  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.
