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

Zhenglong Xiang; Hongrun Wu; Yu Xue; Fei Yu; Yuanxiang Li

cs.NEOct 24, 2026

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

Authors: Zhenglong Xiang, Hongrun Wu, Yu Xue, Fei Yu, Yuanxiang Li

Organizations: School of Computer Science, School of Software, Nanjing University of Information Science and Technology, Nanjing 210044, China · Key Lab of Intelligent Optimization and Information Processing, Minnan Normal University, Zhangzhou 363000, China · State Key Lab. for Novel Software Technology, Nanjing University, Nanjing 210023, China · School of Computer Science, Wuhan University, Wuhan 430072, China

Abstract

The neighborhood topology of particle swarm optimization (PSO) determines information propagation and regulates the exploration-exploitation balance. However, existing evaluation methods based on fitness and spatial diversity fail to characterize the dynamic evolution internal interaction structure of the population. To address this gap, this paper focuses on small-scale PSO (20-50 particles) with regular topologies and models the iterative process as a dynamic weighted influence graph. We construct a multi-scale information-flow evaluation framework covering micro-, meso-, and macro-level metrics, and propose a freeze-point detection method based on effective nodes, a macro-level distributional metric. Experiments on CEC2017 benchmark functions show that the information flow structure freezes early in the search (3 percent - 22 percent for representative configurations and 15 percent-49 percent for global medians) and remains unchanged thereafter. The freeze time follows the order Gbest $<$ Von Neumann $<$ Ring, consistent across standard, LDIW, and TVAC variants. Parameter-adaptive variants delay freezing without altering the topological order, and the freeze point is significantly negatively correlated with final solution quality. Intervention experiments under Gbest show that weakening hub nodes delays freezing and improves performance on multimodal functions, while strengthening hubs produces no additional deterioration due to saturation, revealing a decoupling between structural and performance effects. Among the three metric levels, only macro-level distributional metrics effectively capture freezing under regular topologies, because micro- and meso-level structures are predetermined by the topology. These findings reveal the freezing mechanism of information flow in small-scale PSO and provide structural prior signals for adaptive topology control.

Figures & tables

Figure 1: Overview of the proposed multi-scale information flow framework. The input layer and method layer are detailed in Section 3, and the findings and validation layer is presented in Section 4.

Scale	Core Question	Metric	Symbol	Role
Micro	Who dominates?	Max eigenvector centrality	$e_{\text{max}}$	Core
Micro		Mean/std of weighted degree	$\bar{s},\,\sigma_{s}$	Auxiliary
Micro		Mean clustering coefficient	$\bar{C}_{i}$	Auxiliary
Meso	Subgroups formed?	Inter - community flow	$\bar{E}_{AB}$	Core
Meso		Multi - resolution modularity	$Q_{\gamma}$	Auxiliary
Meso		Number of communities	$n_{c}$	Auxiliary

Table 1: Multi-scale information flow metrics for PSO populations

Scale	Metric	Symbol	Formula	Ref.
Micro	Max eigenvector centrality	$e_{\text{max}}$	$\displaystyle\max_{i}x_{i},\ \text{where }\mathbf{W}_{sym}\mathbf{x}=\lambda_{\text{max}}\mathbf{x}$	This work
Micro	Mean weighted degree	$\bar{s}$	$\displaystyle\frac{1}{N}\sum_{i}s_{i}$	[ 24 ]
Micro	Std. of weighted degree	$\sigma_{s}$	$\displaystyle\sqrt{\frac{1}{N}\sum_{i}(s_{i}-\bar{s})^{2}}$	[ 24 ]
Micro	Mean clustering coefficient	$\bar{C}_{i}$	$\displaystyle\frac{1}{N^{\prime}}\sum_{i:k_{i}\geq 2}\frac{2E_{i}}{k_{i}(k_{i}-1)}$	[ 3 ]
Meso	Inter-community flow	$\bar{E}_{AB}$	$\displaystyle\frac{1}{k(k-1)}\sum_{a\neq b}\frac{\displaystyle\sum_{i\in C_{a}}\sum_{j\in C_{b}}W_{ij}}{\|C_{a}\|\cdot\|C_{b}\|}$	This work
Meso	Multi-resolution modularity	$Q_{\gamma}$	$\displaystyle\frac{1}{2m}\sum_{ij}\left(W_{ij}-\gamma\frac{s_{i}s_{j}}{2m}\right)\delta(c_{i},c_{j})$	[ 13 ]

Table 2: Summary of metric formulas

Parameter	Setting
Benchmark	CEC2017 (10 functions)
Dimension	10, 30
PSO variants	Standard, LDIW, TVAC
Topologies	Gbest, Ring, Von Neumann
Population size	30
Max iterations	$D\times 500$

Table 3: Experimental configuration

Figure 2: Freezing of Effective Nodes under three topologies for a representative configuration (F8,

30D

, standard PSO). Red: Gbest; Blue: Von Neumann; Green: Ring. Dashed lines mark the median freeze points across 15 runs; shaded bands indicate the 25th–75th percentile range. The global freeze-point distribution across 10 functions and 3 variants is shown in Figure 3 .

