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Population Scaling or Data Dilution? Dynamics of Local Topology Evolution in Decentralized Learning
Yin-Kuan Liang, Yan Gao, Yang Long
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Abstract
Abstract:Scaling decentralized learning changes not only the number of clients $N$, but also the dynamics of information propagation and consensus. We argue that the effect of increasing $N$ cannot be understood in isolation, because data allocation, topology-dependent mixing, and communication capacity may change simultaneously. We study these coupled effects on CIFAR-10 with $N\in\{10,50,100,200\}$, comparing a degree-two Ring, a Static Random graph, and Local-First Heuristic Evolution (LFHE), a locally adaptive topology process based on friend-of-friend discovery. The Ring provides an analytically transparent failure mode: its Metropolis spectral gap decays as $\Theta(N^{-2})$, implying progressively slower contraction of model disagreement as the population grows. Experiments show that holding the nominal local dataset size fixed substantially reduces the apparent population penalty observed when a fixed total dataset is divided among more clients. The remaining degradation depends strongly on communication structure: Ring enters a high-disagreement regime, whereas Static Random and LFHE remain close to consensus. Increasing LFHE's degree threshold further improves accuracy and consensus, but at a substantially higher model-transmission cost. These results show that decentralized scaling is governed by coupled learning and communication dynamics, rather than by the number of clients alone.
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- Source: https://arxiv.org/abs/2610.05476v1
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Population Scaling or Data Dilution? Dynamics of Local Topology Evolution in Decentralized Learning Yin-Kuan Liang 1 Yan Gao 2 Yang Long 1 1 Durham University 2 University of Cambridge Abstract Scaling decentralized learning changes not only the number of clientsN, but also the dynamics of information propagation and consensus. We argue that the effect of increasingNcannot be understood in isolation, because data allocation, topology-dependent mixing, and communication capacity may change simultaneously. We study these coupled effects on CIFAR-10 withN ∈ 10,50,100,200, comparing a degree-two Ring, a Static Random graph, and Local-First Heuristic Evolution (LFHE), a locally adaptive topology process based on friend-of-friend discovery. The Ring provides an analytically transparent failure mode: its Metropolis spectral gap decays as Θ(N −2 ), implying progressively slower contraction of model disagreement as the population grows. Experiments show that holding the nominal local dataset size fixed substantially reduces the apparent population penalty observed when a fixed total dataset is divided among more clients. The remaining degradation depends strongly on communication structure: Ring enters a high-disagreement regime, whereas Static Random and LFHE remain close to consensus. Increasing LFHE’s degree threshold further improves accuracy and consensus, but at a substantially higher model-transmission cost. These results show that decentralized scaling is governed by coupled learning and communication dynamics, rather than by the number of clients alone. 1 Introduction Decentralized learning replaces server aggregation with peer-to-peer model exchange, building on distributed optimization over communication graphs [4,6,7,11]. Without a central coordinator, optimization becomes directly coupled to the communication topology: the graph structure and mixing weights determine how rapidly information propagates and local models approach consensus [2,8,9]. When the topology itself adapts during training, this dependence becomes bidirectional. Model states influence which edges are created or removed, the resulting graph determines subsequent mixing, and mixing in turn changes the model states that guide future topology updates. Understanding how such systems scale with the number of clientsNis difficult because several resources may change simultaneously. Under a fixed total dataset, increasingNenlarges the communication network