## 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.

Original language | English |
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Pages (from-to) | 156-169 |

Number of pages | 14 |

Journal | Neural Networks |

Volume | 63 |

DOIs | |

State | Published - 1 Mar 2015 |

## Keywords

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