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LFHE: Local-First Heuristic Evolution for Bounded Local Topology Search in Decentralized Learning with Non-IID Data
Yin-Kuan Liang, Yan Gao, Yang Long
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Abstract
Abstract:Decentralized learning is highly sensitive to communication topology under non-IID data. Adaptive peer-selection methods can exploit local model information, but broader peer discovery may require increasingly large control state, whereas direct spectral optimization typically relies on graph-wide information. We study the intermediate setting of bounded local topology search and propose Local-First Heuristic Evolution (LFHE), a representation-driven rewiring framework whose candidate discovery and scoring use only ego-neighborhood and friend-of-a-friend (FoF) information. The structural score admits an exact interpretation through graph Dirichlet energy: its sum across clients equals twice the representation Dirichlet energy, which under standard linear consensus dynamics governs the instantaneous dissipation of representation disagreement. LFHE combines this state-dependent structural signal with early exploration and degree control, while algebraic connectivity remains an offline graph diagnostic. Under bounded sparse degree, its FoF candidate state remains local rather than expanding toward population-wide peer tracking. Across four image, speech, and text benchmarks, LFHE achieves competitive decentralized learning performance. Matched-protocol controls identify the structural term as the principal empirical topology-selection signal, while comparison with broader peer discovery exposes a trade-off between predictive performance and discovery-state locality. Together, these results motivate state-aware bounded local topology search between pairwise peer selection and globally informed topology optimization.
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- Source: https://arxiv.org/abs/2610.08176v1
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LFHE: Local-First Heuristic Evolution for Bounded Local Topology Search in Decentralized Learning with Non-IID Data Yin-Kuan Liang Affiliation: Durham University Yan Gao Affiliation: University of Cambridge Yang Long Affiliation: Durham University October 2026 Abstract Decentralized learning is highly sensitive to communication topology under non-IID data. Adaptive peer-selection methods can exploit local model information, but broader peer discovery may require increasingly large control state, whereas direct spectral optimization typically relies on graph-wide information. We study the intermediate setting of bounded local topology search and propose Local-First Heuristic Evolution (LFHE), a representation-driven rewiring framework whose candidate discovery and scoring use only ego-neighborhood and friend-of-a-friend (FoF) information. The structural score admits an exact interpretation through graph Dirichlet energy: its sum across clients equals twice the representation Dirichlet energy, which under standard linear consensus dynamics governs the instantaneous dissipation of representation disagreement. LFHE combines this state-dependent structural signal with early exploration and degree control, while algebraic connectivity remains an offline graph diagnostic. Under bounded sparse degree, its FoF candidate state remains local rather than expanding toward population-wide peer tracking. Across four image, speech, and text benchmarks, LFHE achieves competitive decentralized learning performance. Matched-protocol controls identify the structural term as the principal empirical topology-selection signal, while comparison with broader peer discovery exposes a trade-off between predictive performance and discovery-state locality. Together, these results motivate state-aware bounded local topology search between pairwise peer selection and globally informed topology optimization. 1 Introduction Decentralized federated learning (DFL) enables collaborative model training without a central aggregation server. Instead, clients exchange and average model information over a peer-to-peer communication graph [11, 15]. Removing a central coordinator makes the communication graph part of the optimization dynamics: its connectivity and mixing properties strongly influence how quickly information propagates and local models move toward consensus [5, 17, 19]. The topology problem becomes more difficult under non-IID data. Sparse or poorly mixed graphs can sustain disagreement between local models and slow decentralized optimization [3, 7, 13]. Existing adaptive approaches often use model similarity or dissimilarity to choose communication partners [6]. These signals are locally useful, but a sequence of locally attractive pairwise choices does not by itself determine whether the realized sparse graph supports efficient graph-wide information propagation. Conversely, methods that directly optimize mixing matrices or spectral graph objectives are more explicit about global communication structure, but commonly require graph-wide information or centralized computation [4, 25, 19]. We study a middle ground: bounded local topology search. Rather than reconstructing a global graph objective, each client searches only its ego-neighborhood and current FoF frontier. With maximum local degree d¯ d, the two-hop candidate state is O(d¯2)O( d^2) per client and O(Nd¯2)O(N d^2) system-wide, reducing to O(1)O(1) and O(N)O(N) under fixed sparse degree. By contrast, gossip-based discovery that progressively approaches population-wide peer coverage, as in Morph [6], can grow toward O(N)O(N) state per client and O(N2)O(N^2) system-wide. This comparison concerns discovery/control state rather than total communication volume. Based on this view, we propose Local-First Heuristic Evolution (LFHE). At each topology update, a client samples an FoF candidate and evaluates an edge addition or one-edge swap using structural disagreement, early exploration, and degree control. The structural score is a node-local contribution to the representation Dirichlet energy; under standard linear consensus dynamics, this global energy governs the instantaneous dissipation of representation disagreement. LFHE does not compute algebraic connectivity λ2 _2; spectral quantities remain offline diagnostics. The reference implementation serializes topology operations, so our decentralization claim concerns local information requirements rather than concurrent or asynchronous execution guarantees. Our contribution is threefold. First, we formulate adaptive communication as bounded ego-neighborhood topology search, separating local online information from global offline diagnostics. Second, we establish an exact graph-theoretic interpretation of LFHE’s structural score: its node-wise values decompose the representation Dirichlet energy, which under standard linear consensus dynamics governs instantaneous disagreement dissipation. Third, we evaluate LFHE across image, speech, and text tasks using matched mechanism controls, sensitivity and communication analyses, update-order diagnostics, and comparison with broader peer-discovery strategies. The results identify the structural term as the principal topology-selection signal in the tested regime while exposing the trade-off between localized discovery and broader peer exposure. 2 Related Work Decentralized learning and heterogeneous data. Federated learning usually aggregates client updates through a server [16], with methods such as FedProx [14] and SCAFFOLD [10] addressing optimization under heterogeneity. DFL instead relies on peer-to-peer communication [11, 15]. Classical distributed optimization and gossip results make clear that convergence and consensus depend strongly on the communication graph [18, 22, 5, 17]. D-Cliques [3] and STL-FW [13] explicitly incorporate heterogeneity into sparse communication design, but rely on broader graph or mixing-matrix construction than the local online search considered here. Adaptive peer discovery and topology evolution. Epidemic Learning [7] uses randomized sparse communication. Morph [6] performs gossip-based peer discovery and model-dissimilarity-driven replacement, progressively broadening discovered-peer state to enlarge peer-selection exposure. LFHE instead restricts candidate exposure to the current FoF frontier and evaluates the resulting ego-topology after an add or swap. The distinction is primarily in search scope: under bounded sparse degree, LFHE operates over a bounded local candidate frontier, whereas gossip-based peer discovery can progressively expose a client to a much broader portion of the network. Connectivity-oriented graph design. Graph-theoretic results show that locally constructed sparse graphs need not have favorable global connectivity, particularly as the system grows [1, 2, 26]. More generally, decentralized averaging and consensus depend strongly on global graph and mixing properties [4, 20, 19, 21]. Direct optimization of graph structure or mixing matrices explicitly targets these properties [4, 25], including in DFL-specific sparse overlays [9], but typically requires graph-wide state or centralized optimization. Positioning. LFHE occupies the space between pairwise peer-selection methods and direct global graph optimization. Its novelty is not dynamic topology adaptation itself, nor the use of model dissimilarity. It is the combination of restricted FoF candidate discovery, post-operation ego-topology scoring, and local add-or-swap updates without online global graph statistics. The structural score is state-dependent rather than a topology-only spectral objective: it is a node-local contribution to the representation Dirichlet energy, whose aggregate governs instantaneous disagreement dissipation under standard linear consensus dynamics, whereas λ2 _2 characterizes the graph’s worst-case normalized dissipation over non-consensus directions. LFHE does not estimate or optimize λ2 _2. 