while reducing the amount of data assigned to each client. Adaptive sparse graphs introduce a second scaling dimension: if their admissible degree grows withN, larger populations also receive greater communication and mixing capacity. Consequently, an observed scaling curve may reflect data allocation, topology-dependent mixing, communication capacity, or a combination of all three. Distinguishing these effects is essential because the convergence of decentralized optimization depends directly on the nontrivial spectrum of the communication operator [1,2,9,12]. Recent work has explored adaptive peer selection and communication topology under heterogeneous data, including performance-based neighbor selection [10] and diversity-driven rewiring [3]. Our focus is complementary: we study how population size, data allocation, and communication capacity jointly shape the dynamics of decentralized learning. We compare three communication structures: a degree-two Ring, which provides an analytically transparent mixing bottleneck; a Static Random graph, which serves as a fixed sparse reference; and Local-First Heuristic Evolution (LFHE), a locally adaptive topology process based on two-hop friend-of-friend (FoF) discovery. For LFHE, we separately study a fixed degree threshold and a population-scaled threshold, allowing us to distinguish the effects of population growth from those of increasing communication capacity. Our contributions are threefold. First, we show that the apparent population penalty depends strongly on data allocation. For LFHE, the accuracy loss fromN= 50 toN= 200 decreases from 14.14 to 3.68 percentage points when moving from a fixed-total regime to one with a fixed nominal local dataset size. Second, we characterize a locality–communication trade-off: bounded local degree limits communication, but causes the 1 arXiv:2610.05476v1 [cs.LG] 4 Oct 2026 Figure 1: LFHE mechanism. Clientidiscovers a two-hop FoF candidatek, evaluates a locally improving addition or edge replacement, and sequentially commits accepted updates. fraction of the population visible through two-hop discovery to shrink withN; increasing the degree threshold expands both local search and mixing capacity, at a rapidly growing communication cost. Third, topology, model-disagreement, and communication-normalized trajectories reveal how these resource choices produce distinct graph–model regimes. Together, these results shift the scaling question from how accuracy changes with N to which data and communication resources must scale with N to preserve effective decentralized learning. 2 Local Topology Evolution Under Scaling LetG t = (V,E t ) denote the communication graph andW t = (w 1,t ,...,w N,t ) the client models. Each communi- cation round performs local optimization followed by graph-dependent mixing, f W t =L(W t ;ξ t ), W t+1 = P (G t ) f W t .(1) EveryH= 5 rounds, LFHE additionally appliesG t+1 =R t (G t ,W t+1 ); otherwise the graph is unchanged. The learning system is therefore coupled:G t determines model mixing, whileW t provides the local signals used to evolve the topology. Local topology evolution. Client i evaluates f i = (1− β t )C i + β t D i − 0.1d i /D max , β t = e −0.01t ,(2) whereC i measures representation disagreement with current neighbors andD i measures cosine mismatch with their mean representation. Client i samples h∈N i , then k ∈N h \ (N i ∪i). Improving candidates are added when online degree conditions permit; otherwise LFHE evaluates a one-edge replacement. At each topology epoch, initiators are processed once in a fixed sequential order; each initiator evaluates one FoF candidate and commits at most one accepted add-or-swap operation before the next initiator observes the updated graph. The thresholdD max regulates online edge acquisition and therefore controls the capacity available to topology evolution. Local visibility. If ̄ d t = max i d i (G t ), the two-hop candidate set satisfies |C (2) i (t)|≤ d i (t)( ̄ d t − 1), |C (2) i (t)| N − 1 ≤ ̄ d t ( ̄ d t − 1) N − 1 .