Dynamics of neural networks over undirected graphs. (March 2015)

• Record Type: Journal Article
• Authors: Goles, Eric; Ruz, Gonzalo A.
• Is Part Of: Neural networks. Volume 63(2015:Mar.)
• Page Start: 156
• Page End: 169
• Publication Date: 2015-03
• Subjects: Neural networks -- Undirected graphs -- Discrete updating schemes -- Attractors -- Fixed points -- Cycles; Neural computers -- Periodicals; Neural networks (Computer science) -- Periodicals; Neural networks (Neurobiology) -- Periodicals; Nervous System -- Periodicals; Ordinateurs neuronaux -- Périodiques; Réseaux neuronaux (Informatique) -- Périodiques; Réseaux neuronaux (Neurobiologie) -- Périodiques; Neural computers; Neural networks (Computer science); Neural networks (Neurobiology); Periodicals; 006.32
• DOI: 10.1016/j.neunet.2014.10.013
• Languages: English
• ISSNs: 0893-6080

Abstract

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.