A polynomial eigenvalue approach for multiplex networks

TL;DR

This paper introduces a polynomial eigenvalue approach for multiplex networks, enabling analytical insights into spectral properties as a function of inter-layer coupling, and relating these to aggregated network spectra.

Contribution

It presents a novel formalism that reduces matrix dimensionality while increasing polynomial degree, linking quadratic eigenvalue problems to multiplex network spectra.

Findings

Relates quadratic eigenvalue problems to multiplex network spectra

Provides bounds and analytical insights on eigenvalue behavior

Applies to supra-adjacency, supra-Laplacian, and transition matrices

Abstract

We explore the block nature of the matrix representation of multiplex networks, introducing a new formalism to deal with its spectral properties as a function of the inter-layer coupling parameter. This approach allows us to derive interesting results based on an interpretation of the traditional eigenvalue problem. More specifically, we reduce the dimensionality of our matrices but increase the power of the characteristic polynomial, i.e, a polynomial eigenvalue problem. Such an approach may sound counterintuitive at first glance, but it allows us to relate the quadratic problem for a 2-Layer multiplex system with the spectra of the aggregated network and to derive bounds for the spectra, among many other interesting analytical insights. Furthermore, it also permits to directly obtain analytical and numerical insights on the eigenvalue behavior as a function of the coupling between layers. Our study includes the supra-adjacency, supra-Laplacian, and the probability transition matrices, which enable us to put our results under the perspective of structural phases in multiplex networks. We believe that this formalism and the results reported will make it possible to derive new results for multiplex networks in the future.