3 Method We formulate topology adaptation as a constrained local combinatorial search problem. At each topology-update step, client i restricts its candidate edge operations to those available from its ego-neighborhood and FoF frontier. LFHE performs sampled greedy search over this local action set and commits an operation only when it improves the initiator-local objective. The composite objective is a local decision rule, while its structural component admits an exact interpretation through representation Dirichlet energy and consensus dissipation. 3.1 Problem Setup We consider a decentralized learning system with N clients connected by a time-varying undirected graph Gt=(V,Et),G_t=(V,E_t), where node i∈Vi∈ V denotes a client and edge (i,j)∈Et(i,j)∈ E_t indicates that clients i and j can exchange model information at round t. Let i(t)N_i(t) denote the neighbor set of client i. Each client i holds a private local dataset iD_i and maintains a local model witw_i^t. At round t, client i performs local training on iD_i and then communicates only with neighbors in i(t)N_i(t). Its local objective is Fi(w)=(x,y)∼i[ℓ(w,x,y)],F_i(w)=E_(x,y) _i[ (w;x,y)], and the global learning objective is minwF(w)=∑i=1NpiFi(w), _wF(w)= _i=1^Np_iF_i(w), where pi≥0p_i≥ 0 and ∑i=1Npi=1 _i=1^Np_i=1. The communication graph strongly influences how efficiently information propagates across clients and how quickly local models approach consensus, especially under non-IID data. We therefore treat topology as an adaptive decision variable rather than a fixed background structure. Our setting is subject to three operational constraints: during online topology adaptation, no client requires a global graph statistic; candidate discovery is restricted to ego-neighborhood and FoF information; and online edge acquisition is regulated by a degree threshold DmaxD_ . In the reference implementation, the initial connected Erdős–Rényi graph is generated with expected degree four and may contain nodes whose initial degree exceeds DmaxD_ . Consequently, DmaxD_ should be interpreted as an online degree threshold: a new edge is added only when both endpoints are below the threshold, while a feasible swap preserves the initiator’s degree. Under these conditions, the objective is to perform bounded local topology search without global spectral computation. LFHE evaluates initiator-local improvement only; it does not require an affected peer endpoint to improve its own score, and it does not guarantee preservation or monotonic improvement of global connectivity. Local topology-search problem. For client i, let i(Gt,k)A_i(G_t,k) denote the feasible one-step actions induced by sampled FoF candidate k: keep GtG_t, add (i,k)(i,k) when the degree condition holds, or replace an incident edge (i,j)(i,j) with (i,k)(i,k) when local redundancy and degree conditions permit. For feasible G′G , define Ji(G′,t)=ω1(1−βt)Ci(G′)+ω2βtDi(G′)−ω3degG′(i)Dmax.J_i(G ;t)= _1(1- _t)C_i(G )+ _2 _tD_i(G )- _3 _G (i)D_ . (1) Given k, LFHE selects the highest-scoring feasible local operation and commits it only if Ji(G′,t)>Ji(Gt,t)J_i(G ;t)>J_i(G_t;t). Thus LFHE performs sampled greedy search over a locally observable topology neighborhood rather than optimization over the global graph space. 3.2 Overview of the Framework LFHE performs online topology adaptation through local rewiring. At each topology-update round, client i samples a FoF candidate and tests a direct addition or one-edge swap. A proposal is committed only when it improves the initiator’s score and satisfies the degree and local redundancy checks. Candidate discovery therefore remains local, while online edge acquisition is explicitly regulated. The objective combines three signals with distinct roles. The structural term scores representation disagreement over incident communication edges; Section 3.5 shows that it is exactly a node-local contribution to representation Dirichlet energy. The mismatch term supplies early exploration, encouraging clients to examine alternative local neighborhoods rather than selecting dissimilar peers indiscriminately. The degree term regulates edge acquisition and discourages excessive communication concentration. Beyond the structural component, their combination remains a heuristic design choice whose role is evaluated empirically. As illustrated in Figure 1, LFHE is a bounded local-search layer on top of decentralized learning. Each client observes only its local neighborhood, proposes a FoF add or swap, and accepts it according to initiator-local fitness. The reference experiments serialize proposals within a synchronous topology-update step; fully concurrent proposal conflicts, stale snapshots, and asynchronous commits are outside the evaluated protocol. Figure 1: Overview of LFHE. Each client i observes only its ego-graph, samples a two-hop FoF candidate, and evaluates it using a local objective with structural, exploration, and degree-control terms. If the candidate improves the objective, LFHE adds the edge when direct addition is feasible or performs a feasible one-edge swap otherwise; if no improving feasible operation exists, the candidate is rejected. 3.3 Local Fitness Function Design The local objective in Eq. 1 uses ω1,ω2,ω3>0 _1, _2, _3>0, representation ri∈ℝdr_i ^d, and annealing coefficient βt=β0e−κt, _t= _0e^-κ t, where β0>0 _0>0 controls the initial exploration weight and κ>0κ>0 its decay. Representations are treated as fixed while candidate topology operations are evaluated. The three terms play distinct roles: the structural term targets current representation disagreement over incident communication edges, mismatch supplies early exploration, and degree regularization controls communication concentration. Local structural term. Ci(G)=∑j∈i(G)‖ri−rj‖22.C_i(G)= _j _i(G)\|r_i-r_j\|_2^2. (2) If R=[r1⊤,…,rN⊤]⊤R=[r_1 ,…,r_N ] , then 12∑iCi(G)=Tr(R⊤LR) 12 _iC_i(G)=Tr(R LR), the representation Dirichlet energy of the current graph. Under standard linear consensus dynamics, this global energy equals the instantaneous rate at which centered representation disagreement is dissipated. Thus Ci(G)C_i(G) gives client i a locally computable contribution to a state-dependent consensus quantity without requiring a global graph statistic. It is not a surrogate for λ2 _2: CiC_i depends on both the graph and the current representation field, whereas λ2 _2 is a topology-only worst-case quantity. Neighborhood mismatch term. Di(G)=1−cos(ri,1|i(G)|∑j∈i(G)rj).D_i(G)=1- \! (r_i,\; 1|N_i(G)| _j _i(G)r_j ). (3) This term measures directional mismatch between node i and its mean neighborhood representation. Unlike Ci(G)C_i(G), which has the Dirichlet-energy interpretation above, Di(G)D_i(G) primarily supplies exploration pressure. Large values indicate directional mismatch with the current neighborhood and increase the influence of exploration early in training. It is therefore used to expose alternative local neighborhoods rather than to favor dissimilar peers indiscriminately. Degree regularization. The term deg(i)/Dmax (i)/D_ penalizes high initiator degree, discouraging excessive edge acquisition and communication concentration. Overall behavior. As βt _t decays, the composite rule shifts from mismatch-driven exploration toward the consensus-grounded structural signal. Early in training, Di(G)D_i(G) receives greater weight; later, Ci(G)C_i(G) becomes more influential. The annealing schedule and relative coefficients remain design choices; only the structural component carries the exact consensus-theoretic interpretation developed below. 3.4 Topology Evolution Mechanism LFHE performs bounded local topology search using only ego-neighborhood and FoF information. For client i, candidate peers are sampled from Ωi=k∈j∣j∈i,k≠i,(i,k)∉E. _i=\k _j j _i,\ k≠ i,\ (i,k)∉ E\. This two-hop construction expands peer exposure beyond the current neighborhood without requiring a global peer directory. Let di=|i|d_i=|N_i| and d¯=maxjdj d= _jd_j. Before removing duplicates and already connected peers, the number of distinct two-hop candidates available to client i is bounded by |Ωi(2)|≤di(d¯−1).| _i^(2)|≤ d_i( d-1). Hence the full FoF candidate space is O(d¯2)O( d^2) per client and O(Nd¯2)O(N d^2) system-wide. Under bounded sparse degree, this yields O(1)O(1) per-client and O(N)O(N) system-wide discovery state, so the local search space need not grow with population size. The implementation may sample only part of this frontier; these bounds characterize candidate discovery and control state rather than total communication volume. For a sampled candidate, LFHE first considers direct addition when both endpoints satisfy the online degree threshold. Otherwise, it evaluates feasible one-edge swaps. A common-neighbor check preserves local redundancy when removing an incident edge, while accepted operations are selected according to the initiator-local objective. Two structural properties follow. First, with representations fixed during a topology decision, every committed operation strictly improves the initiator objective JiJ_i. Second, each decision uses only ego/FoF structure, representations, and endpoint degree information, requiring neither a global peer directory nor online computation of L or λ2 _2. Thus LFHE provides monotone local improvement within a population-independent per-client search space under bounded degree. The reference implementation commits proposals sequentially within each topology-update step, so later clients observe earlier modifications and the realized graph can depend on processing order. The local redundancy check is not a global connectivity certificate, and improvement is guaranteed for the initiating client rather than every affected endpoint. 