Random Matrix Theory and Networks

Workshop Report

On 7-18 June 2021 the MPIPKS hosted a virtual workshop on Random Matrices and Networks.  The workshop counted 112 participants from 19 countries.  

The aim of the workshop was to bring together  researchers working on applications of random matrix theory to different branches of complex systems theory.
Random matrix theory was initiated about 80 years ago as a new mathematical tool to study many-body systems, such as, heavy nuclei or atoms. Standard models of random matrix theory rely on independent and identically distributed matrix entries. In recent years, new random matrix models have been developed that incorporate features of real-world systems, such as, network architecture, modularity, and recurrent motifs. These network models appear in the study of complex systems, such as, financial markets, signalling networks, neural networks, or ecosystems.    Therefore, we thought that it is a good idea to bring scientists together and discuss the recent progress.

A recurrent theme in the workshop was the linear stability of complex systems, as inspired by the seminal paper of Robert May in 1972. For example, Jean-Phillipe Bouchaud presented novel results on the stability of economies, Giorgio Carugno discussed the stability of complex fluids, Ariel Amir and Srdjan Ostojic examined the stability of gene regulatory networks and neural networks, respectively, and Tim Rogers analysed the fluctuation spectra of stable ecosystems.  Novel approaches that go beyond the linear stability analysis were also explored. For instance, Boris Khoruzenko explained how to compute the number of fixed-points in a nonlinear system, and Stefano Allesina discussed the feasibility of equilibria in ecosystems described by Lotka-Volterra equations. 

Another central theme of the workshop was the localisation of eigenvectors in random matrices.  David Nelson and Grace Zhang discussed localisation of right eigenvectors in one-dimensional chains, Paolo Barucca presented novel results for the eigenvector moments in time series analysis, Ivan Khaymovich discussed the multifractality of eigenvectors in random matrix models, Federico Ricci-Tersenghi presented recent results on the eigenvector localisation of the Hessian matrix of spin-glass models, and Diego Tapias discussed localisation phenomena in trap models on networks.  

Applications of random matrix theory are diverse, and this was also apparent from the wide variety of topics covered in the conference.   Other topics that were touched upon are the slow relaxation of non-equilibrium driven systems (Peter Sollich), new results to ranking nodes in a complex system (Fabio Caccioli), novel random matrix models for quasi-Hermitian and pseudo-Hermitian systems (Joshua Feinberg), the capacity of neural networks to store low dimensional manifolds (Rémi Monasson), the applications of random matrix theory to study exploration and search in complex networks (Reimer Kühn), the representation of complex data by simple models (Matteo Marsili), and different types of statistical inference problems on networks (Antoine Maillard, Carlo Lucibello, Andrea de Matino, and Rok Cestnik), and many others. Maciej Nowak concluded the conference with an inspiring talk on the links between random matrix theory and classical mechanics.  The conference also hosted a social event in terms of a talk by Giuseppe Mussardo on the enigma of J. Robert Oppenheimer that was joined by several members of the Max Planck Institute.  

The balance between experts and newcomers was an important characteristic of the event, which stimulated the discussion between the participants on the social media platform Gather. Taken together, the diversity of topics explored in the event provided a very interesting  overview of the most recent applications of random matrix theory to complex systems, which will hopefully inspire new researchers to enter in this field.