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

TL;DR

This paper introduces a novel approach using the cavity method to analyze the equilibrium and non-equilibrium dynamics of sparse Boolean networks with self-interactions and multi-node interactions, overcoming previous technical limitations.

Contribution

It presents a new mapping technique to handle self-interactions in Boolean networks, enabling detailed analysis of their long-term dynamics and attractor structures.

Findings

Mapping to bipartite systems simplifies analysis of self-interactions.

Systems with bidirectional interactions support multiple attractors.

The method allows studying transient and long-time behaviors.

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.