Symmetries in the time-averaged dynamics of networks: reducing unnecessary complexity through minimal network models

TL;DR

This paper demonstrates that network symmetries influence a wide range of dynamical processes, enabling the development of minimal models that simplify simulations while preserving essential behavior.

Contribution

It extends the understanding of network symmetries beyond synchronization, proposing reduction techniques for efficient simulation of complex network dynamics.

Findings

Nodes related by symmetry have identical time-averaged properties

Symmetry-based reductions are effective for various dynamical models

Minimal models optimize computational resources in network simulations

Abstract

Complex networks are the subject of fundamental interest from the scientific community at large. Several metrics have been introduced to characterize the structure of these networks, such as the degree distribution, degree correlation, path length, clustering coefficient, centrality measures etc. Another important feature is the presence of network symmetries. In particular, the effect of these symmetries has been studied in the context of network synchronization, where they have been used to predict the emergence and stability of cluster synchronous states. Here we provide theoretical, numerical, and experimental evidence that network symmetries play a role in a substantially broader class of dynamical models on networks, including epidemics, game theory, communication, and coupled excitable systems. Namely, we see that in all these models, nodes that are related by a symmetry relation show the same time-averaged dynamical properties. This discovery leads us to propose reduction techniques for exact, yet minimal, simulation of complex networks dynamics, which we show are effective in order to optimize the use of computational resources, such as computation time and memory.