Figure 3: Boxplots of the freeze point distributions under the three topologies (

30D

, based on runs with successful freezing). The detection rates are given in parentheses: Gbest (78.2%), Von Neumann (36.9%), and Ring (41.6%). Runs without detection indicate that the information flow structure had not fully frozen by the end of the search.

Topology	Median FP ( $\times 10$ gens)	Iteration ratio	Detection rate
Gbest	225	15.0%	78.2%
Von Neumann	455	30.3%	36.9%
Ring	738	49.2%	41.6%

Table 4: Statistics of freezing points (FP) grouped by topology (

30D

, across 10 functions and 3 variants)

Topology	Standard	LDIW	TVAC
Gbest	87	156	201
Von Neumann	152	509	284
Ring	334	594	615

Table 5: Median freeze points under different variants and topologies (F8,

30D

; unit: sampling index

\times 10

generations)

Figure 4: Scatter plot of freeze point versus final fitness (

30D

, all configurations). X-axis: normalized freeze point (freeze point / total iterations); Y-axis: min-max normalized final fitness per function. Each point represents one independent run (705 points in total), colored by topology: red for Gbest, blue for Von Neumann, and green for Ring. The black dashed line shows the linear trend, with Spearman correlation

r_{s}=-0.294

(

p<0.001

) . Runs without a detected freeze point are excluded.

Figure 5: Comparison of Effective Nodes evolution under intervention experiments (F8,

30D

, Gbest). Black: Control; Green: Group A (Hub Weakening); Red: Group B (Hub Strengthening). Dashed lines mark the median freeze points across 15 runs; shaded bands indicate the 25th–75th percentile range.

Function	Group	Final fitness (mean $\pm$ std)	vs Control (p-value)
F8	Control	$813.23\pm 4.45$	—
F8	A (weaken)	$807.88\pm 4.24$	$0.0009\downarrow$
F8	B (strengthen)	$808.35\pm 3.34$	$0.9973$
F15	Control	$1256.7\pm 466.7$	—
F15	A (weaken)	$1092.8\pm 466.2$	$0.18$
F15	B (strengthen)	$1133.3\pm 418.0$	$0.71$

Table 6: Intervention experiment results

Figure 6: Evolution of maximum eigenvector centrality

e_{\max}

under three topologies (F8,

30D

, standard PSO). Red: Gbest; Blue: Von Neumann; Green: Ring. Shaded bands indicate the 25th–75th percentile range across 15 runs.

Figure 7: Evolution of inter-community information flow

\bar{E}_{AB}

under three topologies (F8,

30D

, standard PSO). Red: Gbest; Blue: Von Neumann; Green: Ring. Shaded bands indicate the 25th–75th percentile range across 15 runs.

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Evolutionary Optimization Particle Swarm Optimization

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Particle swarm optimization (PSO) has been widely applied to solve complex optimization problems from real-world applications due to its efficient exploration of large solution spaces and the ability to converge towards optimal solutions without requiring gradient information. Common swarm topologies in standard PSO and its variants, e.g., Ring and Star, can be regarded as graphs, where each edge connects only two particles. Such topology structures allow direct interactions only between connected particle pairs, and thus often fail to directly capture the higher-order social relationships that are necessary for navigating complex search landscapes. Therefore, this article proposes a novel PSO variant termed Hypergraph-assisted Particle Swarm Optimization (HPSO). In HPSO, the topology of the particles in a swarm is modeled by a hypergraph, in which hyperedges are used to connect multiple particles. This allows multiple particles within a hyperedge to interact directly. Furthermore, an adaptive hypergraph updating strategy is designed to periodically reconstruct the topology based on cumulative average particle displacement, thereby maintaining swarm diversity throughout the evolutionary process. In the experiments, the effectiveness of HPSO is verified on the IEEE CEC'17 benchmark suite, and the results demonstrate that HPSO achieves promising performance across various types of functions. Furthermore, the ablation experiment demonstrates that HPSO has excellent search capabilities.

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