发表机构
College of Sciences, China Jiliang University（中国计量大学理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文从理论上分析DomiRank中心性，揭示其与图自同构、熵的关系，并研究图变换的影响，证明其优势，为复杂网络节点重要性评估提供新工具。
AI 中文摘要
DomiRank是一种用于无权网络的节点重要性评估算法，其定义基于一个动力系统模型，该模型的稳态解由竞争强度参数、支配阈值和自然衰减率共同决定。本文研究了DomiRank与图自同构之间的内在联系：由自同构相互映射的顶点具有相同的DomiRank值，且所有顶点DomiRank值两两不同的图必定是非对称的；我们进一步推导了正则图和顶点传递图的DomiRank性质，并揭示了轨道数量与不同DomiRank值数量之间的定量关系。对于DomiRank熵，我们证明了连通图的最大熵仅由正则图取得，且熵随竞争参数单调递减，将识别从“重要节点”转变为“主导关键节点”。我们还研究了图变换（顶点相似性、顶点划分、边交换和m-积）对DomiRank向量的影响，并解析地表征了σ的敏感性和极限行为：归一化DomiRank分布是σ不变的当且仅当度向量是邻接矩阵的特征向量，且σ在度中心性和最小特征向量中心性之间连续插值；在四个真实世界网络上的数值实验证实了这些结果。我们进一步表明，DomiRank相对于主特征向量中心性具有独特优势，为复杂网络中的节点重要性评估提供了新工具。
英文摘要
DomiRank is a node-importance algorithm for unweighted networks, defined by a dynamical-system model whose steady state is governed by a competition-strength parameter, a dominance threshold, and a natural decay rate. We study its intrinsic relations with graph automorphism: vertices mapped to each other by an automorphism share the same DomiRank value, and a graph whose DomiRank values are pairwise distinct must be asymmetric; we derive DomiRank properties of regular and vertex-transitive graphs and bound the number of orbits by that of distinct DomiRank values. For DomiRank entropy, the maximum over connected graphs is attained only by regular graphs, and under sufficient conditions (rigorously in the low-competition regime) the entropy decreases monotonically with the competition parameter, a behavior observed on all tested networks and conjectured to hold generally; tuning sigma shifts the identification from important to dominant key nodes. We also study how graph transformations (vertex similarity, vertex partitions, edge swaps, m-products) affect the DomiRank vector, and characterize analytically the sensitivity and limiting behavior of sigma: the normalized DomiRank distribution is sigma-invariant iff the degree vector is an eigenvector of the adjacency matrix, and sigma interpolates continuously between degree and least-eigenvector centrality; experiments on four real networks confirm these results. These results position DomiRank as a tunable complement to principal-eigenvector centrality, with distinctive behavior under strong competition and new tools for node-importance evaluation. Because the parameterization by sigma is a structural property of the measure, not a guarantee of advantage on a downstream task, 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 an explicit degree correction.
Comments34 pages, 14 figures, 2 tables, 34 references