(3) With bounded local degree, relative FoF visibility therefore shrinks asO(N −1 ). For example, withD max = 4, a client can access at most 12 two-hop candidates, corresponding to at most 24.5% of the population atN= 50, 12.1% atN= 100, and 1.2% atN= 1000. This formalizes a locality–coverage tension: bounded degree controls communication, but the fraction of the population visible through two-hop discovery vanishes asNgrows. Increasing degree expands the search frontier but also increases communication, making population size and degree budget distinct scaling axes. 2 Mixing and disagreement. For symmetric Metropolis–Hastings mixingP t =P(G t ), letJ= 1 N 11 ⊤ denote global averaging and define ρ t =∥P t − J∥ 2 . Then one communication step satisfies ∥(I − J )P t X∥ F ≤ ρ t ∥(I − J )X∥ F ,(4) soρ t directly controls how quickly model disagreement contracts: smallerρ t implies faster mixing [1,2,9,12]. Under non-IID data, local optimization continually reintroduces disagreement, making the learning dynamics a balance between local drift and network mixing. This dependence becomes explicit for a degree-two Ring, whose mixing eigenvalues are μ k = 1 + 2 cos(2πk/N ) 3 ,1− μ 1 = 2 3 1− cos 2π N = Θ(N −2 ).(5) Intuitively, information must travel around an increasingly long cycle, so local updates are averaged across the population progressively more slowly. Hence, asNgrows, the Ring spectral gap vanishes quadratically and disagreement contracts increasingly slowly. The Ring therefore provides a concrete population-dependent mixing failure mode. Static Random offers a contrasting sparse regime with approximately constant local degree but without the deterministic cycle bottleneck. 3 Controlled Population and Degree Scaling We use CIFAR-10 [5] with class-wise Dirichlet heterogeneityα= 0.3,N ∈10,50,100,200, and seeds42,43,44. Training runs for 300 communication rounds with one local epoch per round, SGD with learning rate 0.05, and batch size 32. Evaluation and LFHE topology updates occur every five rounds. We report means and sample SD; further details appear in Appendix B. We compare Ring, a fixed cycle; Static Random, a fixed connected sparse graph; and LFHE. Static Random and LFHE share seed-matched connected Erdős–Rényi initializations with expected degree four; all methods use degree-aware Metropolis–Hastings mixing. Data and degree regimes. Under fixed-total, the 50,000 CIFAR-10 training examples are partitioned across the population, so per-client data decreases withN. Under fixed-per-client, each client instead receives a local dataset of 500 sampled indices, keeping local dataset size constant as the population grows. Full sampling semantics are given in Appendix B. LFHE uses either a fixed online threshold D max = 4 or the population-dependent schedule D max = max(2,⌊2 lnN⌋) =4, 7, 9, 10. The fixed-threshold setting tests population growth under approximately constant local communication capacity; the scaled-threshold setting measures the benefit and cost of increasing that capacity with population. Model disagreement and cumulative directed model transmissions are D t = 1 N N X i=1 θ i,t − 1 N X j θ j,t 2 2 , M (T ) = 2 X t<T |E t |.(6) Communication accounting includes directed model exchange; topology-control traffic and wall-clock latency are not included in this metric. 4 What Drives Population Scaling? Data allocation changes the scaling regime. With fixed-total data, fixed-D max LFHE drops from 74.65% to 68.64% overN= 50→100 and from 68.64% to 60.51% overN= 100→200, losses of 6.01 and 8.13 points. Under fixed-per-client training, the corresponding losses are 1.71 points (70.52%→68.81%) and 1.97 points (68.81%→66.84%). Preserving local dataset size therefore reduces the apparent penalties by 4.30 and 6.16 points, respectively. 3 1050100200 Clients, N 50 60 70 80 Aligned checkpoint accuracy (%) (a) Fixed-total data 1050100200 Clients, N 50 60 70 80 Final accuracy (%) (b) Fixed-per-client data 1050100200 Clients, N 62.5 65.0 67.5 70.0 72.5 75.0 Final accuracy (%) (c) LFHE degree budget 200400 Directed transmissions 1e3 0.1 0.2 0.3 0.4 0.5 0.6 Final disagreement N=50 N=100 N=200 N=50 N=100 N=200 (d) Consensus-communication RingStatic RandomLFHE, fixed Dmax=4LFHE, scaled degree Figure 2: Controlled scaling on CIFAR-10. (a) Fixed-total population scaling; (b) fixed-per-client scaling withD max = 4; (c) LFHE under fixed and population-scaled degree thresholds; (d) disagreement versus realized communication. Panel (a) uses the