Matteo Tanzi (University of Victoria)

Title: Heterogeneously Coupled Maps: Rigorous Mean Field Reduction and Dynamics Reconstruction

Abstract: 
A graph with heterogeneous degrees has most of its nodes making a small number of connections, while the remaining nodes, called hubs, have very high degree. This type of graph is ubiquitously found in models of natural and artificial systems made of interacting components such as, among others, neuronal networks, gene-regulatory networks, and power grids. I will report recent results addressing the ergodic theoretical properties of uniformly expanding maps coupled on such graphs, focusing the attention on the case where the number of nodes in the graph is very large. These high-dimensional systems have a regime of the coupling strength where the interaction is small for poorly connected systems, but large for the hub nodes so that global hyperbolicity might be lost. It is shown that, under certain hypotheses on the network, the dynamics of the hubs  can be generically well approximated by a low-dimensional hyperbolic system for exponentially long time in the size of the network, and even if this describes only a long transient, it could be the only behaviour observed in experiments. 
The results above justify the emergence of macroscopic behaviour such as coherence of dynamics among hubs with the same number of connections. They also suggest an algorithm to reconstruct the network structure and dynamical features from observations of the dynamics at every node. Tested on computer simulations, the algorithm is able to effectively recover the degree distribution of the network, community structures, local dynamics and effective coupling. Furthermore, it is shown to work in a much wider variety of cases than that covered by the rigorous results.