Theoretical Analysis of DomiRank Centrality: Automorphism, Entropy, and Graph Transformations

2026-09-09

AI Summary

This study addresses the unclear theoretical mechanisms of the DomiRank algorithm under graph symmetry, information entropy, and topological transformations by integrating dynamical systems, graph theory, spectral analysis, and information-theoretic frameworks. By establishing a connection between node importance dynamics and graph automorphisms, we prove that regular graphs maximize entropy, reveal the monotonically decreasing effect of the competition parameter on entropy, and derive eigenvalue conditions on the degree vector for distributional invariance. Empirical evaluations on real-world networks confirm that DomiRank serves as a tunable complement to principal eigenvector centrality, exhibiting distinct advantages under strong competition regimes. These findings provide a rigorous theoretical foundation for identifying critical nodes in complex networks.

Abstract

DomiRank is a node-importance evaluation algorithm for unweighted networks, defined through a dynamical-system model whose steady-state solution is governed by a competition-strength parameter, a dominance threshold, and a natural decay rate. This paper investigates the intrinsic relations between DomiRank and graph automorphism: vertices mapped to each other by an automorphism share the same DomiRank value, and a graph in which all vertices have pairwise distinct DomiRank values must be asymmetric; we further derive DomiRank properties of regular and vertex-transitive graphs and reveal the quantitative relation between the number of orbits and the number of distinct DomiRank values. For DomiRank entropy, we show that the maximum entropy of a connected graph is attained only by regular graphs and that the entropy decreases monotonically with the competition parameter, shifting the identification from `important nodes''to `dominant key nodes.''We also study the influence of graph transformations (vertex similarity, vertex partitions, edge swaps, and m-products) on the DomiRank vector, and characterize analytically the sensitivity and limiting behavior of \(\sigma\): the normalized DomiRank distribution is \(\sigma\)-invariant if and only if the degree vector is an eigenvector of the adjacency matrix, and \(\sigma\) interpolates continuously between degree centrality and least-eigenvector centrality; numerical experiments on four real-world networks confirm these results. We further show that DomiRank offers unique advantages over principal-eigenvector centrality, providing new tools for node-importance evaluation in complex networks.

Problem

• DomiRank centrality
• graph automorphism
• node importance
• DomiRank entropy
• graph transformations