3.5 Consensus-Dissipation Interpretation Let R=[r1⊤,…,rN⊤]⊤R=[r_1 ,…,r_N ] stack client representations and let L=D−AL=D-A denote the graph Laplacian. The structural scores exactly decompose the representation Dirichlet energy: ℰG(R)=Tr(R⊤LR)=∑(i,j)∈E‖ri−rj‖22=12∑iCi(G).E_G(R)=Tr(R LR)= _(i,j)∈ E\|r_i-r_j\|_2^2= 12 _iC_i(G). (4) Thus, Ci(G)C_i(G) is a node-local contribution to a global state-dependent consensus energy. Under standard linear consensus dynamics R˙=−LR R=-LR, define centered representation disagreement as V(R)=12‖PR‖F2,P=I−1N⊤.V(R)= 12\|PR\|_F^2, P=I- 1N11 . Then V˙=−ℰG(R). V=-E_G(R). (5) Hence the representation Dirichlet energy is exactly the instantaneous rate at which centered disagreement is dissipated. For fixed R, replacing edge (i,j)(i,j) by (i,k)(i,k) changes this energy by ℰG′(R)−ℰG(R)=‖ri−rk‖22−‖ri−rj‖22.E_G (R)-E_G(R)=\|r_i-r_k\|_2^2-\|r_i-r_j\|_2^2. Therefore, a positive structural-energy change increases the instantaneous rate of disagreement dissipation for the current representation field. Rayleigh–Ritz further gives ℰG(R)≥λ2(L)‖PR‖F2.E_G(R)≥ _2(L)\|PR\|_F^2. (6) Thus, λ2 _2 characterizes the graph’s worst-case normalized dissipation over non-consensus directions, whereas LFHE’s structural score evaluates the representation field currently realized on the graph. This establishes an exact consensus-theoretic interpretation for the structural component; the full LFHE objective additionally trades this signal against exploration and degree control and does not directly optimize λ2 _2. The full derivation is provided in Appendix D. 3.6 Algorithm Description Each communication round consists of local training and decentralized aggregation, with an additional topology-adaptation step every K rounds. Local training. Each client performs E local epochs on its private dataset using the task-specific local optimizer. Decentralized aggregation. Each node aggregates neighbor models using degree-aware weights aij=11+max(deg(i),deg(j)),aii=1−∑j∈iaij.a_ij= 11+ ( (i), (j)), a_i=1- _j _ia_ij. Topology adaptation. Every K rounds, each client samples a FoF candidate, evaluates the feasible addition and one-edge-swap operations induced by that candidate, and commits the highest-scoring feasible operation only when it improves the initiator-local objective. Full pseudocode and implementation details are provided in Appendix E. 4 Experimental Setup We evaluate LFHE on four datasets across three modalities: CIFAR-10 and CIFAR-100 [12], Google Speech Commands [24], and Sentiment140 [8]. Unless otherwise stated, data are partitioned with class-wise Dirichlet α=0.1α=0.1, the main system has N=30N=30 clients, and LFHE uses online degree threshold Dmax=4D_ =4. The main comparison includes FedAvg [16], Fully Connected, Ring [15], Static Random, Epidemic Learning [7], and DissDL, a legacy dissimilarity-guided implementation retained from our original benchmark. For shared learning components, comparisons use matched data partitions, model initialization, local optimization budgets, and evaluation schedules where applicable. Topology-specific discovery, edge direction, and communication semantics follow each method’s native design. Main results use five seeds 42,43,44,45,46\42,43,44,45,46\; diagnostic studies use three seeds or a single controlled seed where explicitly stated. Complete model, optimizer, representation, initialization, and communication details are provided in Appendix F. The connected Erdős–Rényi initializer targets expected degree four and may contain nodes above DmaxD_ ; accordingly, DmaxD_ denotes an online edge-acquisition threshold rather than a hard maximum degree in the reference LFHE implementation. 5 Main Convergence Results and Topology Analysis We evaluate LFHE with 30 clients across image, speech, and text tasks. Among the decentralized methods reported in Table 1, LFHE reaches the earliest or joint-earliest mean target accuracy on all four datasets and obtains the highest mean final accuracy on CIFAR-10, CIFAR-100, and Speech Commands. On Sentiment140, Epidemic Learning has the highest mean final accuracy. Because several differences are comparable to the reported standard deviations, these results support competitive and often favorable performance rather than uniform statistical dominance. Figure 2 illustrates CIFAR-10 dynamics at α=0.1α=0.1. (a) Training loss (b) Test accuracy Figure 2: Main convergence results on CIFAR-10 under non-IID data. LFHE shows favorable long-run performance relative to the reported decentralized baselines; shaded bands summarize run-to-run variability. Target attainment. As shown in Table 1, LFHE reaches the selected target first among the reported decentralized methods at rounds 225, 125, and 170 on CIFAR-10, CIFAR-100, and Sentiment140, respectively, and jointly reaches the Speech Commands target with Epidemic Learning at round 60. These measurements are based on communication rounds rather than wall-clock time or bytes-to-target and depend on the selected accuracy thresholds. Final performance. Among the decentralized methods in Table 1, LFHE achieves the highest mean final accuracy on CIFAR-10, CIFAR-100, and Speech Commands, at 71.6%, 45.2%, and 78.8%, respectively. On Sentiment140, LFHE reaches 65.3%, slightly below Epidemic Learning at 65.5%65.5\%. The small gaps on several tasks and overlapping run variability indicate strong cross-modal competitiveness rather than consistent superiority over every alternative. Scalability perspective. A preliminary three-seed accuracy study at N∈10,50,100N∈\10,50,100\ is complemented by a discovery-state diagnostic extending to N=500N=500. Across N=30N=30 to N=500N=500, LFHE’s measured mean FoF candidate set remains between 9.76 and 11.86 peers per client. This empirical stability is consistent with the O(1)O(1)-per-client discovery-state bound under sparse degree that does not scale with N, showing that LFHE’s local candidate space need not grow with population size. This result concerns discovery/control state rather than total communication volume or large-system learning speed. Additional moderate-scale accuracy is provided in Appendix G.5. Key insight. Taken together, the results separate the effect of localized candidate exposure from that of structural scoring. Random-FoF reaches accuracy close to LFHE in the matched regime, indicating that FoF exposure itself contributes substantially to the observed learning performance. However, degree-preserving random swaps and the graph-only control show that topology churn or high algebraic connectivity alone do not reproduce LFHE’s behavior, while the structural score produces substantially different realized communication graphs. Together with the Dirichlet-energy interpretation, these controls support the structural term as a principled state-dependent topology-selection signal whose clearest empirical effect is on communication structure rather than a universal accuracy advantage. Full matched-control trajectories are provided in Appendix H, with computational and control-overhead measurements in Appendix I. Table 1: Convergence speed and final accuracy across datasets over five random seeds. R@T denotes the communication round at which the mean accuracy across seeds first reaches the target specified in each dataset column, and “–” indicates that the target was not reached within the training budget. Final accuracy is reported as mean± . FedAvg and Fully Connected are included as reference points. Among decentralized methods, the best result is shown in bold and the second-best in underline. Method CIFAR-10 (≥ 70%) CIFAR-100 (≥ 35%) Speech Commands (≥ 70%) Sentiment140 (≥ 65%) R@T Final Acc. (%) R@T Final Acc. (%) R@T Final Acc. (%) R@T Final Acc. (%) FedAvg 115 78.40± 0.80 85 51.90± 0.30 35 86.80± 0.50 40 70.60± 2.70 Fully Connected 105 78.40± 0.70 85 51.80± 0.50 35 86.40± 1.40 40 70.60± 2.60 Ring – 52.60± 1.20 – 27.00± 0.80 – 53.00± 3.10 – 58.00± 2.70 Static Random – 67.40± 1.70 160 41.70± 1.20 75 74.20± 3.50 – 63.50± 2.30 Epidemic Learning – 70.00± 1.60 130 44.10± 0.70 60 78.10± 3.30 190 65.50± 2.30 DissDL (legacy impl.) – 63.70± 2.10 185 40.20± 0.30 80 74.60± 3.70 230 64.20± 3.80 LFHE (Ours) 225 71.60± 2.10 125 45.20± 0.60 60 78.80± 2.30 170 65.30± 1.70 5.1 Topology Evolution Behavior To examine how LFHE reshapes the communication graph, we analyze global topology statistics and representative graph snapshots under the highly non-IID CIFAR-10 setting (α=0.1α=0.1). Figure 3(a) tracks the evolution of algebraic connectivity and clustering, while Figure 3(b) shows representative graph snapshots over training. Additional consensus and client-level variance diagnostics are provided in Appendix G.7. (a) Evolution of λ2 _2 and clustering (b) Topology snapshots Figure 3: Topology evolution of LFHE on CIFAR-10 under non-IID data (α=0.1α=0.1). Left: algebraic connectivity λ2 _2 and clustering coefficient. Right: representative graph snapshots at different training rounds. Early improvement in global connectivity. As shown in Figure 3(a)–(b), the offline diagnostic λ2 _2 rises early and then stabilizes, showing that the realized graphs become more algebraically connected. This complements the state-dependent interpretation developed in Section 3.5: LFHE scores the current representation field locally, while λ2 _2 independently measures the resulting graph’s worst-case spectral connectivity. Reduced local redundancy. The clustering coefficient decreases early and then stabilizes at a lower level. Together with the increase in λ2 _2, this indicates that LFHE restructures local connectivity rather than simply forming denser neighborhoods. The resulting graphs remain sparse while exhibiting communication patterns compatible with broader information propagation across the network. Exploration-to-refinement dynamics. The trajectories and snapshots are consistent with early bridge-like exploration followed by less frequent rewiring as the annealing schedule shifts weight toward the structural term. By rounds 10–20, new cross-region edges coincide with the sharp rise in λ2 _2; later graphs remain sparse and show no dominant hub. These observations are consistent with the intended exploration-to-refinement behavior, while the component diagnostics below separately examine the roles of the individual objective terms. 