Voter model with non-Poissonian interevent intervals

Recent analysis of social communications among humans has revealed that the interval between interactions for a pair of individuals and for an individual often follows a long-tail distribution. We investigate the effect of such a non-Poissonian nature of human behavior on dynamics of opinion formation. We use a variant of the voter model and numerically compare the time to consensus of all the voters with different distributions of interevent intervals and different networks. Compared with the exponential distribution of interevent intervals (i.e., the standard voter model), the power-law distribution of interevent intervals slows down consensus on the ring. This is because of the memory effect; in the power-law case, the expected time until the next update event on a link is large if the link has not had an update event for a long time. On the complete graph, the consensus time in the power-law case is close to that in the exponential case. Regular graphs bridge these two results such that the slowing down of the consensus in the power-law case as compared to the exponential case is less pronounced as the degree increases.

Figure 1

Schematic of interface on the ring. The local network surrounding an interface (pointed by the arrow) is depicted. Open and solid circles represent voters with opinions 0 and 1, respectively.

Figure 2

Time course of the fraction of 1 voters $m$ (gray lines) and the number of interfaces $E_{\mathrm{if}}$ (black lines) in the (a) exponential and (b) power-law voter models on the ring with 100 nodes. The values of $E_{\mathrm{if}}$ shown are those normalized by the total number of links in the ring, which is equal to $N$.

Figure 3

Schematics of the movement of an interface on the ring. In (a), no update event occurs on link $\ell$ between time 0 and time $t$. In (b), at least one update event occurs between time 0 and time $t$. In (b), the time of the last update event before $t$ is denoted as $t_{\mathrm{last}}$. The nodes and links near link $\ell$ on the ring are depicted above the time lines of the update events. Open and solid circles represent voters with opinion 0 and 1, respectively.

Figure 4

(a) Average consensus time of the exponential voter model (circles) and power-law voter model with $\alpha =3.5$ (squares) on the ring (main panel). The solid line indicates $T\propto N^2$. For the results shown in the main panel, $T^{\mathrm{power}}/T^{\exp }$ (squares) and $\langle t-t_{\mathrm{last}}\rangle$ for the power-law voter model (circles) are plotted against $N$ in the inset. (b) Average consensus time of the power-law voter model with $\alpha =2.5$ on the ring as a function of the number of simulation runs. The error bars indicate the standard deviation. We set $N=100$.

Figure 5

Average consensus time of the exponential (circles) and power-law (squares) voter models on the complete graph with $N$ nodes. The solid line is the analytical solution given by Eq. (22) for the exponential voter model; i.e., $T=N\ln 2$.

Figure 6

(a) Average number of the effective events $\langle M_{\mathrm{e}}\rangle$ and (b) average interval between successive effective events $\langle {\tau }_{\mathrm{e}}\rangle$ on the complete graph for the exponential (circles) and power-law (squares) voter models.

Figure 7

(Color online) Expected interval between effective update events ${\tau }_{\mathrm{e}}$ for the exponential and power-law voter models, i.e., Eqs. (26) and (28).

Figure 8

$T^{\mathrm{power}}/T^{\exp }$ for the extended ring (filled symbols) and for the RRG (empty symbols) with degree $k$. We set (a) $N=500$ and (b) $N=1000$.