发表机构
Dipartimento di Informatica, Università degli Studi di Milano（米兰大学计算机科学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究谱中心性度量的半单调性，证明特征向量中心性、Katz指数和PageRank在连通无向图上满足严格排名半单调性和得分半单调性，并回答了一个开放问题。
AI 中文摘要
得分单调性和排名单调性是描述在网络中添加一条弧时中心性度量行为的性质：前者要求该弧的目标节点的得分应增加，后者要求其相对于其余节点的重要性不应恶化。在有向网络中，几乎所有经典中心性度量都满足这两个性质，但在无向网络中，大多数度量都不满足：添加一条边可能会降低其一个端点的得分或排名。半单调性是最近针对无向网络引入的一个较弱的性质，要求新边的两个端点中至少有一个具有单调性，已知该性质对接近中心性、谐波中心性、距离衰减中心性和介数中心性成立。在本文中，我们研究了三种经典谱中心性度量的半单调性：特征向量中心性、Katz指数和PageRank。我们证明了在连通无向图上，所有这些度量都严格满足排名半单调性，并且它们也都满足得分半单调性（唯一的例外是当通过将常数向量投影到主特征空间来归一化得分时的特征向量中心性，对此我们提供了一个反例）。特别地，添加一条边永远不会同时降低其两个端点的PageRank，这回答了先前工作中留下的一个问题。
英文摘要
Score monotonicity and rank monotonicity are properties describing the behavior of a centrality measure when an arc is added to a network: the former requires that the score of the target of the arc should increase, the latter that its importance with respect to the remaining nodes should not deteriorate. While in directed networks almost all classical centrality measures satisfy both properties, in undirected networks they fail for most measures: adding an edge can reduce the score or the rank of one of its endpoints. Semi-monotonicity is a recently introduced weaker property for undirected networks, requiring that at least one of the two endpoints of the new edge enjoys monotonicity, and it is known to hold for closeness, harmonic centrality, distance-decay centralities and betweenness. In this paper we study semi-monotonicity for three classical spectral centrality measures: eigenvector centrality, Katz's index, and PageRank. We prove that all of them are strictly rank semi-monotone on connected undirected graphs, and that all of them are also score semi-monotone, (the only exception being eigenvector centrality when scores are normalized by projecting the constant vector on the dominant eigenspace, for which we provide a counterexample). In particular, adding an edge can never decrease the PageRank of both its endpoints, which answers a question left open in previous work.