5.2 Component and Mechanism Diagnostics The original removal ablation on CIFAR-10 with α=0.3α=0.3 shows that full LFHE reaches the target first and obtains the highest final mean among the reported removal variants. A stricter matched-score diagnostic at α=0.1α=0.1, however, shows that Structural-only slightly exceeds full LFHE: final λ2 _2 is 2.3062.306 versus 2.2472.247, and final accuracy is 75.08%75.08\% versus 74.82%74.82\%. Exploration-only rewires more frequently (3.4063.406 versus 1.3331.333 accepted rewires per update for LFHE) while producing weaker connectivity and accuracy. These results identify the structural term as the principal empirical topology-selection signal in this regime, with exploration and degree control acting as regime-dependent search regulators rather than universally necessary components. Additional matched controls reinforce this interpretation without reducing topology quality to a single graph statistic. Random-FoF reaches 74.62±2.25%74.62± 2.25\% final accuracy across three seeds, close to LFHE, but produces lower final algebraic connectivity (1.6961.696 versus 2.2472.247). A degree-preserving random-swap control performs many topology changes but reaches only 66.24%66.24\% accuracy, while a representation-free graph-only control reaches a higher final λ2=2.403 _2=2.403 but only 73.52%73.52\% accuracy. Together, these controls show that neither topology churn nor maximizing spectral connectivity alone explains LFHE’s behavior, and they support the role of state-dependent structural scoring. Full removal ablations are provided in Appendix A, and matched mechanism controls and trajectories in Appendix H. 6 Discussion LFHE’s structural signal is not merely a heuristic proxy for algebraic connectivity. For the current representation field, CiC_i is exactly a node-local contribution to graph Dirichlet energy, whose value gives the instantaneous dissipation rate of centered disagreement under standard linear consensus. LFHE therefore performs state-aware local topology search rather than approximating a global spectral objective. The experiments support this distinction. Structural-only slightly exceeds full LFHE in the matched-score diagnostic, while Random-FoF, graph-only, and random-swap controls show that neither FoF exposure, high λ2 _2, nor topology churn alone explains the observed behavior. The structural term therefore acts as the principal topology-selection signal in the tested regime, while exploration and degree control regulate how that signal is searched and constrained. A separate three-seed comparison with Morph (Appendix G.2) further illustrates this trade-off. Morph achieves stronger predictive performance, whereas LFHE restricts search to a bounded FoF frontier. Under bounded sparse degree, LFHE maintains O(1)O(1) candidate state per client and O(N)O(N) system-wide, emphasizing discovery locality rather than predictive superiority or lower total communication. The reference implementation serializes topology proposals, so realized graphs may depend on update order; Appendix G.3 quantifies this sensitivity. Fully concurrent or asynchronous execution remains outside the evaluated protocol. 7 Conclusion We introduced LFHE, a framework for bounded local topology search in decentralized learning with non-IID data. LFHE evaluates local add-or-swap operations using structural disagreement, exploration, and degree control without online global spectral computation. Its structural scores exactly decompose representation Dirichlet energy, which under standard linear consensus determines instantaneous disagreement dissipation. Across the reported baselines, LFHE shows favorable target attainment and competitive final accuracy, while matched diagnostics identify the structural term as the main empirical topology-selection signal. LFHE should therefore be viewed as state-aware localized topology search with bounded FoF discovery, rather than as an approximation to global spectral optimization. Limitations. The theoretical result applies to the structural component with fixed representations during each topology decision and under standard linear consensus. The full objective still depends on representation choice, coefficient settings, and the annealing schedule. DmaxD_ is an online edge-acquisition threshold rather than a strict global degree cap, acceptance is initiator-local, and topology updates are serialized. Evaluation remains limited to controlled benchmarks and moderate training populations. Future work. Future directions include the analysis of the discrete weighted aggregation operator, the development of learned exploration schedules, the investigation of concurrent topology updates, and large-scale evaluation under communication-normalized settings. Reproducibility statement The main protocol, optimization settings, and comparison scope are summarized in Section 4. Complete experimental settings and architectures are provided in Appendix F, full algorithmic details in Appendix E, and sensitivity and diagnostic protocols in Appendices G.4–G.7. AI use statement Generative AI tools were used to assist with language editing, restructuring portions of the manuscript, experiment planning, and software/code edits. AI-assisted text, code changes, and experiment configurations were reviewed by the authors, while numerical results were taken from the executed experimental pipelines and checked against saved outputs. The authors take responsibility for the final text, claims, code, and reported artifacts produced with AI assistance. References [1] P. Balister, B. Bollobás, A. Sarkar, and M. Walters (2005) Connectivity of random k-nearest-neighbour graphs. Advances in Applied Probability 37 (1), p. 1–24. 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Appendix A Full Component Ablation A.1 Original removal ablation and topology diagnostics To assess the role of each component, we conduct an ablation on CIFAR-10 with α=0.3α=0.3. Under this specific protocol, full LFHE reaches the target first and has the highest final mean among the removal variants. This experiment tests removal from the full objective; it should be distinguished from the matched-score diagnostic at α=0.1α=0.1, where structural-only is slightly stronger than full LFHE. Figure 4: Ablation study on CIFAR-10 showing test accuracy for LFHE and its ablated variants. Method R@Target Final Acc. Static Random 85 0.7766 LFHE 55 0.8079 w/o Structural 65 0.8019 w/o Exploration 65 0.8017 w/o Degree Reg. 60 0.8063 Table 2: Ablation results on CIFAR-10 with α=0.3α=0.3. R@Target denotes the round at which accuracy first reaches 0.65, and Final Acc. denotes the final test accuracy. Effect of the local structural term. Removing the structural term causes the clearest degradation in topology quality. In Figure 5, the variant without CiC_i attains the weakest and most unstable λ2 _2, together with slower convergence and lower final accuracy in Figure 4 and Table 2. This pattern is consistent with the Dirichlet-energy interpretation in Appendix D: removing CiC_i removes the objective component that directly scores current representation disagreement across incident communication edges. Effect of exploration. Removing the exploration term produces a mild degradation in this ablation. Without the early exploratory signal, nodes are more likely to remain in locally convenient neighborhoods rather than expose bridge-forming opportunities across heterogeneous regions. The small difference, together with the structural-only result in Appendix H, indicates that exploration is a regime-dependent search regulator rather than a universally necessary source of final accuracy. Effect of degree regularization. Removing the degree penalty increases λ2 _2 the most, but does not yield the best accuracy. The resulting graph also maintains higher clustering, indicating denser and less controlled connectivity. This indicates that stronger connectivity alone is insufficient; without degree control, the topology becomes denser and less communication-constrained, while the additional connectivity does not translate into improved learning performance [23]. Topology dynamics and insight. Figure 5 identifies the structural term as the clearest source of the measured topology change. Exploration modestly affects search in this regime, while removing degree regularization yields denser connectivity without a corresponding accuracy gain. Importantly, the matched α=0.1α=0.1 diagnostic reports final λ2=2.306 _2=2.306 and accuracy 0.75080.7508 for structural-only, compared with 2.2472.247 and 0.74820.7482 for full LFHE. We therefore do not claim that all three terms are jointly optimal in every regime; full LFHE is the reference composite rule, whereas structural-only can be stronger under the matched-score setting. Figure 5: Topology evolution under ablation. Left: algebraic connectivity (λ2 _2). Right: clustering coefficient. Appendix B