A tensor formalism for multilayer network centrality measures using the Einstein product

TL;DR

This paper introduces a tensor-based formalism using the Einstein product to extend centrality measures to multilayer networks, enabling analysis of complex systems with multiple interaction types.

Contribution

It develops a tensor formalism with the Einstein product for multilayer networks and extends centrality measures, including subgraph centrality, with computational methods for large networks.

Findings

Tensor formalism effectively models multilayer networks.

Extended centrality measures to multilayer contexts.

Krylov subspace methods enable scalable computations.

Abstract

Complex systems that consist of different kinds of entities that interact in different ways can be modeled by multilayer networks. This paper uses the tensor formalism with the Einstein tensor product to model this type of networks. Several centrality measures, that are well known for single-layer networks, are extended to multilayer networks using tensors and their properties are investigated. In particular, subgraph centrality based on the exponential and resolvent of a tensor are considered. Krylov subspace methods are introduced for computing approximations of different measures for large multilayer networks.