Structural Properties of DomiRank Centrality: Symmetry Axioms, an Entropy Bound, and Spectral Interpolation
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 establish three structural results. First, on symmetry. Automorphism-related vertices share the same DomiRank value, and a graph whose values are pairwise distinct must be asymmetric. The orbit count bounds the number of distinct values, and regular and vertex-transitive graphs admit closed forms. Second, on entropy. The maximum of the DomiRank entropy over connected graphs, log_2 N, is attained if and only if the graph is regular. The initial entropy derivative is the degree-log-degree edge covariance, making the decrease rigorous near vanishing competition for structurally disassortative graphs, and an assortative counterexample shows it does not extend to all graphs; a sufficient-condition theorem and a refined conjecture cover the rest. Third, on spectrum. The normalized distribution is sigma-invariant if and only if the degree vector is an adjacency-matrix eigenvector. The parameter sigma interpolates continuously between degree and smallest-eigenvector centrality within the resolvent (Bonacich-type) family, its sensitivity being a second-order suppression operator on the degree vector. Graph transformations are also characterized, including the m-product. Experiments on four real networks and synthetic families up to 10^4 vertices confirm the interpolation and exhibit monotone entropy decrease, continuous ranking interpolation, decaying positive fraction and ranking reversals; where the positive fraction has decayed far from one, the limit describes the solution vector rather than the induced ranking. DomiRank is thus a tunable complement to principal-eigenvector centrality. A companion study quantifies how much of its task-level edge over a degree baseline survives degree correction.
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BibTeX
@misc{zhang2026structural,
title = {{Structural Properties of DomiRank Centrality: Symmetry Axioms, an Entropy Bound, and Spectral Interpolation}},
author = {Yingying Zhang and Chengye Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00107},
eprint = {2610.00107},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00107},
}
RIS
TY - UNPB
TI - Structural Properties of DomiRank Centrality: Symmetry Axioms, an Entropy Bound, and Spectral Interpolation
AU - Yingying Zhang
AU - Chengye Zhao
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.00107
UR - https://arxiv.org/abs/2610.00107
AB - 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 establish three structural results. First, on symmetry. Automorphism-related vertices share the same DomiRank value, and a graph whose values are pairwise distinct must be asymmetric. The orbit count bounds the number of distinct values, and regular and vertex-transitive graphs admit closed forms. Second, on entropy. The maximum of the DomiRank entropy over connected graphs, log_2 N, is attained if and only if the graph is regular. The initial entropy derivative is the degree-log-degree edge covariance, making the decrease rigorous near vanishing competition for structurally disassortative graphs, and an assortative counterexample shows it does not extend to all graphs; a sufficient-condition theorem and a refined conjecture cover the rest. Third, on spectrum. The normalized distribution is sigma-invariant if and only if the degree vector is an adjacency-matrix eigenvector. The parameter sigma interpolates continuously between degree and smallest-eigenvector centrality within the resolvent (Bonacich-type) family, its sensitivity being a second-order suppression operator on the degree vector. Graph transformations are also characterized, including the m-product. Experiments on four real networks and synthetic families up to 10^4 vertices confirm the interpolation and exhibit monotone entropy decrease, continuous ranking interpolation, decaying positive fraction and ranking reversals; where the positive fraction has decayed far from one, the limit describes the solution vector rather than the induced ranking. DomiRank is thus a tunable complement to principal-eigenvector centrality. A companion study quantifies how much of its task-level edge over a degree baseline survives degree correction.
ER -