Voter model on heterogeneous graphs

We study the voter model on heterogeneous graphs. We exploit the nonconservation of the magnetization to characterize how consensus is reached. For a network of N nodes with an arbitrary but uncorrelated degree distribution, the mean time to reach consensus T-N scales as Nμ(2)(1)/μ(2), where μ(k) is the kth moment of the degree distribution. For a power-law degree distribution n(k)∼ k(-ν), T-N thus scales as N for ν> 3, as N/lnN for ν=3, as N(2ν-4)/(ν-1) for 2<ν< 3, as (lnN)(2) for ν=2, and as O(1) for ν< 2. These results agree with simulation data for networks with both uncorrelated and correlated node degrees.