An accurate mean-field equation for voter model dynamics on scale-free networks

TL;DR

This work tackles the inadequate performance of classical mean-field theory for the continuous-time noisy voter model on scale-free networks by introducing a modified mean-field equation (mMFE) that uses degree-weighted shares $ar{c}$ and a network-specific correlation factor $\alpha$. The mMFE is derived from first-order moment dynamics under a correlation and interchangeability framework, and it reduces to $\frac{d}{dt} \bar{c}_a = \sum_{b\neq a} \bar{c}_a(\alpha r_{a,b}\bar{c}_b + \tilde{r}_{a,b})(\mathbf{e}_b - \mathbf{e}_a)$; for $M=2$ this yields $\frac{d}{dt} \bar{c} = \alpha r_{0,1}(1-\bar{c})\bar{c} + \tilde{r}_{0,1}(1-\bar{c}) - \alpha r_{1,0}\bar{c}(1-\bar{c}) - \tilde{r}_{1,0}\bar{c}$. Error analysis shows small, robust errors in regimes with similar imitation rates and moderate noise (A/B), while time-varying correlations in other regimes (C/D) reduce accuracy unless a time-dependent $\alpha$ is considered. Numerically, the optimal $\alpha$ is network-dependent (e.g., $\alpha\approx0.81$ for BA with $m=3$, $\alpha\approx0.75$ for configuration model), and the mMFE outperforms standard MFE and PA with substantially fewer variables, offering a practical, scalable macroscopic description for CNVM on realistic networks. The work underscores the value of degree-weighted collective variables and data-driven insights for reduced modeling, with public code and potential extensions to time-varying or community-aware formulations.

Abstract

Understanding the emergent macroscopic behavior of dynamical systems on networks is a crucial but challenging task. One of the simplest and most effective methods to construct a reduced macroscopic model is given by mean-field theory. The resulting approximations perform well on dense and homogeneous networks but poorly on scale-free networks, which, however, are more realistic in many applications. In this paper, we introduce a modified version of the mean-field approximation for voter model dynamics on scale-free networks. The two main deviations from classical theory are that we use degree-weighted shares as coarse variables and that we introduce a correlation factor that can be interpreted as slowing down dynamics induced by interactions. We observe that the correlation factor is only a property of the network and not of the state or of parameters of the process. This approach achieves a significantly smaller approximation error than standard methods without increasing dimensionality.

Figures (10)

• Figure 1: Errors $\delta_{\text{cor}}(t; \alpha^*)$, $\delta_{\text{int}}(t)$ and $L_1(T; \alpha^*)$, as defined in \ref{['eq:dcor']}, \ref{['eq:dint']}, and \ref{['eq:L1_error']}, respectively, in the example systems of the different regimes A, B, C, D. Estimated from numerical simulations.
• Figure 2: Trajectories of $\bar{c}$, given in \ref{['def:wsh']}, estimated from simulations, for the example systems of the different regimes A B,C,D (left), compared to the mMFE \ref{['eq:modified_mfe_M2']}. The mMFE uses the optimal $\alpha^*$ as in \ref{['eq:optimal_alpha']}, shown on the right (green dashed). The right plot also shows the $\alpha$ that minimizes $\delta_{\text{cor}}(t;\alpha)$ for each $t$ (orange solid).
• Figure 3: Interchangeability error $\delta_{\text{int}}$, see \ref{['eq:dint']}, when using the regular shares $c$ instead of the degree-weighted shares $\bar{c}$, for regimes A and B (with different initial conditions than in Fig. \ref{['fig:errors_assumptions']}).
• Figure 4: $L_1$-error \ref{['eq:L1_error']} of the mMFE for different initial conditions and parameters in regime A and B (top). Even for the two trajectories (a) and (b) with largest errors (bottom) the approximation is very good. Solid lines indicate true weighted shares and dashed/dotted lines the solution of the mMFE.
• Figure 5: The factor $\alpha$ in the mMFE \ref{['eq:modified_mfe']} depends on the preferential attachment parameter $m$ of the BA graph.