aligned cross-method checkpoint; the remaining panels report final endpoints. Error bars show sample SD over seeds 42–44. Figure 3: Dynamics atN= 100 under fixed-per-client training. (a) Ring enters a high-disagreement regime, while Static Random and fixed-D max LFHE remain near consensus. (b) Communication-normalized accuracy: fixed-D max LFHE leads within the shared budget, while scaled-threshold LFHE reaches a higher endpoint at greater cost. Residual scaling depends on communication structure. Under fixed-per-client training, Ring falls from 61.34% to 44.46% over N = 10→ 200, while Static Random remains comparatively stable (64.95%→ 63.24%). Fixed-D max LFHE reaches 66.84% atN= 200, 3.60 points above Static Random with approximately 9.7% more model transmissions. Disagreement trajectories distinguish the same regimes: Ring moves toward high disagreement, whereas Static Random and LFHE remain near consensus (Figure 3a); topology trajectories are reported in Appendix E. Communication capacity sets a second scaling frontier. Increasing LFHE’s degree threshold raises accuracy from 70.52% to 72.49% atN= 50, from 68.81% to 72.31% atN= 100, and from 66.84% to 72.04% at N= 200, while model transmissions increase by 47.1%, 88.1%, and 104.1%, respectively (Appendix F). At a common 115.8k-transmission budget, fixed-D max LFHE reaches 67.77% versus 65.98% for Static Random, whereas scaled-threshold LFHE reaches its higher endpoint only after approximately 247k transmissions (Figure 3b). Additional degree therefore extends the attainable learning regime at increasing communication cost. 5 Discussion and Conclusion Two resource effects shape decentralized scaling. First, data allocation: preserving per-client data reduces LFHE’s losses from 6.01 to 1.71 points overN= 50→100 and from 8.13 to 1.97 overN= 100→200, revealing different behavior across structures. Ring deteriorates, Static Random remains comparatively stable, and LFHE maintains higher accuracy with low disagreement. The second effect is communication capacity. Increasing LFHE’s degree threshold improves mixing and raises the achievable endpoint, but the benefit comes with rapidly increasing model exchange. The fixed- threshold results show that LFHE retains a performance advantage over Static Random even when their realized communication is close, while the scaled-threshold results expose how additional degree can trade communication for accuracy. 4 Together, these findings suggest a broader view of decentralized scaling. Population size is not an isolated systems variable: its effect depends on how data, topology, and communication resources scale with it. For the tested finite-population regime, preserving local data substantially reduces the apparent population penalty, topology determines the remaining consensus pressure, and additional degree determines how far that pressure can be reduced. Scaling studies should therefore report these resources jointly rather than summarize behavior through client count and endpoint accuracy alone. References [1]Stephen Boyd, Persi Diaconis, and Lin Xiao. Fastest mixing markov chain on a graph. SIAM Review, 46(4): 667–689, 2004. doi: 10.1137/S0036144503423264. [2]Stephen Boyd, Arpita Ghosh, Balaji Prabhakar, and Devavrat Shah. Randomized gossip algorithms. IEEE Transactions on Information Theory, 52(6):2508–2530, 2006. doi: 10.1109/TIT.2006.874516. [3] Bart Cox, Antreas Ioannou, and Jérémie Decouchant. Dynamic topology optimization for non-iid data in decentralized learning. arXiv preprint arXiv:2602.03383, 2026. [4]Anastasia Koloskova, Nicolas Loizou, Sadra Boreiri, Martin Jaggi, and Sebastian Stich. A unified theory of decentralized SGD with changing topology and local updates. In Proceedings of the 37th International Conference on Machine Learning, volume 119 of Proceedings of Machine Learning Research, pages 5381–5393. PMLR, 2020. [5]Alex Krizhevsky. Learning multiple layers of features from tiny images. Technical report, University of Toronto, 2009. [6]Xiangru Lian, Ce Zhang, Huan Zhang, Cho-Jui Hsieh, Wei Zhang, and Ji Liu. Can decentralized algorithms outperform centralized algorithms? a case study for decentralized parallel stochastic gradient descent. In Advances in Neural Information Processing