bibkey: zhangzhao2026domirank authors: Yingying Zhang and Chengye Zhao year: 2026 title: “Theoretical Analysis of DomiRank Centrality: Automorphism, Entropy, and Graph Transformations” doi: 10.48550/arXiv.2610.00107 url: https://arxiv.org/abs/2610.00107v1 claim: “the DomiRank entropy of a connected non-regular graph decreases monotonically with σ” strata_touched:

• D5/S3/Combinatorics/Graph/DomiRankEntropyRefutation license: citation-only triage: anchor

DomiRank entropy and competition

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• DOI: https://doi.org/10.48550/arXiv.2610.00107
• Version: https://arxiv.org/abs/2610.00107v1
• Primary text: https://arxiv.org/html/2610.00107v1
• Target: Section 4.1, Remark 4.6.

Remark 4.6 states:

This contrast indicates that the conditions of Theorem 4.5 are conservative, and the unconditional version — “the DomiRank entropy of a connected non-regular graph decreases monotonically with σ” — deserves further study as an open problem.

For a finite unweighted undirected simple graph with adjacency matrix $A$, degree vector $d=A1$, and competition parameter $\sigma$, the source equilibrium with $\theta =1$ is

$$\Gamma (\sigma )=\sigma (I+\sigma A{)}^{-1}d,0<\sigma <-1/{\lambda }_{\min }(A).$$

The entropy uses the positive part of the scores and normalization:

$$P_i(\sigma )=\frac{\max ({\Gamma }_i(\sigma ),0)}{{\sum }_j\max ({\Gamma }_j(\sigma ),0)},H(\sigma )=-\sum _i{P}_i(\sigma ){\log }_2{P}_i(\sigma ).$$

The source’s graph product replaces each vertex by $m$ independent clones and each edge by all edges between its two clone classes. This is the graph on $V\times \mathrm{Fin}(m)$ whose adjacency depends only on the original vertices. The entropy shift is $H_{G_m}(s)=H_G(ms)+{\log }_{2}m$.

Theorem 4.5 supplies conditional sufficient criteria for entropy decrease. Remark 4.6 asks whether those criteria can be removed; the unconditional assertion is the target here. Conjecture 4.7, that stars minimize DomiRank entropy among connected graphs, is a separate question and is excluded.

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