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Uncovering the non-equilibrium stationary properties in sparse Boolean networks

Research output: Contribution to journalArticlepeer-review

Original languageEnglish
Article number053303
JournalJournal of Statistical Mechanics: Theory and Experiment
Issue number5
Early online date13 May 2022
Accepted/In press28 Mar 2022
E-pub ahead of print13 May 2022

Bibliographical note

Funding Information: G T is supported by the EPSRC Centre for Doctoral Training in Cross-Disciplinary Approaches to Non-Equilibrium Systems (CANES EP/L015854/1). Publisher Copyright: © 2022 IOP Publishing Ltd and SISSA Medialab srl.

King's Authors


Dynamic processes of interacting units on a network are out of equilibrium in general. In the case of a directed tree, the dynamic cavity method provides an efficient tool that characterises the dynamic trajectory of the process for the linear threshold model. However, because of the computational complexity of the method, the analysis has been limited to systems where the largest number of neighbours is small. We devise an efficient implementation of the dynamic cavity method which substantially reduces the computational complexity of the method for systems with discrete couplings. Our approach opens up the possibility to investigate the dynamic properties of networks with fat-tailed degree distribution. We exploit this new implementation to study properties of the non-equilibrium steady-state. We extend the dynamic cavity approach to calculate the pairwise correlations induced by different motifs in the network. Our results suggest that just two basic motifs of the network are able to accurately describe the entire statistics of observed correlations. Finally, we investigate models defined on networks containing bi-directional interactions. We observe that the stationary state associated with networks with symmetric or anti-symmetric interactions is biased towards the active or inactive state respectively, even if independent interaction entries are drawn from a symmetric distribution. This phenomenon, which can be regarded as a form of spontaneous symmetry-breaking, is peculiar to systems formulated in terms of Boolean variables, as opposed to Ising spins.

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