title: Dynamics of neural networks over undirected graphs.
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 block-sequential 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 α(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, α(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 block-sequential scheme such that each block is of minimum cardinality two) we exhibit cycles with non-polynomial periods.
authors.affiliation: Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Av. Diagonal Las Torres 2640, Santiago, Chile. Electronic address: eric.chacc@uai.cl.
authors.forename: Eric
authors.initials: E
authors.lastname: Goles
authors.affiliation: Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Av. Diagonal Las Torres 2640, Santiago, Chile; Center of Applied Ecology and Sustainability (CAPES), Santiago, Chile. Electronic address: gonzalo.ruz@uai.cl.
authors.forename: Gonzalo A
authors.initials: GA
authors.lastname: Ruz
mesh_terms: D000465:Algorithms; D016571:Neural Networks, Computer
keywords: Attractors; Cycles; Discrete updating schemes; Fixed points; Neural networks; Undirected graphs