Neural Network Spectral Robustness under Perturbations of the Underlying Graph

Recent studies have been using graph-theoretical approaches to model complex networks (such as social, infrastructural, or biological networks) and how their hardwired circuitry relates to their dynamic evolution in time. Understanding how configuration reflects on the coupled behavior in a system of dynamic nodes can be of great importance, for example, in the context of how the brain connectome is affecting brain function. However, the effect of connectivity patterns on network dynamics is far from being fully understood. We study the connections between edge configuration and dynamics in a simple oriented network composed of two interconnected cliques (representative of brain feedback regulatory circuitry). In this article our main goal is to study the spectra of the graph adjacency and Laplacian matrices, with a focus on three aspects in particular: (1) the sensitivity and robustness of the spectrum in response to varying the intra- and intermodular edge density, (2) the effects on the spectrum of perturbing the edge configuration while keeping the densities fixed, and (3) the effects of increasing the network size. We study some tractable aspects analytically, then simulate more general results numerically, thus aiming to motivate and explain our further work on the effect of these patterns on the network temporal dynamics and phase transitions. We discuss the implications of such results to modeling brain connectomics. We suggest potential applications to understanding synaptic restructuring in learning networks and the effects of network configuration on function of regulatory neural circuits.