Overcoming the complexity barrier of the dynamic message-passing method in networks with fat-tailed degree distributions

TL;DR

This paper introduces a dynamic programming algorithm that reduces the computational complexity of the dynamic cavity method in networks with fat-tailed degree distributions, enabling analysis of more complex systems.

Contribution

It presents a novel algorithm that lowers the complexity from exponential to quadratic for systems with fat-tailed degree distributions, broadening the applicability of the dynamic cavity method.

Findings

Efficient analysis of Boolean networks with fat-tailed degree distributions.

Revealed heterogeneity in node activation patterns due to noise.

Demonstrated the algorithm's effectiveness in a case study.

Abstract

The dynamic cavity method provides the most efficient way to evaluate probabilities of dynamic trajectories in systems of stochastic units with unidirectional sparse interactions. It is closely related to sum-product algorithms widely used to compute marginal functions from complicated global functions of many variables, with applications in disordered systems, combinatorial optimization and computer science. However, the complexity of the cavity approach grows exponentially with the in-degrees of the interacting units, which creates a de-facto barrier for the successful analysis of systems with fat-tailed in-degree distributions. In this manuscript, we present a dynamic programming algorithm that overcomes this barrier by reducing the computational complexity in the in-degrees from exponential to quadratic, whenever couplings are chosen randomly from (or can be approximated in terms of) discrete, possibly unit-dependent, sets of equidistant values. As a case study, we analyse the dynamics of a random Boolean network with a fat-tailed degree distribution and fully asymmetric binary $\pm J$ couplings, and we use the power of the algorithm to unlock the noise dependent heterogeneity of stationary node activation patterns in such a system.