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 Nμ²₁/μ₂, 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/ln(N for ν=3, as N^{(2ν-4)/(ν-1)} for 2<ν<3, as (ln(N)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.