  Competition between alternative states is an essential process in social and
biological networks. Neutral competition can be represented by an unbiased
random drift process in which the states of vertices (e.g., opinions,
genotypes, or species) in a network are updated by repeatedly selecting two
connected vertices. One of these vertices copies the state of the selected
neighbor. Such updates are repeated until all vertices are in the same
"consensus" state. There is no unique rule for selecting the vertex pair to be
updated. Real-world processes comprise three limiting factors that can
influence the selected edge and the direction of spread: (1) the rate at which
a vertex sends a state to its neighbors, (2) the rate at which a state is
received by a neighbor, and (3) the rate at which a state can be exchanged
through a connecting edge. We investigate how these three limitations influence
neutral competition in networks with two communities generated by a stochastic
block model. By using Monte Carlo simulations, we show how the community
structure and update rule determine the states' success probabilities and the
time until a consensus is reached. We present a heterogeneous mean-field theory
that agrees well with the Monte Carlo simulations. The effectiveness of the
heterogeneous mean-field theory implies that quantitative predictions about the
consensus are possible even if empirical data (e.g., from ecological fieldwork
or observations of social interactions) do not allow a complete reconstruction
of all edges in the network.