Systems, volume 30, 2017. [7]Angelia Nedić and Asuman Ozdaglar. Distributed subgradient methods for multi-agent optimization. IEEE Transactions on Automatic Control, 54(1):48–61, 2009. doi: 10.1109/TAC.2008.2009515. [8]Angelia Nedić, Alex Olshevsky, and Michael G. Rabbat. Network topology and communication-computation tradeoffs in decentralized optimization. Proceedings of the IEEE, 106(5):953–976, 2018. doi: 10.1109/ JPROC.2018.2817461. [9]Reza Olfati-Saber, J. Alex Fax, and Richard M. Murray. Consensus and cooperation in networked multi-agent systems. Proceedings of the IEEE, 95(1):215–233, 2007. doi: 10.1109/JPROC.2006.887293. [10]Noa Onoszko, Gustav Karlsson, Olof Mogren, and Edvin Listo Zec. Decentralized federated learning of deep neural networks on non-iid data. arXiv preprint arXiv:2107.08517, 2021. [11] John N. Tsitsiklis, Dimitri P. Bertsekas, and Michael Athans. Distributed asynchronous deterministic and stochastic gradient optimization algorithms. IEEE Transactions on Automatic Control, 31(9):803–812, 1986. doi: 10.1109/TAC.1986.1104412. [12]Lin Xiao and Stephen Boyd. Fast linear iterations for distributed averaging. Systems & Control Letters, 53 (1):65–78, 2004. doi: 10.1016/j.sysconle.2004.02.022. 5 A Implementation-Faithful LFHE For the controlled-scaling CIFAR-10 experiments, LFHE uses a 10-dimensional summary of the final classifier weight matrix as the client representation. LetW t i ∈R 10×256 denote clienti’s final classifier weight matrix. The implemented representation is z t i = 1 256 W t i 1 256 , obtained by averaging each output class’s 256 incoming weights. Its local terms are C i (G,t) = X j∈N i (G) ∥z t i − z t j ∥ 2 2 ,(7) D i (G,t) = 1− cos z t i , 1 |N i (G)| X j∈N i (G) z t j ,(8) with f i (G,t) = (1− β t )C i (G,t) + β t D i (G,t)− 0.1 d i (G) D max , β t = e −0.01t . The structural term is representation-dependent and is not a direct estimator of algebraic connectivity. At each topology epoch, initiators are processed once in a fixed sequential order. Each initiator samples h∼ Unif(N i ), k ∈N h \ (N i ∪i), and evaluates that FoF candidate against its current graph state. A simple addition is available when the relevant online degree conditions are satisfied; otherwise, LFHE evaluates feasible one-edge replacements. Each initiator commits at most one accepted add-or-swap operation before processing moves to the next client, which then observes the already updated graph. The implementation therefore uses sequential rather than concurrent topology commits. The thresholdD max regulates online edge acquisition. Because the connected Erdős–Rényi initializer is specified by expected degree, some nodes may initially exceed the threshold; these degrees are preserved unless subsequently changed by topology updates. Consequently, fixed-D max = 4 runs have mean degree near 4.4, while endpoint maximum degree can exceed four. B Experimental Details Controlled scaling uses CIFAR-10 [5], class-wise Dirichletα= 0.3,N ∈10,50,100,200, seeds42,43,44, 300 communication rounds, one local epoch per round, SGD with learning rate 0.05, batch size 32, evaluation every five rounds, and LFHE topology updates every five rounds. All methods use the same CIFAR-10 convolutional model: two convolutional stages with 32 and 64 channels, batch normalization and max pooling, followed by a 256-dimensional fully connected hidden representation, dropout 0.3, and a 10-way classifier. In the controlled-scaling runs, LFHE uses the 10-dimensional class-wise mean of the final classifier weight matrix as its topology representation signal. The separate matched-FoF diagnostic uses the complete flattened 10×256 final classifier weight matrix. The architecture, preprocessing pipeline, and optimizer configuration are held identical across Ring, Static Random, and LFHE. The controlled conditions vary data allocation and communication topology; because training uses one local epoch, the data regime also determines the per-round local SGD workload. Ring uses a cycle. Static Random and LFHE use seed-matched connected Erdős–Rényi initializations with expected degree four. For neighboring clients, a ij = 1 1 + max(d i ,d j ) , a i = 1− X j∈N i a ij . Under fixed-total training, all 50,000 CIFAR-10 training examples are partitioned