Symmetry and Symmetry-Breaking of the Emergent Dynamics of the Discrete Stochastic Majority-Voter Model

We analyse the emergent dynamics of the so called majority voter model evolving
on complex networks. In particular we study the influence of three characteristic
types of networks, namely Random Regular, Erd¨os-R´enyi (ER), Watts and Strogatz
(WS, small-world) and Barabasi (scale- free) on the bifurcating stationary coarsegrained
solutions. We first prove analytically some simple properties about the
symmetry and symmetry breaking of the macroscopic dynamics with respect to the
network topology. We also show how one can exploit the Equation-free framework to
bridge in a computational strict manner the micro to macro scales of the dynamics of
stochastic individualistic models on complex random graphs. In particular, we show
how systems-level tasks such as bifurcation analysis of the coarse-grained dynamics
can be performed bypassing the need to extract macroscopic models in a closed
form. A comparison with the mean-field approximations is also given illustrating the
merits of the Equation-Free approach, especially in the case of scale-free networks
exhibiting a heavy-tailed connectivity distribution.