Related Work Comparison Table Table 3: Comparison of representative communication-structure approaches. The last column refers to online information requirements rather than concurrent execution semantics. ✓∗ ^* denotes a state-dependent Dirichlet structural signal rather than direct optimization of λ2 _2. Method Connectivity-aware Similarity-aware Ego-graph local No global graph statistics Spectral methods ✓ × × × D-PSGD / Gossip × × × ✓ D-Cliques / STL-FW ✓ ✓ × × Epidemic Learning × × ✓ ✓ DissDL (legacy impl.) × ✓ ✓ ✓ Morph × ✓ × ✓ FedAMP / Ditto × ✓ × × LFHE ✓∗ ✓ ✓ ✓ Appendix C Graph Laplacian, Dirichlet Energy, and Algebraic Connectivity For an undirected graph G=(V,E)G=(V,E) with |V|=n|V|=n, let A be its adjacency matrix, D=diag(d1,…,dn)D=diag(d_1,…,d_n) its degree matrix, and L=D−AL=D-A its combinatorial Laplacian. For a scalar graph signal x∈ℝnx ^n, x⊤Lx=12∑i=1n∑j=1nAij(xi−xj)2=∑(i,j)∈E(xi−xj)2.x Lx= 12 _i=1^n _j=1^nA_ij(x_i-x_j)^2= _(i,j)∈ E(x_i-x_j)^2. (7) For vector-valued representations R=[r1⊤,…,rn⊤]⊤∈ℝn×dR=[r_1 ,…,r_n ] ^n× d, summing over dimensions gives ℰG(R):=Tr(R⊤LR)=∑(i,j)∈E‖ri−rj‖22,E_G(R):=Tr(R LR)= _(i,j)∈ E\|r_i-r_j\|_2^2, (8) the graph Dirichlet energy of the current representation field. The eigenvalues satisfy 0=λ1≤λ2≤⋯≤λn0= _1≤ _2≤·s≤ _n. By the Courant–Fischer theorem, λ2(L)=minx≠0x⟂x⊤Lx⊤x. _2(L)= _ subarraycx≠ 0\\ x 1 subarray x Lxx x. (9) Thus λ2 _2 is not the energy of the current model state; it is the smallest normalized Dirichlet energy over all non-consensus directions, explaining its role as a topology-only worst-case connectivity and consensus quantity [20, 19]. Appendix D Structural Signal as Local Consensus Dissipation LFHE uses Ci(R,G)=∑j∈i(G)‖ri−rj‖22.C_i(R,G)= _j _i(G)\|r_i-r_j\|_2^2. (10) Proposition 1 (Local Dirichlet decomposition). For any undirected G and fixed R, ℰG(R)=12∑i=1nCi(R,G).E_G(R)= 12 _i=1^nC_i(R,G). (11) Proof. Summing CiC_i counts every undirected edge disagreement once from each endpoint: ∑iCi=2∑(i,j)∈E‖ri−rj‖22=2Tr(R⊤LR). _iC_i=2 _(i,j)∈ E\|r_i-r_j\|_2^2=2\,Tr(R LR). Dividing by two proves the result. □ Proposition 2 (Consensus-dissipation identity). Consider standard continuous-time linear consensus R˙=−LR. R=-LR. (12) Let P=I−1n⊤P=I- 1n11 and V(R)=12‖PR‖F2V(R)= 12\|PR\|_F^2. Then V˙(R)=−ℰG(R)=−12∑iCi(R,G). V(R)=-E_G(R)=- 12 _iC_i(R,G). (13) Proof. Since L=0L1=0, PL=LP=LPL=LP=L, and therefore V˙=Tr((PR)⊤PR˙)=−Tr(R⊤LR)=−ℰG(R). V=Tr\! ((PR) P R )=-Tr(R LR)=-E_G(R). Proposition 1 gives the second equality. □ Thus CiC_i is client i’s local contribution to the instantaneous dissipation of the currently observed representation disagreement. LFHE can therefore allocate limited communication using current model state without reconstructing a global spectral objective. Corollary (One-edge swap). Let G′G replace (i,j)(i,j) in G by (i,k)(i,k), with R fixed during the topology decision. Then ℰG′(R)−ℰG(R)=‖ri−rk‖22−‖ri−rj‖22.E_G (R)-E_G(R)=\|r_i-r_k\|_2^2-\|r_i-r_j\|_2^2. (14) A positive structural change therefore strictly increases the instantaneous dissipation rate in Proposition 2. This statement concerns the structural component; the full objective may trade structural change against exploration or degree terms. Finally, because every column of PRPR is orthogonal to 1, Eq. 9 gives ℰG(R)≥λ2(L)‖PR‖F2=2λ2(L)V(R),E_G(R)≥ _2(L)\|PR\|_F^2=2 _2(L)V(R), (15) hence V˙≤−2λ2(L)V V≤-2 _2(L)V. Algebraic connectivity is therefore a worst-case graph-level lower bound on dissipation, whereas CiC_i evaluates the representation field currently present during training. LFHE does not approximate λ2 _2; it performs state-aware local structural search within the same Dirichlet-energy framework. Relation to the implemented averaging rule. For fixed G, the degree-aware aggregator is W=I−LaW=I-L_a, with edge weights aij=1/(1+max(di,dj))a_ij=1/(1+ (d_i,d_j)). Its weighted Dirichlet energy is ℰa(R)=∑(i,j)∈Eaij‖ri−rj‖22.E_a(R)= _(i,j)∈ Ea_ij\|r_i-r_j\|_2^2. If d¯=maxidi d= _id_i, then ℰG(R)1+d¯≤ℰa(R)≤12ℰG(R). E_G(R)1+ d _a(R)≤ 12E_G(R). Thus bounded degree links LFHE’s structural score to the energy induced by the implemented aggregation weights, up to degree-controlled factors. The experiments use discrete degree-aware averaging rather than the continuous-time model above. The propositions establish the structural score’s consensus-theoretic meaning, while the matched FoF controls test its relevance under the implemented learning dynamics. There, Structural-only and LFHE produce the largest measured λ2 _2 gains and lowest model divergence among matched score variants. Appendix E Full Algorithm Description In this appendix, we provide the procedural description of the decentralized learning framework with LFHE topology adaptation. Local training and aggregation are synchronous. Within a topology-update call, client proposals are evaluated and committed sequentially on a copied graph, matching the reference implementation used in the experiments. E.1 Decentralized Federated Learning Procedure Algorithm 1 Decentralized Federated Learning with LFHE Topology Updates 1: local batch size B, local epochs E, task-specific local optimizer O, total rounds T, topology update interval K 2: final local models wiTi∈V\w_i^T\_i∈ V 3: Initialize a connected communication graph G0=(V,E0)G^0=(V,E^0) 4: Initialize local models wi0i∈V\w_i^0\_i∈ V 5: for each round t=0,1,…,T−1t=0,1,…,T-1 do 6: // Local training 7: for each client i∈Vi∈ V in parallel do 8: for e=1,…,Ee=1,…,E do 9: for mini-batch b sampled from client i do 10: wi←OptimizerStep(wi,b,)w_i← OptimizerStep(w_i,b;O) 11: end for 12: end for 13: end for 14: // Synchronous decentralized aggregation 15: for each client i∈Vi∈ V in parallel do 16: Ni←i(Gt)N_i _i(G^t) 17: for each j∈Nij∈ N_i do 18: aij←11+max(deg(i),deg(j))a_ij← 11+ ( (i), (j)) 19: end for 20: aii←1−∑j∈Niaija_i← 1- _j∈ N_ia_ij 21: w~i←aiiwi+∑j∈Niaijwj w_i← a_iw_i+ _j∈ N_ia_ijw_j 22: end for 23: for each client i∈Vi∈ V do 24: wi←w~iw_i← w_i 25: end for 26: // Periodic serialized topology update 27: if tmodK=0t K=0 then 28: Gt+1←LFHE-Update(Gt,wi,t)G^t+1← LFHE-Update(G^t,\w_i\,t) 29: else 30: Gt+1←GtG^t+1← G^t 31: end if 32: end for 33: return wiTi∈V\w_i^T\_i∈ V E.2 LFHE Topology Update LFHE performs local graph rewiring using only ego-network information. For each client i, a candidate node is discovered through a FoF mechanism. The candidate edge is accepted if it improves the node-level fitness under the degree constraint. When direct addition is not feasible, LFHE attempts an atomic swap by replacing one existing neighbor with the candidate. Algorithm 2 LFHE Topology Update with Degree Constraint and Atomic Swap 1: graph G=(V,E)G=(V,E), client models wi\w_i\, online degree threshold DmaxD_ , fitness weights ω1,ω2,ω3 _1, _2, _3, round t 2: updated graph G′G 3: G′←G ← G 4: for each client i∈Vi∈ V do 5: Ni←i(G′)N_i _i(G ) 6: if |Ni|=0|N_i|=0 then 7: continue 8: end if 9: // Friend-of-Friend candidate discovery 10: Sample j∼Unif(Ni)j (N_i) 11: i←k∈j(G′)∣k≠i,(i,k)∉E(G′)C_i←\k _j(G ) k≠ i,\ (i,k)∉ E(G )\ 12: if i=∅C_i= then 13: continue 14: end if 15: Sample k∼Unif(i)k (C_i) 16: fold←Fitness(i,G′,wi,t)f_old← Fitness(i,G ,\w_i\,t) 17: if deg(i)<Dmax (i)<D_ and deg(k)<Dmax (k)<D_ then 18: Add edge (i,k)(i,k) to G′G 19: fnew←Fitness(i,G′,wi,t)f_new← Fitness(i,G ,\w_i\,t) 20: if fnew≤foldf_new≤ f_old then 21: Remove edge (i,k)(i,k) from G′G 22: end if 23: else 24: fbest←−∞f_best←-∞ 25: j⋆←Nonej 26: for each j′∈Nij ∈ N_i do 27: if deg(i)≤1 (i)≤ 1 then 28: continue 29: end if 30: if deg(j′)≤1 (j )≤ 1 then 31: continue 32: end if 33: if deg(k)≥Dmax (k)≥ D_ then 34: continue 35: end if 36: if not AlternativePath(G′,i,j′)(G ,i,j ) then 37: continue 38: end if 39: Temporarily remove (i,j′)(i,j ) and add (i,k)(i,k) 40: fswap←Fitness(i,G′,wi,t)f_swap← Fitness(i,G ,\w_i\,t) 41: if fswap>fbestf_swap>f_best then 42: fbest←fswapf_best← f_swap 43: j⋆←j′j ← j 44: end if 45: Restore the original edges 46: end for 47: if fbest>foldf_best>f_old then 48: Remove edge (i,j⋆)(i,j ) from G′G 49: Add edge (i,k)(i,k) to G′G 50: end if 51: end if 52: end for 53: return G′G E.3 Fitness Function For each client i, LFHE evaluates the quality of its local neighborhood using a node-level fitness function. Let ri∈ℝdr_i ^d denote the representation vector extracted from client i’s local model. The annealing coefficient is defined as βt=β0e−κt, _t= _0e^-κ t, where β0>0 _0>0 is the initial annealing weight and κ>0κ>0 is the decay constant. The first component is a local structural term, Ci(G)=∑j∈i(G)‖ri−rj‖22,C_i(G)= _j _i(G)\|r_i-r_j\|_2^2, which measures the incident-edge contribution to representation Dirichlet energy. As shown in Appendix D, its network-wide sum has an exact consensus-dissipation interpretation; it is state-dependent and does not require computing a global spectral statistic. The second component is a neighborhood mismatch term, implemented through cosine similarity to the mean neighbor representation: r¯i=1|i(G)|∑j∈i(G)rj, r_i= 1|N_i(G)| _j _i(G)r_j, simi(G)=cos(ri,r¯i).sim_i(G)= (r_i, r_i). The corresponding mismatch signal 1−simi(G)1-sim_i(G) acts primarily as an exploration term. The third component is a degree penalty, deg_normi=deg(i)Dmax,deg\_norm_i= (i)D_ , which discourages overly dense local connectivity. The final fitness used in the implementation is Fitness(i,G,wi,t)=w1(1−βt)Ci(G)+w2βt(1−simi(G))−w3deg(i)Dmax. Fitness(i,G,\w_i\,t)=w_1(1- _t)C_i(G)+w_2 _t (1-sim_i(G) )-w_3 (i)D_ . This formulation yields a time-varying trade-off. In early rounds, when βt _t is relatively large, the mismatch term 1−simi(G)1-sim_i(G) has stronger influence and encourages exploration. In later rounds, as βt _t decays, the local structural term becomes more influential, promoting topology refinement while