Freezing Mechanism and Topology Dependence of Information Flow Structure in Small-Scale Particle Swarm Optimization

1. Introduction

• The paper models small-scale PSO with regular topologies and conventional variants to isolate information-flow concentration and examine structural evolution during search.The scope covers 20–50 particles, Gbest, Ring, and Von Neumann, and standard, LDIW, and TVAC variants.Source: §1 · p. 4
• A dynamic weighted influence-graph framework evaluates information flow at micro, meso, and macro levels, with effective nodes, Strength Gini, and Strength Entropy characterizing macro-level distributions.The freeze point is the first sample at which the effective-nodes series remains statistically constant over a sliding window.Source: §1 · pp. 4–6
• Only macro-level distributional metrics effectively capture freezing under regular topologies because micro- and meso-level structures are predetermined by topology.The introduction frames this as a response to limited understanding of topology’s structural influence and gaps in population-level analyses.Source: §1 · pp. 3, 5
• Information flow freezes early and remains unchanged; freeze time is earliest under Gbest, followed by Von Neumann and Ring, across standard, LDIW, and TVAC variants.Parameter-adaptive variants delay freezing without changing that topology order.Source: §1 · pp. 5–6
• The freeze point is significantly negatively correlated with final solution quality, while Gbest interventions show that weakening hubs delays freezing and improves multimodal performance.Strengthening hubs produces no additional deterioration because of saturation, indicating decoupling between structural and performance effects.Source: §1 · pp. 5–7

2.1. PSO Neighborhood Topologies and Convergence Analysis

• Topology studies report differing performance patterns across problem types, while traditional convergence analyses often simplify topology and focus on particle trajectories rather than population-level information flow.Prior comparisons associate Von Neumann with consistency and Gbest with unimodal and separable problems, while Ring is preferred for multimodal, non-separable, and composition problems.Source: §2.1 · pp. 7–8

2.2. Network Science Perspective on PSO Information Flow

• Influence graphs and network metrics capture interactions and diagnose stagnation, while static network analyses describe small-world structure and hubs; systematic study of dynamic freezing and topology dependence remains lacking.The review identifies missing causal evidence linking structural evolution to search performance.Source: §2.2 · pp. 8–9

2.3. Adaptive Topology Mechanisms

• Adaptive-topology methods vary connectivity or restructure neighborhoods, but adjustment rules often rely on heuristics or predefined triggers rather than a structural-state signal.The review identifies detecting when information flow has ceased evolving as a key gap for intervention guidance.Source: §2.3 · pp. 9–10

2.4. Search-State Diagnosis and Stagnation Detection

• Fitness and spatial-diversity diagnostics can miss population interaction structure, while network diagnostics have limited quantitative mapping, early-warning capability, and multiscale analysis.Fitness-based detection may occur only after stagnation, and threshold settings can depend on experience.Source: §2.4 · pp. 10–11

2.5. Summary of Research Gaps

• The review consolidates gaps in characterizing freezing, comparing topology and variant effects on freeze time, establishing causal links to performance, and identifying effective metric levels.These gaps motivate the paper’s analyses of structural evolution and freezing.Source: §2.5 · pp. 11–12

3.1. Problem Description and Modeling Motivation

• For small-scale PSO populations, common network metrics can fluctuate substantially and modularity optimization has a resolution limit, motivating tailored metric screening.The stated typical population size is 20–50 particles.Source: §3.1 · p. 12
• In the influence graph, directed edges are based on the selection of a particle’s neighborhood-best learning exemplar, rather than simply the static communication topology.The PSO update description defines the personal best and neighborhood best used in particle updates.Source: §3.1 · p. 12
• The proposed analysis models particles as a dynamic weighted influence graph and organizes measures into micro, meso, and macro levels.Macro-level distributional metrics support freeze-point detection, while micro- and meso-level metrics provide auxiliary verification.Source: §3.1 · p. 12

3.2.1. Swarm Influence Graph

• A directed edge j→i records when particle i uses particle j’s personal best as a learning exemplar; self-loops represent cognitive influence and the best neighbor represents social influence.Self-loops guarantee that every node has an incident edge and help avoid isolated-node distortions in network metrics.Source: §3.2.1 · p. 13

3.2.2. Weighted Influence Graph with Sliding Window

• A sliding-window weighted influence graph captures short-term interaction statistics that a single-iteration graph misses because it reflects only instantaneous, random exchange.The graph is normalized by its maximum weight to preserve relative edge strength, then symmetrized for undirected graph tools such as Laplacian spectral analysis.Source: §3.2.2 · pp. 13–14

