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.