across the population. Under fixed-per-client training, each client draws 500 local-dataset indices only from its own Dirichlet pool and reuses the resulting local dataset throughout training. This holds the nominal number of local training entries 6 fixed at 500 per client. Sampling is without replacement when the client’s pool contains at least 500 indices and with replacement otherwise; repeated indices therefore do not introduce data from another client’s pool. The two LFHE threshold regimes are fixed: D max = 4,scaled: D max = max(2,⌊2 lnN⌋) =4, 7, 9, 10. The controlled fixed-threshold runs store exact graph snapshots every five communication rounds together with the final topology. Algebraic connectivity is computed from the unweighted combinatorial Laplacian L=D− Aof these exact edge lists. Historical scaled-threshold runs without evolving adjacency snapshots are not used to construct spectral trajectories. Model disagreement is D t = 1 N N X i=1 θ i,t − ̄ θ t 2 2 , ̄ θ t = 1 N X i θ i,t . All uncertainty values reported for the controlled study are sample SD over the three paired seeds. Table 1: Matched-FoF diagnostic atN= 30,α= 0.1, over seeds 42–44. Runs use 300 rounds, topology updates every five rounds,D max = 6, and a budget of six proposals per topology update. Adaptive variants share initialization, candidate generation, update frequency, and proposal budget; only the local score changes. ∆λ 2 is measured relative to Static at the final endpoint. Ops./update is the system-wide number of accepted topology operations per update, averaged over updates and seeds. MethodFinal λ 2 ∆λ 2 Final Acc. Final Div. Ops./update Static0.5920.0000.66750.7810.000 Similarity-only0.578 -0.0140.67040.7360.183 Exploration-only 1.0380.4470.73560.2083.406 Structural-only2.3061.7150.75080.1361.111 LFHE2.2471.6550.74820.1351.333 Exploration-only accepts substantially more topology operations per update than Structural-only or LFHE yet reaches lower algebraic connectivity and lower accuracy. Operation frequency alone therefore does not explain the resulting trajectory; changing the local scoring rule materially changes the graph–model regime produced by the same FoF update process. C Controlled Scaling Results Table 2: Final fixed-per-client accuracy (%, mean ± sample SD over seeds 42–44). MethodN = 10N = 50N = 100 N = 200 Ring61.34±0.77 56.00±1.43 51.77±0.57 44.46±0.27 Static Random64.95±1.40 66.69±1.56 66.10±0.43 63.24±0.50 LFHE, fixed (D max = 4) 65.43±1.92 70.52±0.07 68.81±0.35 66.84±0.27 LFHE, scaled threshold 65.43±1.92 72.49±0.77 72.31±0.55 72.04±0.46 The 0.07 sample SD for fixed-D max LFHE at N = 50 is computed directly across the three completed seeds. Table 3: Final fixed-D max = 4 LFHE endpoints at N ∈50, 100, 200 (mean ± sample SD). RegimeMetricN = 50N = 100N = 200 Fixed-total Accuracy (%) 74.65±1.89 68.64±0.59 60.51±0.96 Disagreement 0.474±0.116 0.354±0.004 0.257±0.029 λ 2 0.603±0.142 0.465±0.124 0.523±0.063 Fixed-per-client Accuracy (%) 70.52±0.07 68.81±0.35 66.84±0.27 Disagreement 0.348±0.137 0.445±0.027 0.557±0.030 λ 2 0.550±0.217 0.426±0.076 0.497±0.058 7 Table 4: Final fixed-per-client model disagreement. MethodN = 10 N = 50 N = 100 N = 200 Ring0.43927.48254.39167.414 Static Random0.1290.5780.6110.803 LFHE, fixed (D max = 4) 0.1800.3480.4450.557 LFHE, scaled threshold0.1800.1100.0950.123 D Communication Accounting Table 5: LFHE cumulative directed model transmissions under fixed-per-client training (mean over seeds 42–44). Threshold regime N = 50 N = 100 N = 200 Fixed (D max = 4) 66,710 131,419 265,519 Scaled threshold98,160 247,254 541,831 Increase47.1%88.1%104.1% Table 6: Fixed-per-client communication endpoints atN= 100 andN= 200. Counts include directed model transmissions and exclude separately unlogged topology-control traffic. MethodN = 100 N = 200 Ring60,000120,000 Static Random120,600 242,000 LFHE, fixed (D max = 4) 131,419 265,519 LFHE, scaled threshold 247,254 541,831 AtN= 100 andN= 200, fixed-threshold LFHE uses approximately 9% and 9.7% more model transmissions than Static Random, respectively. Scaling the LFHE threshold from four to nine atN= 100 increases communication by 88.1%, while scaling it from four to ten at N = 200 increases communication by 104.1%. 