3.2.3. Adaptation for Small-Scale Populations

• For populations of 20–50, disconnected graphs are analyzed on their largest connected subgraph, components are recorded separately, and a 10-iteration window balances stability and temporal resolution.Weights are divided by Wmax rather than the window length to preserve relative edge strength.Source: §3.2.3 · p. 14

3.3. Multi-Scale Information Flow Metrics

• The three metric scales address particle-level influence, subgroup differentiation, and the overall distribution of information flow; core metrics have full formulas in the main text, with auxiliary formulas summarized in Table 2.Table 1 distinguishes core from auxiliary metrics.Source: §3.3 · p. 14, §3.3.4 · p. 17, Table 1 · p. 18

3.3.1. Micro-Scale: Who Dominates the Information Flow?

• Maximum eigenvector centrality emax identifies hub dominance by measuring particles’ global influence; it supplements hub diagnosis, while weighted-degree and clustering measures are auxiliary.For Von Neumann, late-stage disconnection can degrade eigenvector centrality, so the paper does not pursue micro-level analysis for that topology.Source: §3.3.1 · pp. 14–15

3.3.2. Meso-Scale: Has the Population Formed Subgroups?

• Inter-community information flow measures exchange between communities, with values near zero indicating isolated subgroups; under regular topologies, community structure does not evolve, so related metrics mainly verify structural invariance.Under Gbest, modularity approaches zero after freezing, consistent with a star graph’s lack of community structure.Source: §3.3.2 · pp. 15–16

3.3.3. Macro-Scale: Is Information Flow Concentrated? When Does It Freeze?

• Effective nodes Eeff, the inverse Simpson index of node strengths, is the most stable small-network measure and the primary freeze-detection metric; Gs and Hs verify the distributional pattern from other perspectives.Eeff near N indicates uniform flow, while Eeff much smaller than N indicates concentration on a few hubs; Gs is sensitive to middle-range inequality and Hs to the distribution tail.Source: §3.3.3 · pp. 16–17

3.3.4. Summary of Multi-Scale Metrics

• Table 1 identifies each metric’s role, while Eeff is the primary freeze-point basis and Gs and Hs provide robustness checks.The paper points to Figure S2 and Table S2 for a full comparison.Source: §3.3.4 · p. 17, Table 1 · p. 18

3.3.5. Summary of Metric Formulas

• Table 2 compiles formulas for all 15 metrics; core metrics are detailed in the main text, and auxiliary metrics follow standard network-science definitions.The table’s notation includes population size N, symmetrized edge weight Wij, node strength si, degree ki, shortest-path distance d(i,j), Laplacian L, and total weight m.Source: §3.3.5 · pp. 17–18

3.4.1. Statistical Characteristics of Small-Scale Networks

• In small networks, integer-valued node degrees restrict degree-distribution shapes, the modularity resolution limit can omit small communities, and spectral gaps are unstable when λ2 approaches zero.For N = 30 and average degree 6, the modularity threshold is approximately 6.7 edges.Source: §3.4.1 · pp. 18, 20, Table 2 · p. 19

3.4.2. Metric Screening and Selection Logic

• Screening selects Eeff as the sole freeze-point basis because it responds to search-state changes but not local connectivity changes; emax supplements hub analysis, while EAB is not primary under regular topologies.Gs and Hs are retained as robustness checks; the five core metrics are Eeff, Gs, Hs, emax, and EAB.Source: §3.4.2 · p. 20

3.5.1. Definition

• The freeze point is the first sample index meeting the stated stability criterion for Eeff, marking the transition from structural evolution to structural stagnation.The criterion uses a detection window W = 50 and tolerance ε = 10^-3.Source: §3.5.1 · pp. 20–21

3.5.2. Parameter Sensitivity

• Across tested W values of 30, 50, and 100 and ε values of 10^-4, 10^-3, and 10^-2, freeze times retain the order Gbest < Von Neumann < Ring.Freeze-point values show some parameter sensitivity, especially Ring’s sensitivity to window length.Source: §3.5.2 · p. 21

3.5.3. Intuitive Interpretation

• When Eeff stabilizes, the same hub nodes continue to dominate information flow, indicating converged learning relationships and a population that has structurally stopped exploring.Source: §3.5.3 · p. 21

3.6.1. Benchmark Functions and Algorithm Configuration

• Ten CEC2017 functions were tested in 10D and 30D with 30 particles, three topologies, three PSO variants, and 15 runs per configuration.The maximum was D×500 iterations; the variants were included to assess whether parameter-adaptive strategies affect freezing.Source: §3.6.1 · p. 21

