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

Yingying Zhang; Chengye Zhao

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arXiv · 2610.00107

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

Yingying Zhang·Chengye Zhao

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Abstract

DomiRank is a node-importance algorithm for unweighted networks, defined by a dynamical model governed by a competition-strength parameter, a dominance threshold, and a decay rate. We study its relations with graph automorphism: automorphism-related vertices share the same DomiRank value, and a graph whose values are pairwise distinct must be asymmetric; we derive DomiRank properties of regular and vertex-transitive graphs and bound the number of distinct values by that of the orbits. For DomiRank entropy, the maximum over connected graphs is attained only by regular graphs, and on the effective (truncated) distribution it decreases monotonically with the competition parameter on all four tested networks, while a sufficient-condition theorem and a refined conjecture cover the theoretical entropy; we further identify the initial entropy derivative with the degree--log-degree edge covariance, making the decrease rigorous near vanishing competition for structurally disassortative graphs (including all tested networks) and yielding an assortative counterexample; tuning sigma shifts node identification from important to dominant. We study how graph transformations (vertex similarity, partitions, edge swaps, m-products) affect the DomiRank vector, and characterize sigma's sensitivity and limits: the normalized distribution is sigma-invariant iff the degree vector is an adjacency-matrix eigenvector, and sigma interpolates between degree and least-eigenvector centrality; experiments on four real networks confirm it. DomiRank is thus a tunable complement to principal-eigenvector centrality. Since parameterization by sigma is a structural property of the measure, not a guarantee of downstream advantage, we also relate these results to the companion null-model study of how much of DomiRank's task-level edge over a degree baseline survives a degree correction.

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Yingying Zhang, Chengye Zhao. 2026-10-13. Theoretical Analysis of DomiRank Centrality: Automorphism, Entropy, and Graph Transformations. https://arxiv.org/abs/2610.00107

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