applicability to alert prediction where new IP addresses appear continuously. Our work extends NBFNet [12] to the hyper-relational setting, achieving strong inductive generalisation while incorporating qualifier context at every propagation step, and further supports disjunctive query templates absent from existing benchmarks. 6 Discussion This work demonstrates that hyper-relational knowledge graph completion is a viable and effective framework for network alert prediction, with the embedding-based paradigm (AlertStar, MT-AlertStar) consistently outperforming graph propagation (HR-NBFNet, MT-HR-NBFNet) in both accuracy and efficiency. The surprising dominance of embedding-based models suggests that for dense alert graphs with rich qualifier context, local compositional reasoning in embedding space captures attack patterns more effectively than explicit multi-hop path traversal. This challenges the prevailing assumption in KGC that path-based models are inherently superior for inductive settings. The qualifier density ablation (A4) reveals a dataset-dependent effect: qualifiers improve prediction on UNSW-NB15 but introduce noise on Warden, where the limited four-category taxonomy renders the triple structure sufficient. This finding highlights the importance of dataset diversity when benchmarking hyper-relational models and motivates the construction of richer alert datasets for future evaluation. The gate trajectory analysis (A2) provides interpretable evidence that AlertStar autonomously learns to favour qualifier context over structural composition, converging stably within 12–14 epochs regardless of the inductive or transductive setting. HR-NBFNet-CQ demonstrates that complex first-order logic queries are feasible over hyper-relational alert graphs, with intersection queries (2i) benefiting from Bellman-Ford propagation. However, two-hop queries remain challenging at low qualifier coverage, pointing to the need for denser qualifier annotation and richer graph connectivity in future alert datasets. The current framework assumes a closed-world attack taxonomy and does not model temporal dynamics between alerts. AlertStar processes each triple independently, missing cross-alert correlations that propagation-based models can capture in principle. The inductive protocol evaluates generalisation to unseen IPs but not to entirely new attack categories. Several directions merit exploration. First, incorporating temporal ordering of alerts as qualifier context would enable time-aware prediction of attack progression. Second, extending the framework to open-set attack classification would address the closed-world limitation. Third, combining the efficiency of AlertStar with the path-reasoning capacity of HR-NBFNet via a hybrid architecture — for instance, using AlertStar as a fast candidate ranker and HR-NBFNet-CQ for complex query verification — could yield both accuracy and scalability. Finally, applying the framework to larger operational datasets from real CERT deployments would validate its practical utility beyond the Warden and UNSW-NB15 benchmarks. 7 Conclusion We presented a hyper-relational knowledge graph framework for network alert prediction, modelling each alert as a qualified statement (h,r,t,)(h,r,t,Q) and formulating prediction as an HR-KGC task. Five models were proposed across three paradigms: HR-NBFNet and MT-HR-NBFNet for qualifier-aware Bellman-Ford path propagation, and AlertStar and MT-AlertStar for embedding-based gated attention and path fusion. HR-NBFNet-CQ extended the framework to complex first-order logic queries over the alert graph. Experiments on Warden and UNSW-NB15 across three qualifier-density regimes show that MT-AlertStar achieves state-of-the-art performance in both inductive and transductive settings, outperforming HR-NBFNet by up to 30% in MRR while achieving up to 50×50× per-epoch speedup. Multi-task supervision provides strong regularisation for the embedding-based paradigm but marginal benefit for graph propagation. Qualifier context improves prediction on diverse datasets but introduces noise on simpler taxonomies, highlighting the importance of dataset complexity for hyper-relational modelling. HR-NBFNet-CQ demonstrates the feasibility of complex threat queries including lateral movement detection (2p), coordinated attack identification (2i), and campaign victim set retrieval (2u). These results establish hyper-relational KGC as a principled and practical framework for proactive threat intelligence in SOC and CERT deployments. References [1] D. Alivanistos, M. Berrendorf, M. Cochez, and M. 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Karniadakis (2024) TransformerG2G: adaptive time-stepping for learning temporal graph embeddings using transformers. Neural Networks 172, p. 106086. Cited by: §3.3.1, §3.3.1. [11] B. Xiong, M. Nayyeri, S. Pan, and S. Staab (2023) Shrinking embeddings for hyper-relational knowledge graphs. In Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), p. 13306–13320. Cited by: §2.4, §5. [12] Z. Zhu, Z. Zhang, L. Xhonneux, and J. Tang (2021) Neural bellman-ford networks: a general graph neural network framework for link prediction. Advances in neural information processing systems 34, p. 29476–29490. Cited by: §1, §2.3, §2.3, §2.3, Figure 2, §3.1, §3.2, §3.2, §3, §5. Zahra Makki Nayeri is a Ph.D. candidate at Shahrood University of Technology and currently a visiting researcher at the University of Stuttgart, Germany, where she conducts research on knowledge graph foundation models. Her research focuses on graph representation learning, temporal and dynamic graph neural networks, and machine learning–based modeling of computer networks, with an emphasis on data-driven analysis of large-scale, evolving interaction graphs. During her Master’s studies, she investigated machine learning techniques for fog, edge, and cloud computing environments, with a particular focus on distributed data processing, resource-aware learning, and system-level optimization. Mohsen Rezvani is an Associate Professor in the Faculty of Computer Engineering at the Shahrood University of Technology. He received his Ph.D. at the School of Computer Science and Engineering at the UNSW Sydney. He holds an M.Sc. in Computer Engineering from Sharif University of Technology and a B.Sc. in Computer Engineering from Amirkabir University of Technology. His research focuses on computer security, intelligent data analysis, and advanced machine learning. He is the director of the Computer Emergency Response Team (CERT) at Shahrood University of Technology.