3.6.2. Data Collection and Freeze-Point Detection

• The influence graph was sampled every 10 generations using tw = 10, and Effective Nodes series were used to detect freeze points.Source: §3.6.2 · p. 21

3.6.3. Search State Calibration

• Search states were divided into exploration before freeze/2, exploitation from freeze/2 to freeze, and frozen after freeze.The freeze point provides the basis for this stage division rather than an arbitrary time split.Source: §3.6.3 · p. 22
• The methodology combines dynamic network modeling, multi-scale information-flow evaluation, and Effective Nodes-based freeze-point detection for small-scale PSO.The next section validates freezing, its topology dependence, and its relationship with search performance on CEC2017 benchmarks.Source: §3.6.3 · p. 22

4.2. The Freezing Phenomenon of Information Flow Structure

• Across 10 functions and three variants, global median freeze points were Gbest 225, Von Neumann 455, and Ring 738 sampling points, with the same order in representative F8.For representative F8, medians were 87, 152, and 334 sampling points, respectively; the effective-node curves then remained flat for the remaining 86%–95% of the search.Source: §4.2 · pp. 23–24, Figure 2 · p. 23, Figure 3 · p. 24

4.3. Topology Dependence of Freeze Time

• Freeze detection rates were Gbest 78.2%, Von Neumann 36.9%, and Ring 41.6%; detected medians were 225, 455, and 738 sampling points, respectively.More uniform topologies had more runs without detected freezing, consistent with weaker hub dominance and slower structural evolution; the Gbest-to-Von Neumann-to-Ring order also held for F8, F20, and F25.Source: Table 4 · p. 25, §4.3 · pp. 25–26

4.3.1. Effect of PSO Variants on Freeze Time

• On F8 (30D), LDIW and TVAC delayed freezing relative to standard PSO across all three topologies, while Gbest < Von Neumann < Ring persisted.Because run-to-run coefficients of variation often exceeded 50%, the freeze-point values are trend indicators rather than precise estimates.Source: Table 5 · p. 26, §4.3.1 · p. 26, §4.7 · p. 34

4.4. Correlation Between Freeze Point and Search Performance

• Earlier freezing was significantly associated with worse final solution quality across 10D and 30D and all function types (Spearman rs = −0.294, p < 0.001).The correlation was moderate, and runs without detected freezing were excluded from the plotted analysis.Source: Figure 4 · p. 27, §4.4 · p. 27

4.5.2. Structural Effects

• Under Gbest, hub strengthening froze earlier (median 94 sampling points) than hub weakening (104) or control (110), while weakening had only a small structural effect.Interventions were tested on F8 and F15; Group A replaced the hub’s personal best when effective nodes fell below 85% of its peak, while Group B forced 30% of particles to learn from the hub at 5% of total iterations.Source: §4.5.1 · pp. 27–28, Figure 5 · p. 28, §4.5.2 · pp. 28–29

4.5.3. Performance Effects

• On multimodal F8, hub weakening improved final fitness versus control (807.88 vs 813.23, p = 0.0009), while hub strengthening showed no significant difference (p = 0.9973).On hybrid F15, Group A, Group B, and control final fitnesses were 1092.8, 1133.3, and 1256.7, but neither intervention difference was significant (p = 0.18 and p = 0.71). The weak structural change alongside the F8 gain indicates structural and performance effects can be decoupled under saturated Gbest.Source: Table 6 · p. 29, §4.5.2 · p. 29, §4.5.3 · p. 30

4.5.4. The Saturation Effect

• Under saturated topology, hub weakening barely changed structure (Group A differed from Control by 6 sample points), yet significantly improved final fitness over Control (Table 6, p = 0.0009).Replacing the hub’s pbest gives particles new learning exemplars and lets them escape the original information aggregation path.Source: §4.5.4 · p. 31

4.6.1. Micro-Level Analysis: Hub Formation

• At Gbest’s freeze point, maximum eigenvector centrality rose to 0.7071, the star-graph value, while Ring centrality remained distributed without a dominant hub.For Von Neumann, centrality fell to zero as declining graph connectivity made eigenvector centrality unreliable.Source: Figure 6 · p. 32, §4.6.1 · pp. 32–33

4.7. Summary

• Weakening hubs delays freezing and improves performance, whereas strengthening hubs produces no additional effect due to saturation; freeze point is significantly negatively correlated with performance.Source: §4.7 · p. 34
• Gbest exhibits saturated hub dominance from the start, while Ring and Von Neumann show gradual hub formation; community structure is largely predetermined by regular topologies and evolves little.Source: §4.7 · p. 34