still penalizing excessive degree growth. E.4 Auxiliary Procedures The implementation uses a lightweight local connectivity safeguard before accepting a swap. In particular, the procedure AlternativePath(G,i,j)(G,i,j) returns true if clients i and j share at least one common neighbor: AlternativePath(G,i,j)=true,if i(G)∩j(G)≠∅,false,otherwise. AlternativePath(G,i,j)= casestrue,&if N_i(G) _j(G)≠ ,\\ false,&otherwise. cases This is a local redundancy safeguard when an edge is removed. A shared neighbor certifies a two-hop alternative path between the endpoints of the specific edge at the time of the decision, without requiring a global path search. This statement concerns the serialized reference update; stale or conflicting concurrent removals would require additional validation. E.5 Implementation Notes In the reference implementation, LFHE-Update is executed on a copy of the current graph and applied sequentially over clients in fixed order. Each accepted operation is committed immediately, so later clients observe earlier changes and the realized topology can depend on processing order. Candidate discovery remains restricted to FoF nodes. If direct addition is infeasible, the initiator evaluates feasible one-edge swaps and commits the swap with the largest improvement in its own fitness. The notation wi\w_i\ in Algorithms 1–2 denotes simulator state compactly; an initiator-local fitness evaluation uses only the representations associated with its current ego-neighborhood and the current FoF candidate. The affected endpoints do not separately approve the operation, so acceptance is initiator-local rather than bilaterally Pareto-improving. In a controlled update-order diagnostic, changing the client order substantially changed the final edge set and λ2 _2, while the three observed final accuracies stayed within a 1.51-percentage-point range; details are reported in Appendix G.3. Across the recorded topology audit, none of 2,280 graph states from 38 runs was disconnected. This is empirical evidence about the observed trajectories, not a connectivity theorem for the common-neighbor safeguard. A concurrent realization could forward FoF candidate identifiers and representation summaries through current neighbors, and could use proposal–validate–commit with endpoint-local topology versions to reject stale or conflicting operations. We present this only as a deployment direction. The experiments in this paper do not implement or validate arbitrary asynchronous transaction semantics. Appendix F Additional Experimental Details We provide additional details on datasets, non-IID partitioning, training protocols, model architectures, and implementation hyperparameters used in the experiments. Datasets. CIFAR-10 and CIFAR-100 [12] are standard image classification benchmarks derived from the Tiny Images dataset. CIFAR-10 contains 50,000 training images and 10,000 test images across 10 classes, while CIFAR-100 uses the same train–test split structure but has 100 classes. For Google Speech Commands [24], we use a 10-keyword subset consisting of yes, no, up, down, left, right, on, off, stop, go, following the standard training and testing splits. For Sentiment140 [8], we use a binary sentiment classification setting and apply balanced random subsampling before client partitioning, resulting in 12,000 training tweets and 2,000 test tweets. Non-IID partitioning. To simulate heterogeneous client distributions, data are partitioned using class-wise Dirichlet sampling. For each class c, we sample a client allocation vector pc∼Dir(α)p_c (α) and distribute samples of that class across clients accordingly. Unless otherwise stated, we use α=0.1α=0.1 as the default highly non-IID setting. The component ablation and moderate-scale diagnostic use α=0.3α=0.3 and are reported separately from the main protocol. System setting. Unless otherwise specified, experiments use N=30N=30 clients and online degree threshold Dmax=4D_ =4. Static Random and LFHE start from the same connected Erdős–Rényi graph with expected degree four. Because this initializer is not degree-capped, some initial nodes may exceed DmaxD_ ; the threshold regulates subsequent additions and swaps rather than retroactively enforcing a hard cap. We additionally report a preliminary study at N∈10,50,100N∈\10,50,100\. Ring, Fully Connected, FedAvg, and dynamic baselines retain their respective topology mechanisms. Training protocol. All methods are implemented in PyTorch. CIFAR and Speech Commands experiments use SGD with learning rate 0.05 and batch size 32. Sentiment140 uses Adam with learning rate 10−310^-3 and batch size 16. Unless otherwise stated, each client performs one local epoch per communication round, except Sentiment140 where each client performs two local epochs. LFHE updates the communication topology every K=5K=5 communication rounds. Results are averaged over five random seeds, 42,43,44,45,46\42,43,44,45,46\. Unless otherwise stated, all LFHE experiments use ω1=1.0 _1=1.0, ω2=1.0 _2=1.0, ω3=0.1 _3=0.1, Dmax=4D_ =4, and the annealing schedule βt=exp(−0.01t) _t= (-0.01t). Table 4: Implementation details and hyperparameters used in the main experiments. Item Setting Number of clients N=30N=30 for the main experiments; N∈10,50,100N∈\10,50,100\ for scalability experiments. Online degree threshold Dmax=4D_ =4; the expected-degree initializer is not guaranteed to satisfy a hard cap. Topology update interval K=5K=5 communication rounds. Evaluation interval Every 5 communication rounds. Random seeds 42,43,44,45,46\42,43,44,45,46\. Non-IID partitioning Class-wise Dirichlet partitioning with α=0.1α=0.1 by default; the component ablation and moderate-scale diagnostic use α=0.3α=0.3. Initial sparse graph Static Random and LFHE use the same connected Erdős–Rényi graph with expected degree 4 and seed matched to the experimental seed. Ring uses a cycle graph, Fully Connected uses a complete graph, and FedAvg uses centralized aggregation. Aggregation rule Degree-aware decentralized averaging with aij=1/(1+max(deg(i),deg(j)))a_ij=1/(1+ ( (i), (j))) and aii=1−∑j∈Niaija_i=1- _j∈ N_ia_ij. LFHE weights ω1=1.0 _1=1.0, ω2=1.0 _2=1.0, ω3=0.1 _3=0.1. Annealing schedule βt=β0e−κt _t= _0e^-κ t with β0=1.0 _0=1.0 and κ=0.01κ=0.01. LFHE update rule Each client samples a FoF candidate, evaluates the local fitness, and accepts the candidate only if the fitness improves. If both endpoints are below the online degree threshold, LFHE adds the edge; otherwise, it evaluates feasible one-edge swaps and commits the best improving swap. CIFAR optimizer SGD with learning rate 0.05, batch size 32, and one local epoch per communication round. Speech optimizer SGD with learning rate 0.05, batch size 32, and one local epoch per communication round. Sentiment140 optimizer Adam with learning rate 10−310^-3, batch size 16, and two local epochs per communication round. Training rounds 300 rounds for CIFAR-10, CIFAR-100, and Sentiment140; 100 rounds for Google Speech Commands. CIFAR preprocessing Random crop with padding 4, random horizontal flip, tensor conversion, and dataset-specific normalization. Speech preprocessing 10-keyword subset of Speech Commands using 64-bin Mel spectrograms followed by amplitude-to-dB conversion. Variable-length spectrograms are padded within each batch. Sentiment140 preprocessing Binary labels are mapped from 0,4\0,4\ to 0,1\0,1\, followed by balanced subsampling with 12,000 training tweets and 2,000 test tweets. Text is tokenized with the distilbert-base-uncased tokenizer using maximum length 32. Table 5: Model architectures and representation extraction rules. Dataset Model architecture Representation rir_i CIFAR-10 CNN with two convolutional blocks: Conv(3,32)-BN-ReLU, Conv(32,32)-BN-ReLU, MaxPool; Conv(32,64)-BN-ReLU, Conv(64,64)-BN-ReLU, MaxPool; adaptive average pooling to 4×44× 4; MLP classifier 1024→256→101024→ 256→ 10 with dropout 0.3. Flattened final classifier-layer weights. CIFAR-100 Same CNN backbone as CIFAR-10, with the final classifier dimension changed to 100 classes: 1024→256→1001024→ 256→ 100. Mean-reduced final classifier-layer weights, producing a 100-dimensional representation. Speech Commands CNN over log-Mel spectrograms: Conv(1,32)-BN-ReLU, Conv(32,32)-BN-ReLU, MaxPool; Conv(32,64)-BN-ReLU, Conv(64,64)-BN-ReLU, MaxPool; adaptive average pooling to 4×44× 4; MLP classifier 1024→256→101024→ 256→ 10 with dropout 0.3. Flattened final classifier-layer weights. Sentiment140 Embedding-based text classifier with 128-dimensional token embeddings, attention-mask mean pooling, and MLP classifier 128→128→2128→ 128→ 2 with dropout 0.2. Flattened final classifier-layer weights. Compute resources. All experiments were implemented in PyTorch 2.2.2+cu118 with Python 3.12.0 and run in a Python virtual environment on a Windows workstation equipped with an NVIDIA GeForce RTX 4060 Laptop GPU with 8GB of GPU memory. The installed NVIDIA driver version was 560.76. PyTorch used CUDA 11.8, while the system driver reported CUDA 12.6 support. The main experiments were averaged over five random seeds, and scalability and overhead analyses used three random seeds where specified. The additional runtime overhead of LFHE topology updates is reported separately in Appendix I. Appendix G Additional Experimental Results G.1 Results on Additional Datasets To assess behavior beyond CIFAR-10, we evaluate LFHE on CIFAR-100, Google Speech Commands, and Sentiment140. Figure 6 reports the mean test-accuracy trajectories. LFHE remains competitive across all three modalities. On CIFAR-100 it reaches the 0.35 target at round 125 and finishes at 45.19%, ahead of the reported decentralized baselines. On Speech Commands it reaches 0.70 at round 60, jointly earliest with Epidemic Learning, and