Capability centrality: the next step from scale-free property

2026-05-05

AI Summary

This study addresses the challenge of effectively distinguishing real-world networks from random ones by uncovering interpretable structural characteristics. To this end, it introduces a novel centrality measure—ksi-centrality—whose distribution is consistently right-skewed in empirical networks yet sharply concentrated in both random graphs and canonical generative models. The measure exhibits theoretical connections to fundamental spectral properties, including algebraic connectivity and the Cheeger constant. Notably, the normalized mean of ksi-centrality uniquely determines the attachment parameter $m$—the number of edges added per new node—in the Barabási–Albert model. As a network property independent of scale-free topology, ksi-centrality offers an interpretable and computationally tractable structural signature for network modeling and parameter inference.

Abstract

In this article we present a new centrality measure called ksi-centrality. We show that ksi-centrality distinguishes real networks from random ones, similar to degree centrality: the ksi-centrality distribution is right-skewed for real networks and centered for random Erdos-Renyi networks, and has linear pattern with a heavy tail on a log plot. Furthermore, the ksi-centrality distribution is centered for models simulating real networks: Barabasi-Albert, Watts-Strogatz, and Boccaletti-Hwang-Latora. Thus, this centrality distribution is an additional and independent property with respect to scale-freeness. We also introduce a normalized version of ksi-centrality and show that it is related to algebraic connectivity and the Chegeer's value of a network. Moreover, the average value of this normalized centrality is in bijective correspondence with the relative number of edges that a new node connects to others in the Barabasi-Albert preferential attachment model, thus answering the question of how to choose the parameter $m$ to model a given real-world network.

Problem

• centrality
• scale-free networks
• network topology
• real-world networks
• preferential attachment

Innovation

• ksi-centrality
• network centrality
• scale-free networks
• algebraic connectivity
• preferential attachment