This week, CSH visitor Fernando Lucas Metz (Federal University of Rio Grande do Sul) is presenting a talk on Thursday, May 19, starting at 3PM. The talk will take place on-site as well as online via zoom.
Please send us an email if you would like to attend.
Title: Random matrix theory and complex networks
Understanding the relationship between the structure of complex networks and their spectra is one of the central and long-standing focus of network theory. Spectral properties of networks determine the performance of algorithms, phase transitions, and the linear stability of complex biological systems, such as neural networks and ecosystems. However, studies of network spectra, derived from the adjacency and Laplacian matrices, are greatly complicated by the sparse interaction structure among the network elements, which invalidates the application of traditional tools of Gaussian random matrix theory. In this talk, I will discuss how random matrices provide the natural language to model complex systems, and how certain techniques of random matrix theory, after being adapted to sparse systems, have led to very interesting advances in our understanding of the spectral properties of networks. In the first part of the talk, I will give a basic overview of random matrix theory and discuss some differences between the spectrum of Gaussian and sparse random matrices. In the second part of the talk, I will present simple and exact analytic expressions for the leading eigenvalue and eigenvector of directed complex networks with arbitrary degree distributions. Such expressions uncover the impact of the network structure on eigenvector localization and on the stability of dynamical systems on networks.