finishes at 78.76%. On Sentiment140, LFHE reaches 0.65 at round 170 versus 190 for Epidemic Learning, but Epidemic Learning finishes slightly higher (65.45% versus 65.29%). The text setting is therefore a convergence advantage rather than a final-accuracy win. (a) CIFAR-100 (b) Google Speech Commands (c) Sentiment140 Figure 6: Test accuracy (mean ± std) on additional datasets under non-IID data. LFHE is competitive across image, speech, and text tasks; Sentiment140 illustrates that faster target attainment need not imply the best final accuracy. G.2 Direct Comparison with Morph Morph [6] is the closest recent adaptive comparison, but it does not share LFHE’s discovery protocol. We therefore report it as a separate systems comparison rather than fold it into the matched FoF controls. Under a shared three-seed learning protocol, Morph achieves stronger predictive performance than LFHE. Table 6: Three-seed LFHE–Morph comparison under the shared diagnostic protocol. nAUC denotes normalized accuracy area under the learning curve. Method Final accuracy (%) nAUC (%) LFHE 71.37± 2.19 60.57± 1.52 Morph 72.25± 1.92 63.17± 1.23 Morph also reaches 65% accuracy 16.7 rounds earlier on average. We therefore do not claim predictive superiority over Morph. The distinction lies in peer-discovery scope. From N=30N=30 to N=500N=500, LFHE’s measured mean FoF candidate set remains 9.76–11.86 peers per client, whereas Morph’s discovered-peer state grows from 29 to 499 peers per client. The corresponding system-wide discovery entries grow from 480 to 8,000 for LFHE and from 3,480 to 998,000 for Morph. This comparison concerns discovery/control state rather than total communication traffic: broader discovery can improve peer selection, and the present evidence does not show that LFHE uses less total communication overall. G.3 Update-Order and Endpoint Effects The reference simulator serializes topology changes, so we test fixed, reverse, and independently randomized per-update client orders on a controlled CIFAR-10 seed while holding the data partition, initialization, training configuration, candidate generation, acceptance rule, and topology-update frequency fixed. Table 7: Single-seed update-order diagnostic. Edge Jaccard is measured against the final graph from fixed order. Order Final acc. (%) nAUC (%) R@65 Final λ2 _2 Edge Jaccard Fixed 69.03 58.82 160 0.936 1.000 Reverse 70.54 60.29 145 0.685 0.291 Random/update 69.61 59.24 155 0.807 0.271 All three final graphs remain connected. The exact topology is substantially order-dependent, whereas the observed final accuracies remain within 1.51 percentage points in this diagnostic. Acceptance is also one-sided: at least one affected non-initiator endpoint has a negative local-fitness change in 33.3%, 31.5%, and 41.5% of accepted operations for fixed, reverse, and randomized processing, respectively; the added-endpoint rates are 33.3%, 29.6%, and 39.0%. Thus, positive initiator gain should not be interpreted as bilateral utility improvement. G.4 Sensitivity and Communication Trade-off We use a separate three-seed CIFAR-10 diagnostic to vary one LFHE design factor at a time while keeping the remaining protocol fixed. Varying either the structural coefficient ω1 _1 or exploration coefficient ω2 _2 from 0.5×0.5× to 2×2× its default keeps final accuracy within 78.01–78.32%, compared with 78.11± 1.84% for the default, and keeps R@65 within 71.7–73.3 rounds. Halving the degree coefficient from 0.1 to 0.05 gives 77.64± 1.46% and R@65 of 71.7. A slower exponential decay gives 78.19% mean accuracy, 67.21% nAUC, and R@65 of 71.7, compared with 78.11%, 67.23%, and 73.3 for the reference schedule. These measurements support a locally stable region around the chosen coefficients; they do not establish cross-dataset hyperparameter invariance. The degree threshold exposes a clearer resource trade-off. Increasing the online threshold from 4 to 6 raises three-seed mean accuracy from 78.11% to 79.70± 1.28%, nAUC from 67.23% to 69.38%, and final λ2 _2 from 1.086 to 1.942, while reducing R@65 from 73.3 to 63.3 rounds. In the seed-42 communication audit, model transmissions increase from 37,810 to 52,540, a 39% increase. Thus, additional degree capacity improves this diagnostic’s learning metrics at a material communication cost. G.5 Moderate-Scale Accuracy Study We report a preliminary three-seed CIFAR-10 study with α=0.3α=0.3 and N∈10,50,100N∈\10,50,100\. Figure 7 reports final test accuracy. LFHE has the highest mean final accuracy among the plotted methods at these three scales. Because the experiment stops at N=100N=100, uses only three seeds, and does not normalize the comparison by total communication, it is evidence of moderate-scale behavior rather than a large-system scaling law. Figure 7: Moderate-scale CIFAR-10 diagnostic under α=0.3α=0.3 with N∈10,50,100N∈\10,50,100\. Bars report final test accuracy averaged over three seeds. G.6 Deployment-Stress Diagnostic We test one compounded communication-disruption condition on seed 42 without retuning LFHE: 80% client participation, 10% link-failure probability, and buffered 0–1-round message delay. The graph remains connected throughout the run and the minimum observed λ2 _2 is 0.670, despite 2,472 dropped model messages, 147 dropped control messages, and 636 delayed or stale control messages. Final accuracy decreases from 79.53% in the clean diagnostic to 60.18% under stress. This result separates topological operation from predictive robustness. The observed topology remains connected, but learning performance degrades substantially. Because this is a single-seed compounded stress test, it is preliminary evidence only and does not establish robustness to arbitrary asynchrony, long delays, or real distributed failures. G.7 Additional Consensus and Client-Variance Diagnostics Figure 8 reports model-consensus and client-accuracy-variance diagnostics for LFHE on CIFAR-10 with α=0.1α=0.1. Model divergence decreases and mean pairwise cosine similarity rises during training; client-level accuracy variance also declines after the early phase. These are descriptive within-method trajectories. Without matched causal interventions, they should not be interpreted as proving that a particular topology statistic causes the learning improvement. (a) Consensus dynamics. (b) Client accuracy variance. Figure 8: Additional LFHE diagnostics on CIFAR-10 with α=0.1α=0.1. Left: model-consensus quantities. Right: client-level test-accuracy variance. Appendix H Bridge Evidence Under Matched Rewiring Budget To more directly test whether the proposed structural term is associated with improved communication structure rather than topology churn alone, we conduct a controlled bridge experiment on CIFAR-10 under highly non-IID data (α=0.1α=0.1). The goal is not to re-evaluate the full learning system against all baselines, but to isolate how different local rewiring objectives affect topology quality and optimization behavior under a matched local update protocol. H.1 Protocol We compare five methods: Static, Similarity-only, Exploration-only, Structural-only, and LFHE. All adaptive variants use the same FoF candidate-generation rule, the same maximum rewiring budget per update, the same initial random graph, and the same training and aggregation protocol. Thus, the compared methods differ only in the local score used to rank candidate updates; candidate access, update frequency, and communication budget are held fixed. The compared local rewiring scores are Similarity-only: Si−ω3deg(i)Dmax,Exploration-only: Di−ω3deg(i)Dmax,Similarity-only: S_i- _3 (i)D_ , -only: D_i- _3 (i)D_ , Structural-only: Ci−ω3deg(i)Dmax,LFHE: ω1(1−βt)Ci+ω2βtDi−ω3deg(i)Dmax.Structural-only: C_i- _3 (i)D_ , : _1(1- _t)C_i+ _2 _tD_i- _3 (i)D_ . All methods therefore operate under the same local rewiring mechanism and differ only in the objective used to evaluate candidate updates. H.2 Results Table 8 summarizes the matched-budget bridge evidence, while Figures 9 and 10 show the full trajectories. Because all adaptive variants use the same candidate-generation rule, rewiring budget, and training protocol, the observed differences are attributable to the local rewiring objective rather than to protocol asymmetries. Table 8: Summary of matched-budget bridge evidence on CIFAR-10 with α=0.1α=0.1. Final λ2 _2 reports the final algebraic connectivity. Δλ2 _2 is computed relative to the shared initial graph used by all matched-budget variants. Mean divergence is computed over the last 20% of evaluation rounds. Method Final λ2 _2 Δλ2 _2 Final Acc. Mean Div. Mean Rewires Static 0.592 0.000 0.6675 0.766 0.000 Similarity-only 0.578 -0.014 0.6704 0.727 0.183 Exploration-only 1.038 0.447 0.7356 0.221 3.406 Structural-only 2.306 1.715 0.7508 0.136 1.111 LFHE 2.247 1.655 0.7482 0.133 1.333 First, the two objectives containing the structural term (Structural-only and LFHE) produce the largest gains in the offline algebraic-connectivity diagnostic. Structural-only is slightly stronger than full LFHE in this regime: final λ2 _2 is 2.3062.306 versus 2.2472.247, and final accuracy is 0.75080.7508 versus 0.74820.7482. This identifies the structural term as the main empirical signal in the matched protocol and rules out a claim that the complete objective is universally optimal. Second, the connectivity gains are not explained by rewiring frequency alone. As shown in Table 8 and Figure 10, Exploration-only performs the most accepted rewires on average (3.4063.406 per update), yet it does not achieve the strongest connectivity, the lowest model divergence, or the best final accuracy. Conversely, Structural-only and LFHE achieve substantially stronger topology and optimization outcomes with far fewer rewires. This