5.1. Mechanistic Interpretation of Structural Freezing

• 3%–22% of iterations in representative configurations and 15%–49% across configurations precede freezing, leaving 51%–85% of the budget redundant.Gbest becomes a fixed star once its global best stabilizes, while local topologies fix edges once best neighbors are determined; micro-, meso-, and macro-level metrics transition together near the freeze point.Source: §5.1 · p. 35

5.2. Topology Dependence and Its Implications

• Freeze time follows Gbest < Von Neumann < Ring, reflecting propagation speed; LDIW and TVAC delay freezing, while later freezing preserves exploration longer.The discussion interprets early freezing as an exploitation bias that can harm multimodal search, whereas rapid freezing may benefit unimodal search.Source: §5.2 · pp. 35–36

5.3. Causal Evidence and the Saturation Effect

• Under Gbest, weakening hubs improves multimodal performance, but strengthening them adds no deterioration because hub dominance is saturated.On F8, weakening hubs significantly improves performance (p = 0.0009); similar freeze points and macro-level metrics despite better fitness indicate a behavioral rather than structural intervention benefit.Source: §5.3 · pp. 36–37

5.4. Why Only Macro-Level Metrics Capture the Freezing Phenomenon

• Under regular topologies, only macro-level distributional metrics effectively capture freezing; fixed degree and community structures constrain micro- and meso-level metrics.Small networks also limit node-level observations and destabilize community detection; hub dominance is a distributional property, and the discussion recommends non-regular topologies or larger populations for evolving micro- and meso-level structures.Source: §5.4 · pp. 37–38

5.5. Comparison with Existing Work

• Unlike prior network diagnosis and static topology comparisons, this study characterizes dynamic freezing and its topology dependence, which prior studies had not reported.It does not introduce new topology rules; it proposes the freeze point as a potential trigger for topology adjustment.Source: §5.5 · pp. 38–39

5.6. Scope and Generalizability

• The study restricts its analysis to 20–50 particles, regular topologies, and standard, LDIW, and TVAC variants; larger or irregular networks remain an extension.The selected regime reflects common population sizes and provides controlled comparisons; the framework is described as extensible to larger populations and non-regular topologies.Source: §5.6 · pp. 39–40

5.7. Limitations

• The authors identify missing real-world validation, sensitivity analysis, and intervention tests beyond Gbest, alongside degraded eigenvector analysis after Von Neumann disconnection.They note that extending intervention experiments to Von Neumann and Ring would provide a more complete picture.Source: §5.7 · p. 40

6.1. Main Findings

• Effective Nodes stabilizes within 3%–22% of iterations for representative configurations and 15%–49% for global medians, then remains unchanged.The phenomenon occurs across tested functions and variants, with micro- and meso-level transitions synchronized near the macro-level freeze point.Source: §6.1 · pp. 40–41
• Mean freeze points are approximately 15% for Gbest, 30.3% for Von Neumann, and 49.2% for Ring; LDIW and TVAC delay freezing.The order is Gbest < Von Neumann < Ring.Source: §6.1 · p. 41
• Earlier freezing is associated with worse final solution quality (Spearman rs = −0.294, p < 0.001) across dimensions and function types.Source: §6.1 · p. 41
• Under Gbest, hub weakening improves multimodal performance (p = 0.0009 on F8), while strengthening adds no deterioration; benefits occur primarily through behavior.Group A differed from Control by only 6 sample points in freeze time, with near-zero differences in Eeff, Gs, and Hs despite significant performance improvement.Source: §6.1 · p. 41

6.2. Methodological Implications

• Under regular topologies, macro-level distributional metrics capture freezing more effectively than micro- and meso-level metrics because degrees and communities are predetermined.Effective Nodes, the Gini coefficient, and Entropy are identified as distributional measures; non-regular topologies or larger populations may let micro- and meso-level structures evolve.Source: §6.2 · pp. 41–42

6.3. Limitations and Future Work

• Generalization beyond small-scale regular-topology PSO remains unverified; the authors also call for systematic parameter sensitivity analysis and interventions on other topologies.Future directions include dynamic optimization, multi-swarm PSO, theoretical modeling of topology and freeze time, and larger irregular networks.Source: §6.3 · p. 42

• Zhenglong Xiang handled conceptualization, methodology, original-draft writing, and funding acquisition.Source: CRediT authorship contribution statement · p. 43
• Hongrun Wu contributed methodology, investigation, and funding acquisition.Source: CRediT authorship contribution statement · p. 43
• Yu Xue contributed validation and funding acquisition; Fei Yu contributed validation.Source: CRediT authorship contribution statement · p. 43
• Yuanxiang Li contributed conceptualization.Source: CRediT authorship contribution statement · p. 43