Role of network topology in non-equilibrium dynamics

Determine the role of network topology in the non-equilibrium dynamics of complex systems, specifically identifying how structural features of directed networks affect violations of detailed balance and entropy production across stochastic dynamical processes.

Background

The paper investigates how directedness and hierarchical organization in networks relate to broken detailed balance and entropy production for several dynamical processes (random walks, Ornstein–Uhlenbeck, and Ising dynamics). While it presents analytical and numerical evidence linking directedness to non-equilibrium behavior, it acknowledges that fully understanding the contribution of network topology to non-equilibrium dynamics remains unresolved.

This open problem is framed broadly and motivates the study’s development of directedness measures (irreciprocity, trophic directedness, and non-normality) and its exploration of synthetic, empirical, and inferred networks. The authors’ results provide partial progress but do not close the question, highlighting the need for a comprehensive theory connecting topology to non-equilibrium thermodynamics.

References

Understanding the role of network topology remains an important unsolved problem in the non-equilibrium dynamics of complex systems.

— Broken detailed balance and entropy production in directed networks  (2402.19157 - Nartallo-Kaluarachchi et al., 2024) in Section 1. Introduction

A natural direction for future work is to generalize the present framework from a one-dimensional chain to arbitrary networks, with each node hosting an Ornstein--Uhlenbeck process and interacting non-reciprocally with its connected neighbours. In this setting, the dynamics can be written as \begin{align} \dot{\vec{x}=M\mathbf{x}+\vec{\eta},\qquad M=-K+A, \end{align} where $K=k \delta_{ij}$ contains the stiffness coefficients, and $A$ is the weighted directed adjacency matrix encoding the inter-node non-reciprocal interactions. The underlying network topology then provides an additional control parameter, naturally raising the question of whether non-reciprocity can be harnessed to enhance targeted transport.

— Non-reciprocally interacting Ornstein-Uhlenbeck processes: Exceptional points, Anomalous relaxation, Pseudo-equilibrium and Boundary refrigeration  (2609.05391 - Pal et al., 4 Sep 2026) in Conclusion and Outlook, second bullet point