Dynamics of sparse Boolean networks with multi-node and self-interactions

Abstract

We analyse the equilibrium behaviour and non-equilibrium dynamics of sparse Boolean networks with self-interactions that evolve according to synchronous Glauber dynamics. Equilibrium analysis is achieved via a novel application of the cavity method to the temperature-dependent pseudo-Hamiltonian that characterises the equilibrium state of systems with parallel dynamics. Similarly, the non-equilibrium dynamics can be analysed by using the dynamical version of the cavity method. It is well known, however, that when self-interactions are present, direct application of the dynamical cavity method is cumbersome, due to the presence of strong memory effects, which prevent explicit analysis of the dynamics beyond a few time steps. To overcome this difficulty, we show that it is possible to map a system of $N$ variables to an equivalent bipartite system of $2N$ variables, for which the dynamical cavity method can be used under the usual one time approximation scheme. This substantial technical advancement allows for the study of transient and long-time behaviour of systems with self-interactions. Finally, we study the dynamics of systems with multi-node interactions, recently used to model gene regulatory networks, by mapping this to a bipartite system of Boolean variables with 2-body interactions. We show that when interactions have a degree of bidirectionality such systems are able to support a multiplicity of diverse attractors, an important requirement for a gene-regulatory network to sustain multi-cellular life.

arXiv comment39 pages, 11 figures,

References in corpus (16)

• The Bethe lattice spin glass revisited
• Simplest random K-satisfiability problem
• Exact solutions for diluted spin glasses and optimization problems
• Parallel dynamics of disordered Ising spin systems on finitely connected random graphs
• The Cavity Approach to Parallel Dynamics of Ising Spins on a Graph
• Dynamic mean-field and cavity methods for diluted Ising systems
• Effect of dilution in asymmetric recurrent neural networks
• The Little-Hopfield model on a Random Graph
• Parallel dynamics of disordered Ising spin systems on finitely connected directed random graphs with arbitrary degree distributions
• Diluted Mean-Field Spin-Glass Models at Criticality
• A message-passing scheme for non-equilibrium stationary states
• Inference of kinetic Ising model on sparse graphs
• Loop expansion around the Bethe approximation through the $M$-layer construction
• One+Infinite Dimensional Attractor Neural Networks
• Phase transitions and memory effects in the dynamics of Boolean networks
• Vaccination with partial transmission and social distancing on contact networks

Cited by in corpus (1)

• Nonequilibrium dynamics of the Ising model on heterogeneous networks with an arbitrary distribution of threshold noise