Spectra of random networks in the weak clustering regime

¹ Institute of Mathematics and Computer Science, University of São Paulo - São Carlos, SP 13566-590, Brazil
² Institute of Science and Technology for Brain-inspired Intelligence, Fudan University - Shanghai 200433, PRC
³ Potsdam Institute for Climate Impact Research (PIK) - 14473 Potsdam, Germany
⁴ Department of Physics, Humboldt University - 12489 Berlin, Germany
⁵ Mathematics Institute, University of Warwick - Gibbet Hill Road, Coventry CV4 7AL, UK
⁶ Centre for Complexity Science, University of Warwick - Coventry CV4 7AL, UK

Received: 28 March 2018
Accepted: 23 April 2018

Abstract

The asymptotic behavior of dynamical processes in networks can be expressed as a function of spectral properties of the corresponding adjacency and Laplacian matrices. Although many theoretical results are known for the spectra of traditional configuration models, networks generated through these models fail to describe many topological features of real-world networks, in particular non-null values of the clustering coefficient. Here we study effects of cycles of order three (triangles) in network spectra. By using recent advances in random matrix theory, we determine the spectral distribution of the network adjacency matrix as a function of the average number of triangles attached to each node for networks without modular structure and degree-degree correlations. Implications to network dynamics are discussed. Our findings can shed light in the study of how particular kinds of subgraphs influence network dynamics.