Conservation laws for the voter model in complex networks

Abstract

We consider the voter model dynamics in random networks with an arbitrary\ndistribution of the degree of the nodes. We find that for the usual node-update\ndynamics the average magnetization is not conserved, while an average\nmagnetization weighted by the degree of the node is conserved. However, for a\nlink-update dynamics the average magnetization is still conserved. For the\nparticular case of a Barabasi-Albert scale-free network the voter model\ndynamics leads to a partially ordered metastable state with a finite size\nsurvival time. This characteristic time scales linearly with system size only\nwhen the updating rule respects the conservation law of the average\nmagnetization. This scaling identifies a universal or generic property of the\nvoter model dynamics associated with the conservation law of the magnetization.\n