  We study simple interacting particle systems on heterogeneous networks,
including the voter model and the invasion process. These are both two-state
models in which in an update event an individual changes state to agree with a
neighbor. For the voter model, an individual "imports" its state from a
randomly-chosen neighbor. Here the average time T_N to reach consensus for a
network of N nodes with an uncorrelated degree distribution scales as N
mu_1^2/mu_2, where mu_k is the kth moment of the degree distribution. Quick
consensus thus arises on networks with broad degree distributions. We also
identify the conservation law that characterizes the route by which consensus
is reached. Parallel results are derived for the invasion process, in which the
state of an agent is "exported" to a random neighbor. We further generalize to
biased dynamics in which one state is favored. The probability for a single
fitter mutant located at a node of degree k to overspread the population--the
fixation probability--is proportional to k for the voter model and to 1/k for
the invasion process.
