A polynomial eigenvalue approach for multiplex networks

Author(s)

de Arruda, Guilherme Ferraz ^{[1, 2]} ; Cozzo, Emanuele ^{[3, 4]} ; Rodrigues, Francisco A. ^[1] ; Moreno, Yamir ^{[2, 3, 4]}

Affiliation

^[1] Univ Sao Paulo, Inst Ciencias Matemat & Comp, Dept Matemat Aplicada & Estat, Campus Sao Carlos, Caixa Postal 668, BR-13560970 Sao Carlos, SP - Brazil

^[2] ISI Fdn, Via Chisola 5, I-10126 Turin - Italy

^[3] Univ Zaragoza, Inst Biocomputat & Phys Complex Syst BIFI, E-50018 Zaragoza - Spain

^[4] Univ Zaragoza, Dept Theoret Phys, E-50018 Zaragoza - Spain

• Document type: Journal article
• Source: NEW JOURNAL OF PHYSICS; v. 20, SEP 20 2018.
• Web of Science Citations: 4

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. Specifically, our formalism is based on the reduction of the dimensionality of a matrix of interest but increasing the power of the characteristic polynomial, i.e, a polynomial eigenvalue problem. This approach may sound counterintuitive at first, but it enable us to relate the quadratic eigenvalue problem for a 2-Layer multiplex network with the spectra of its respective aggregated network. Additionally, it also allows us to derive bounds for the spectra, among many other interesting analytical insights. Furthermore, it also permits us 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 enables 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. (AU)

• FAPESP's process: 13/26416-9 - Modelling of dynamical processes in complex networks
• Grantee: Francisco Aparecido Rodrigues
• Support Opportunities: Regular Research Grants
• FAPESP's process: 12/25219-2 - Modeling, analysis and simulation of dynamic process on complex networks
• Grantee: Guilherme Ferraz de Arruda
• Support Opportunities: Scholarships in Brazil - Doctorate