Local iterative algorithms for approximate symmetry guided by network centralities

TL;DR

The paper addresses robust approximate network symmetry, defined via $E(A)=\frac{1}{4}\min_{P}\|A-PAP^{\mathrm{T}}\|_{1}$ and the normalized $S(A)$, noting the combinatorial difficulty of finding exact automorphisms. It introduces a centrality-guided annealing heuristic that uses a similarity matrix based on centralities such as $\Gamma$ (e.g., eigenvector, PageRank, betweenness) to bias transpositions toward structurally similar vertices, implemented via $m_{ij}=(|\Gamma(i)-\Gamma(j)|+\beta)^{-1}$ and a move rule with $\Delta E_b=\max(\Delta M_b,\phi)$. The method is evaluated across grid, ER, BA, and DD models, with significant improvements in grid, BA, and DD—particularly for larger graphs and sparser BA/DD instances—while ER shows limited gains. The results demonstrate that centrality-guided search can more effectively navigate the permutation space to reveal near-automorphisms, suggesting practical utility for analyzing symmetry in noisy or uncertain networks and guiding future metaheuristic extensions. The study highlights how automorphism-preserving centralities can inform scalable symmetry detection in complex networks and points to model-dependent effects and future work on broader centrality choices and optimization strategies.

Abstract

Recently, the influence of potentially present symmetries has begun to be studied in complex networks. A typical way of studying symmetries is via the automorphism group of the corresponding graph. Since complex networks are often subject to uncertainty and automorphisms are very sensitive to small changes, this characterization needs to be modified to an approximate version for successful application. This paper considers a recently introduced approximate symmetry of complex networks computed as an automorphism with acceptance of small edge preservation error, see Liu 2020. This problem is generally very hard with respect to the large space of candidate permutations, and hence the corresponding computation methods typically lead to the utilization of local algorithms such as the simulated annealing used in the original work. This paper proposes a new heuristic algorithm extending such iterative search algorithm method by using network centralities as heuristics. Centralities are shown to be a good tool to navigate the local search towards more appropriate permutations and lead to better search results.

Figures (4)

• Figure 1: Comparison of the performance of simulated annealing guided by different centralities across various random network models. The results are visualized in violin plots, where each color represents a version of annealing guided by a different centrality. For each configuration of parameters, we conduct 50 simulations. The first subplot presents results for grid graphs with 100 nodes in 2D and 3D. The second subplot presents results for Erdős–Rényi graphs with 100 nodes and edge density of 0.1. The third subplot presents results for Barabási–Albert graphs with 150 nodes and $k=5$. The final subplot presents results for the Duplication–Divergence model with 150 nodes and a divergence probability of 0.1.
• Figure 2: Comparison of the performance of original simulated annealing and simulated annealing guided by eigenvector centrality, BA model, where $k$ is the number of links of a new node.
• Figure 3: Comparison of the performance of original simulated annealing and simulated annealing guided by eigenvector centrality, DD model.
• Figure 4: Comparison of measured symmetry and Cohen’s d values for larger BA and DD graphs.