Phase transitions in stochastic non-linear threshold Boolean automata networks on Z²: the boundary impact

Jacques Demongeot (1) , Sylvain Sené (2, 3)

1 AGEIS - Autonomie, Gérontologie, E-santé, Imagerie & Société [Grenoble]
2 IXXI - Institut Rhône-Alpin des systèmes complexes
3 CANA - Calcul Naturel

This paper addresses the question of the impact of the boundary on the dynamical behaviour of finite Boolean automata networks on Z 2. The evolution over discrete time of such networks is governed by a specific stochastic threshold non-linear transition rule derived from the classical rule of formal neural networks. More precisely, the networks considered in this paper are finite but the study is done for arbitrarily large sizes. Moreover, the boundary impact is viewed as a classical definition of a phase transition in probability theory, characterising in our context the fact that a network admits distinct asymptotic behaviours when different boundary instances are assumed. The main contribution of this paper is the highlight of a formula for a necessary condition for boundary sensitivity, whose sufficiency and necessity are entirely proven with natural constraints on interaction potentials.

Mots clés: Boolean automata networks; non-linearity; stochastic pro- cesses; phase transitions; boundary sensitivity

Domaines: Informatique [cs]

Jacques Demongeot, Sylvain Sené. Phase transitions in stochastic non-linear threshold Boolean automata networks on Z²: the boundary impact. Advances in Applied Mathematics, 2018, 98, pp.77--99. ⟨hal-01785459⟩