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 $\theta$, we derive a rich phase diagram, with the consensus time exhibiting various scaling laws depending on $\theta$ and on the exponent of the degree distribution $\gamma$. Numerical simulations give very good agreement for small values of $\left|\theta \right|$. An additional analytical treatment (heterogeneous pair approximation) improves the agreement with numerics, but the theoretical understanding of the behavior in the limit of large $\left|\theta \right|$ remains an open challenge.

Figure 1

Phase diagram of the voter model (left-hand side) and Moran process (right-hand side) on weighted scale-free networks.

Figure 2

Exit probability $E_1(k)$ starting from a single $+1$ spin in a vertex of degree $k$, for the voter model (filled symbols) and the Moran process (open symbols). Dashed lines represent the expected theoretical scaling with $k$, circles refer to the case $\theta =0$ and squares to $\theta =1$. Data from quenched networks of size $N={10}^3$ with $\gamma =2.5$ (voter model) and $\gamma =2.2$ (Moran process).

Figure 3

Scaling with $N$ for the voter model (left-hand side) and Moran process (right-hand side) on scale-free weighted networks in different regions of the corresponding phase diagrams, Fig. 1. Squares represent data from simulations run on annealed networks, while circles concern quenched graphs. Dashed lines represent the theoretical scaling predicted by HMF theory.

Figure 4

Consensus time for the voter model (left-hand side) and Moran process (right-hand side) in slices of the phase diagrams at two fixed values of $\gamma$ and varying $\theta$, compared with the corresponding HMF predictions, Eqs. (23) and (25), respectively (full lines). Results from simulations performed on quenched networks of size $N=3\times {10}^3$.

Figure 5

Probability ${\rho }_k^S$ that an edge connected to a node of degree $k$ and selected for the dynamics is active as a function of $k$, in the quasisteady state for $x=1/2$. The solid line is the result of the heterogeneous pair approximation, while symbols are results of numerical simulations. Binning has deliberately been avoided to show the large variability of numerical results for larger degrees. Data from voter dynamics on quenched networks of size $N={10}^5$, with $\gamma =3.25$ and $\theta =1.5$.

Figure 6

Comparison of the consensus time $T_N$ as a function of $N$ obtained in numerical simulations for voter dynamics on quenched networks (open circles) and the results of the numerical evaluation of the HMF, Eq. (23) (filled circles), and HPA, Eq. (37) (open squares) predictions. Data correspond to networks with $\gamma =2.75$, $\theta =0.5$ (top), $\gamma =3.25$, $\theta =1.5$ (center), and $\gamma =4$, $\theta =4$ (bottom).