8 E Graph–Model Dynamics Under a Fixed Degree Budget Figure 4: Graph–model dynamics atN= 100 under fixed-per-client training withD max = 4. LFHE maintains stronger algebraic connectivity than Ring while mean degree remains near 4.4, and its disagreement converges to the low-disagreement regime shared with Static Random. Curves show means over seeds 42–44; shading is sample SD. AtN= 100, LFHE’s algebraic connectivity changes from 0.394 to 0.426 while its mean degree remains near 4.4. Static Random remains near 0.35, whereas the Ring has substantially weaker algebraic connectivity. At the same time, LFHE’s model disagreement contracts from a large initial value to 0.445, Static Random stabilizes near 0.6, and Ring enters a substantially higher-disagreement regime. Structural and optimization changes are concentrated in the early part of training, after which both topology and disagreement become comparatively stable. The exact trajectories show that topology evolution and consensus formation occur on related but non-identical time scales, reinforcing the view of LFHE as a coupled graph–model process rather than a single spectral optimization step. 9 F Degree-Budget Diagnostic 1050100200 Clients, N 64 66 68 70 72 Accuracy (%) (a) Final accuracy 1050100200 Clients, N 0.0 0.2 0.4 0.6 Model disagreement (b) Final disagreement 1050100200 Clients, N 4 5 6 7 8 9 Mean degree (c) Final mean degree 1050100200 Clients, N 0 100 200 300 400 500 Directed transmissions 1e3 (d) Model transmissions LFHE, fixed Dmax=4LFHE, scaled degree Figure 5: LFHE degree-budget sensitivity under fixed-per-client training. Increasing the online threshold produces higher endpoint accuracy and lower disagreement, together with higher realized degree and communication. AtN= 50, changingD max from 4 to 7 raises accuracy from 70.52% to 72.49%, reduces disagreement from 0.348 to 0.110, and increases transmissions from 66,710 to 98,160. AtN= 100, changingD max from 4 to 9 raises accuracy from 68.81% to 72.31%, reduces disagreement from 0.445 to 0.095, and increases transmissions from 131,419 to 247,254. AtN= 200, changingD max from 4 to 10 raises accuracy from 66.84% to 72.04%, reduces disagreement from 0.557 to 0.123, and increases transmissions from 265,519 to 541,831. This comparison quantifies LFHE’s sensitivity to additional degree capacity. The corresponding communication- normalized behavior is reported in Appendix G. G Communication-Normalized Learning Figure 6: Accuracy versus cumulative directed model transmissions atN= 100 under fixed-per-client training. Scaled- threshold LFHE attains the highest endpoint after extending substantially beyond the fixed-threshold communication range. 10 At a common budget of 115.8k transmissions, Static Random, fixed-D max LFHE, and scaled-threshold LFHE achieve 65.98%, 67.77%, and 63.58%, respectively. The transmissions required to first reach 65% accuracy are approximately 106.3k, 88.6k, and 130.7k. Fixed-threshold LFHE therefore reaches stronger learning within the shared communication range, whereas scaled-threshold LFHE trades a larger total budget for a higher eventual endpoint. H Scope and Extensions The controlled study focuses on CIFAR-10, populations up toN= 200, and three paired seeds, with commu- nication measured through directed model transmissions. Within this setting, the experiments characterize how data allocation, mixing structure, local search visibility, and communication capacity shape the observed scaling regime. LFHE uses final classifier weights as its representation signal throughout; evaluating alternative representation layers or learned signals would further characterize the sensitivity of the topology process. Extending the analysis to larger populations and broader learning tasks would test whether the same graph–model regimes persist beyond the finite-population setting studied here. An equally dense fixed-graph comparison would further separate additional connectivity from topology adaptation when degree grows, while concurrent topology updates would extend the present sequential protocol toward asynchronous decentralized execution. Comparisons with additional adaptive-topology processes such as PENS and Morph are also natural extensions under the same controlled data and communication regimes. 11