Dynamics of neural networks over undirected graphs

Eric Goles, Gonzalo A. Ruz

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


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.

Original languageEnglish
Pages (from-to)156-169
Number of pages14
JournalNeural Networks
StatePublished - 1 Mar 2015


  • Attractors
  • Cycles
  • Discrete updating schemes
  • Fixed points
  • Neural networks
  • Undirected graphs


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