Fixed-Point Centrality for Networks

TL;DR

This paper introduces fixed-point centralities, a new family of network centrality measures based on fixed points of permutation-equivariant mappings, applicable to both finite and infinite graphs, with connections to various network models.

Contribution

It defines a novel centrality concept using fixed points, extending it to infinite graphs via graphons, and establishes their variation bounds and connections to existing network models.

Findings

Defined fixed-point centralities for finite networks.

Extended the concept to infinite graphs using graphons.

Established bounds on how these centralities vary with network changes.

Abstract

This paper proposes a family of network centralities called fixed-point centralities. This centrality family is defined via the fixed point of permutation equivariant mappings related to the underlying network. Such a centrality notion is immediately extended to define fixed-point centralities for infinite graphs characterized by graphons. Variation bounds of such centralities with respect to the variations of the underlying graphs and graphons under mild assumptions are established. Fixed-point centralities connect with a variety of different models on networks including graph neural networks, static and dynamic games on networks, and Markov decision processes.