Abstract
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.
Highlights
Numerous social and biological phenomena in complex networks can be modeled as dynamic processes in which vertices update their states by copying their neighbors
As well as in any other regular network, no difference exists between the sender-limited process (SLP), recipient-limited process (RLP), and connection-limited process (CLP); every vertex is likely to be a sender and likely to be a recipient
The purpose of this study is to demonstrate the differences between the SLP, RLP, and CLP in networks with a community structure
Summary
Numerous social and biological phenomena in complex networks can be modeled as dynamic processes in which vertices update their states by copying their neighbors. We assume that the system is homogeneous in the sense that all vertices are active senders, they are all active recipients, and all edges are open for transmission with equal rates Under these assumptions, the conventional voter model corresponds to the RLP. Research on random drift models assumed that the network was either a complete graph [18] or a regular lattice [10] In these networks, as well as in any other regular network (i.e., a network in which each vertex has the same number of neighbors), no difference exists between the SLP, RLP, and CLP; every vertex is likely to be a sender and likely to be a recipient.
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