Voter models on weighted networks.

We study the dynamics of the voter and Moran processes running on top of complex network substrates where each edge has a weight depending on the degree of the nodes it connects. For each elementary dynamical step the first node is chosen at random and the second is selected with probability proportional to the weight of the connecting edge. We present a heterogeneous mean-field approach allowing to identify conservation laws and to calculate exit probabilities along with consensus times. In the specific case when the weight is given by the product of nodes' degree raised to a power θ, we derive a rich phase diagram, with the consensus time exhibiting various scaling laws depending on θ and on the exponent of the degree distribution γ. Numerical simulations give very good agreement for small values of |θ|. An additional analytical treatment (heterogeneous pair approximation) improves the agreement with numerics, but the theoretical understanding of the behavior in the limit of large |θ| remains an open challenge.