indicates that the benefit comes from which local objective guides rewiring, rather than from topology churn alone. Taken together, these matched score controls show that topology churn alone does not explain the observed structure and that CiC_i is the dominant score component in this regime. The matched score-factor diagnostic isolates the effect of the local scoring rule under shared FoF access; the next subsection adds score-free Random-FoF, degree-preserving random-swap, and representation-free graph-only controls to test broader alternative explanations. None of these results implies that Ci(t)C_i(t) monotonically tracks λ2 _2. Figure 9: Bridge evidence under matched FoF candidate generation and rewiring budget on CIFAR-10 with α=0.1α=0.1. Methods containing the structural term (Structural-only and LFHE) produce substantially larger and more stable gains in algebraic connectivity than Similarity-only or Exploration-only. The same methods also achieve lower model divergence and stronger test accuracy under the same local update protocol. Figure 10: Accepted rewires per topology update under the same maximum rewiring budget. Exploration-only performs more rewiring than LFHE or Structural-only during much of training, but does not achieve the strongest connectivity or accuracy. This shows that the advantage of structural-term methods is not explained by rewiring frequency alone. H.3 Additional Simple Controls The score-factor study above is complemented by simpler controls designed to test alternative explanations. Across three seeds, a matched Random-FoF variant reaches 74.62±2.25%74.62± 2.25\% final accuracy, close to LFHE’s 74.82%74.82\% in this diagnostic, but has lower final algebraic connectivity (1.6961.696 versus 2.2472.247). On seed 42, Random-FoF commits 81 accepted operations and reaches 73.24% final accuracy with 61.26% nAUC. A degree-preserving random-swap diagnostic accepts 297 double-edge swaps but reaches only 66.24% accuracy. Conversely, a representation-free graph-only control reaches a higher final λ2 _2 of 2.403 but only 73.52% accuracy. These controls rule out two simple explanations. First, more topology activity is not sufficient for better learning. Second, maximizing a graph-only connectivity diagnostic is not sufficient either. The Random-FoF result is also a useful caution: its final accuracy is close to LFHE in the matched regime, so the evidence supports a structural difference more strongly than a large universal accuracy gap. We therefore avoid attributing all LFHE gains uniquely to its score. H.4 Interpretation This experiment complements the removal ablation by holding FoF access fixed and varying only the score. Together with Appendix D, it separates two claims: CiC_i has an exact state-dependent Dirichlet-energy interpretation, and the matched experiment shows that using this signal produces the strongest measured topology change in this regime. Structural-only can still outperform the full objective, so exploration and degree control are best understood as search and resource regulators rather than theoretically necessary components of every optimal update. Appendix I Cost Analysis In this appendix, we analyze the computational overhead of LFHE-based topology optimization in the decentralized training pipeline. Our implementation records both wall-clock topology-update time and a set of control-overhead statistics, including neighbor-list reads, FoF candidate checks, fitness evaluations, accepted rewires, and edge changes. These quantities are accumulated during training and summarized across multiple random seeds. I.1 Measurement Protocol The cost analysis follows the main CIFAR-10 configuration with N=30N=30, T=300T=300, local epoch E=1E=1, topology interval K=5K=5, and online degree threshold Dmax=4D_ =4. LFHE is therefore executed every five communication rounds, resulting in 6060 topology update calls per seed over the full training process. For each topology update, the implementation measures the elapsed wall-clock time of LFHE-Update: τt=tend−tstart, _t=t_end-t_start, where tstartt_start and tendt_end are recorded immediately before and after the topology update routine. The total topology overhead and amortized per-round overhead are defined as τtotal=∑t∈updτt,τround=τtotalT, _total= _t _upd _t, _round= _totalT, where updT_upd denotes the set of update rounds. In addition to runtime, we record several control-overhead statistics: • Neighbor-list reads: the number of accesses to local neighborhood structure; • Candidate checks: the number of FoF candidates examined; • Fitness evaluations: the number of node-level fitness computations; • Accepted rewires: the number of candidate rewiring operations that are committed; • Edge changes: the total number of edge insertions/removals between consecutive graph states. I.2 Asymptotic Cost Discussion LFHE restricts discovery to ego-network and FoF information. Let d¯=maxidi d= _id_i denote the maximum degree actually present during an update, including any degree inherited from the initializer. The candidate-inspection cost is bounded by O(∑i=1Ndi2)⊆O(Nd¯2).O\! ( _i=1^Nd_i^2 ) O(N d^2). If the initializer and online protocol keep d¯ d bounded independently of N, the control computation is linear in the number of clients. Because the reference initializer is specified by expected degree rather than a hard cap, this conditional statement is more precise than replacing d¯ d unconditionally by DmaxD_ . In addition, each candidate test only requires a local fitness computation based on the current representation of a node and its neighbors. Hence, LFHE avoids repeated global eigendecomposition or full-graph optimization during training. I.3 Observed Runtime Overhead Across seeds 42, 43, and 44, the mean total topology-update time is 1.886±0.0381.886± 0.038 seconds for the entire 300-round training run. When amortized over all communication rounds, this corresponds to only 0.00629±0.000130.00629± 0.00013 seconds per round. Since topology evolution is triggered once every five communication rounds, the mean time per topology update is 0.0314±0.00060.0314± 0.0006 seconds. These measurements show small simulator-side topology-update time on the reported workstation. They do not include real network latency or all deployment control traffic, so they should be interpreted as implementation overhead for this synchronous simulator rather than an end-to-end systems cost. Figure 11: Per-update LFHE topology overhead across communication rounds. Thin lines show individual seeds and the bold line shows the mean across seeds. The runtime remains stable throughout training, indicating that LFHE does not incur increasing topology-update cost in later rounds. As shown in Figure 11, the wall-clock cost of LFHE updates fluctuates slightly across rounds and seeds but remains within a narrow range. This suggests that the local rewiring mechanism does not become progressively more expensive as training proceeds. I.4 Control Overhead The control statistics reveal that LFHE performs bounded local exploration while committing only a small number of structural updates. Averaged across seeds, LFHE performs 353.78±48.55353.78± 48.55 candidate checks and the same number of fitness evaluations per update, while accepting only 0.339±0.1220.339± 0.122 rewires per update. Over the full training run, this corresponds to 21,226.67±2,913.0721,226.67± 2,913.07 candidate checks, 21,226.67±2,913.0721,226.67± 2,913.07 fitness evaluations, and 20.33±7.3220.33± 7.32 accepted rewires. The total number of neighbor-list reads is 8,326±562.758,326± 562.75, and the total number of edge changes is 43.33±15.8443.33± 15.84. These results indicate that LFHE explores a large number of local candidates but commits only a very small subset of rewiring operations. This behavior is consistent with a conservative accept-if-improves policy, which favors stability over aggressive topology modification. (a) Candidate checks per topology update. (b) Accepted rewires per topology update. Figure 12: Control overhead of LFHE during topology evolution. Left: Candidate checks increase in the early stage and then stabilize, showing a stable candidate-check workload over later training rounds. Right: Accepted rewiring operations are concentrated in early rounds and become rare as the topology stabilizes. Figure 12 further illustrates this behavior. As shown in Figure 12(a), the number of candidate checks rises during the early phase of training and then stabilizes at around 350350 checks per update. This shows that the recorded candidate-check workload remains approximately stable over later training rounds in these runs; it should not be interpreted as an empirical scaling law in N. Figure 12(b) shows a complementary trend: accepted rewires are more frequent in the early stage but become rare in later rounds, consistent with a stabilizing communication topology. I.5 Acceptance Ratio A useful empirical measure of update selectivity is AcceptRate=# accepted rewiresmax(# candidate checks,1).AcceptRate= \# accepted rewires (\# candidate checks,1). Using the full-run averages, this yields an efficiency on the order of 10−310^-3, indicating that only a very small fraction of evaluated candidates are accepted. Only a small fraction of inspected candidates is therefore committed. This is consistent with a conservative accept-if-improves rule at the initiator, but “accepted” should not be read as globally or bilaterally beneficial. Taken together, Figures 12(a) and 12(b) show that LFHE performs selective rather than aggressive topology modification: it continues to inspect local candidates, but commits only those rewiring operations that improve node-level fitness under the imposed constraints.