  We study the voter dynamics model on heterogeneous graphs. We exploit the
non-conservation of the magnetization to characterize how consensus is reached
on networks with different connectivity patterns. For a network of N sites with
an arbitrary degree distribution, we show that the mean time to reach consensus
T_N scales as N mu_1^2/mu_2, where mu_k is the kth moment of the degree
distribution. For a power-law degree distribution n_k k^{-nu}, we thus find
that T_N scales as N for nu>3, as N/ln N for nu=3, as N^{(2nu-4)/(nu-1)} for
2<nu<3, as (ln N)^2 for nu=2, and as order one for nu<2.
