Finite size effects and symmetry breaking in the evolution of networks of competing Boolean nodes

TL;DR

This paper investigates how finite size influences the evolution of Boolean networks, revealing symmetry breaking and additional selection pressures that differ from infinite networks, with implications for real-world complex systems.

Contribution

It demonstrates that finite size networks exhibit different evolutionary dynamics and symmetry breaking, introducing new selection for input-inverting functions not present in infinite networks.

Findings

Finite size networks evolve differently than infinite ones.

Symmetry breaking leads to selection of input-inverting functions.

Empirical results match analytic predictions using Polya's theorem.

Abstract

The effects of the finite size of the network on the evolutionary dynamics of a Boolean network are analyzed. In the model considered, Boolean networks evolve via a competition between nodes that punishes those in the majority. It is found that finite size networks evolve in a fundamentally different way than infinitely large networks do. The symmetry of the evolutionary dynamics of infinitely large networks that selects for canalizing Boolean functions is broken in the evolutionary dynamics of finite size networks. In finite size networks there is an additional selection for input inverting Boolean functions that output a value opposite to the majority of input values. These results are revealed through an empirical study of the model that calculates the frequency of occurrence of the different possible Boolean functions. Classes of functions are found to occur with the same frequency. Those classes depend on the symmetry of the evolutionary dynamics and correspond to orbits of the relevant symmetry group. The empirical results match analytic results, determined by utilizing Polya's theorem, for the number of orbits expected in both finite size and infinitely large networks. The reason for the symmetry breaking in the evolutionary dynamics is found to be due to the need for nodes in finite size networks to behave differently in order to cooperate so that the system collectively performs as well as possible. The results suggest that both finite size effects and symmetry are important for understanding the evolution of real-world complex networks, including genetic regulatory networks.