Goles, E., & Ruz, G. A. (2015). Dynamics of neural networks over undirected graphs. Neural Netw., 63, 156–169.
Abstract: In this paper we study the dynamical behavior of neural networks such that their interconnections are the incidence matrix of an undirected finite graph G = (V, E) (i.e., the weights belong to {0, 1}). The network may be updated synchronously (every node is updated at the same time), sequentially (nodes are updated one by one in a prescribed order) or in a blocksequential way (a mixture of the previous schemes). We characterize completely the attractors (fixed points or cycles). More precisely, we establish the convergence to fixed points related to a parameter alpha(G), taking into account the number of loops, edges, vertices as well as the minimum number of edges to remove from E in order to obtain a maximum bipartite graph. Roughly, alpha(G') < 0 for any G' subgraph of G implies the convergence to fixed points. Otherwise, cycles appear. Actually, for very simple networks (majority functions updated in a blocksequential scheme such that each block is of minimum cardinality two) we exhibit cycles with nonpolynomial periods. (C) 2014 Elsevier Ltd. All rights reserved.
