Generalized voter-like models on heterogeneous networks

We describe a generalization of the voter model on complex networks that encompasses different sources of degree-related heterogeneity and that is amenable to direct analytical solution by applying the standard methods of heterogeneous mean-field theory. Our formalism allows for a compact description of previously proposed heterogeneous voter-like models, and represents a basic framework within which we can rationalize the effects of heterogeneity in voter-like models, as well as implement novel sources of heterogeneity, not previously considered in the literature.

💡 Research Summary

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The paper presents a unified theoretical framework for voter‑like dynamics on complex networks that incorporates a broad spectrum of degree‑related heterogeneities while remaining analytically tractable through heterogeneous mean‑field (HMF) theory. The authors begin by recalling the classic voter model and the Moran process, both of which describe binary‑state dynamics (σ = ±1) on a graph but differ in the direction of copying: in the voter model the selected node copies its neighbor, whereas in the Moran process the neighbor copies the selected node. Although these two processes are equivalent on regular lattices, they diverge on heterogeneous networks because the order of selection and the probability of picking a node become degree‑dependent.

To capture this richness, the authors introduce two node‑specific quantities: a “fitness” f_i that biases the probability that node i initiates an update, and a pairwise copying probability Q(i,j) that determines whether the selected node i actually copies the state of its neighbor j. The update rule is: (i) pick a source node i with probability proportional to f_i; (ii) pick a random neighbor j of i; (iii) with probability Q(i,j) set σ_i ← σ_j. The standard voter model is recovered by f_i = 1 and Q(i,j) = 1, while the Moran process corresponds to f_i = 1 and Q(i,j) = k_i/k_j. The microscopic copying rate therefore reads C_{ij}=f_i a_{ij} k_i Q(i,j), where a_{ij} is the adjacency matrix.

The authors then apply heterogeneous mean‑field theory, which (i) groups nodes by degree k, assuming all nodes of the same degree share identical dynamical properties, and (ii) replaces the quenched network by an annealed version where the probability that a degree‑k node is connected to a degree‑k′ node is P(k′|k)=k′P(k′)/⟨k⟩ for uncorrelated networks. Averaging the fitness and copying probability over degree classes yields f_k and Q(k,k′). The mesoscopic copying rate becomes C(k,k′)=f(k) P(k′|k) Q(k,k′). To make further progress, the authors factorize Q(k,k′) as a(k) b(k′) s(k,k′), where a(k) and b(k′) depend only on the source and target degrees respectively, while s(k,k′) is a symmetric interaction kernel. This factorization is broad enough to encompass most previously studied voter‑like variants.

Defining x_k(t) as the fraction of degree‑k nodes in state +1, the authors derive transition rates Π(k;±1) and obtain the rate equation
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