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
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
emax
Core
Micro
Mean/std of weighted degree
sˉ,σs
Auxiliary
Micro
Mean clustering coefficient
Cˉi
Auxiliary
Meso
Subgroups formed?
Inter - community flow
EˉAB
Core
Meso
Multi - resolution modularity
Qγ
Auxiliary
Meso
Number of communities
nc
Auxiliary
Table 1: Multi-scale information flow metrics for PSO populations
Scale
Metric
Symbol
Formula
Ref.
Micro
Max eigenvector centrality
emax
imaxxi,where Wsymx=λmaxx
This work
Micro
Mean weighted degree
sˉ
N1i∑si
[ 24 ]
Micro
Std. of weighted degree
σs
N1i∑(si−sˉ)2
[ 24 ]
Micro
Mean clustering coefficient
Cˉi
N′1i:ki≥2∑ki(ki−1)2Ei
[ 3 ]
Meso
Inter-community flow
EˉAB
k(k−1)1a=b∑∣Ca∣⋅∣Cb∣i∈Ca∑j∈Cb∑Wij
This work
Meso
Multi-resolution modularity
Qγ
2m1ij∑(Wij−γ2msisj)δ(ci,cj)
[ 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×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 ( ×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 ×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 rs=−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 ± std)
vs Control (p-value)
F8
Control
813.23±4.45
—
F8
A (weaken)
807.88±4.24
0.0009↓
F8
B (strengthen)
808.35±3.34
0.9973
F15
Control
1256.7±466.7
—
F15
A (weaken)
1092.8±466.2
0.18
F15
B (strengthen)
1133.3±418.0
0.71
Table 6: Intervention experiment results
Figure 6: Evolution of maximum eigenvector centrality emax 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 EˉAB 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.
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.
Wenbin Pei, Xi Luo, Bing Xue +2
School of Computer Science and Technology, Dalian University of Technology, Dalian, 116024, China · Key Laboratory of Social Computing and Cognitive Intelligence (Dalian University of Technology), Ministry of Education, China · Centre for Data Science and Artificial Intelligence & School of Engineering and Computer Science, Victoria University of Wellington, Wellington, 6140, New Zealand +1
Swarm-based optimization algorithms have demonstrated remarkable success in solving complex problems, yet their widespread adoption remains limited due to poor transparency in how algorithmic components influence performance. This work presents a multi-faceted explainability framework for Particle Swarm Optimization (PSO) through two complementary perspectives: landscape-based and algorithmic explainability. From the landscape-based perspective, we develop a comprehensive characterization framework using Exploratory Landscape Analysis (ELA) to quantify problem difficulty, multimodality, and ruggedness, extracting ELA meta-features, dispersion measures, and information content statistics, while a machine learning approach employing Decision Tree and Random Forest classifiers enables prediction of optimal topology-specific hyperparameter configurations for unseen problems. From the algorithmic explainability perspective, we integrate IOHxplainer for temporal convergence profiling and Search Trajectory Networks (STN) for spatial navigation mapping, proposing three novel STN metrics-Connectivity Density, Fragmentation Score, and Search Efficiency-that enhance visual explainability by quantifying topology-specific search organization and transition effectiveness. Through systematic experimentation across 24 benchmark functions in multiple dimensions with Star, Ring, and Von Neumann topologies, we establish practical guidelines for topology selection and parameter configuration. Our findings uncover the black-box nature of PSO, providing greater transparency and interpretability to swarm intelligence systems. The source code is available at https://github.com/GitNitin02/ioh_pso.
Nitin Gupta, Bapi Dutta, Anupam Yadav
Department of Mathematics and Computing Dr. B. R. Ambedkar National Institute of Technology Jalandhar, Jalandhar - 144008, INDIA · Department of Computer Science University of Jaén, Jaén, SPAIN
Particle Swarm Optimization (PSO) frequently suffers from premature convergence. This paper introduces a family of problem-informed diversity-enhancing strategies that manipulate the swarm's social and cognitive components. These include opposing-best strategies that repel particles from optimal regions, negative learning strategies that guide exploration toward poor solutions, and reverse learning strategies that push particles away from inferior regions. These socio-cognitive mechanisms are evaluated against an analogous suite of problem-unaware, explicit randomization strategies that inject randomness either into velocity update components or directly into position updates. The results reveal that the effectiveness of diversity enhancement is determined primarily by how it is embedded within the swarm dynamics, rather than by the mere presence of extraneous problem-informed guidance. Particularly, random perturbations introduced at the velocity-update level consistently outperform those applied directly to particle positions.
Piotr Urbańczyk, Aleksandra Urbańczyk
AGH University of Krakow, al. Mickiewicza 30, 30-059 Kraków, Poland · Jagiellonian University, ul. Gołębia 24, 31-007 Kraków, Poland