From Centrality Measures to Nanowire Networks: disordered systems methods to the rescue

Abstract: This talk is divided into two parts. First, I will address why many seemingly non-local centrality measures, like Katz, PageRank, and Trophic Levels, strongly correlate with local quantities such as the ``degree". Using techniques from disordered systems, such as the replica trick and cavity method, we derive mean field theories in the dense large-N and in the sparse case. These methods are general for certain types of matrix inverses. Next, we shift focus to show why one of these techniques can be used for self-assembled nanowire "memristive" networks. At the nanoscale, memory effects in I-V characteristics enable memristive behavior. By treating these systems as networks governed by Kirchhoff's laws, we use disordered systems methods to derive a mean field theory for their dynamics in the annealed approximation. We’ll conclude by discussing the implications for neuromorphic computing and why some of these techniques are useful for understanding the asymptotic dynamics of these systems.