A Large Diffusion and Small Amplification Dynamics for Density Classification on Graphs

The density classification problem on graphs consists in finding a local dynamics such that, given a graph and an initial configuration of 0's and 1's assigned to the nodes of the graph, the dynamics converge to the fixed point configuration of all 1's if the fraction of 1's is greater than the critical density (typically 1/2) and, otherwise, it converges to the all 0's fixed point configuration. To solve this problem, we follow the idea proposed in [R. Briceno, P. M. de Espanes, A. Osses and I. Rapaport, Physica D 261, 70 (2013)], where the authors designed a cellular automaton inspired by two mechanisms: diffusion and amplification. We apply this approach to different well-known graph classes: complete, regular, star, Erdos-Renyi and Barabasi-Albert graphs.