Effects of communication burstiness on consensus formation and tipping points in social dynamics

Current models for opinion dynamics typically utilize a Poisson process for speaker selection, making the waiting time between events exponentially distributed. Human interaction tends to be bursty though, having higher probabilities of either extremely short waiting times or long periods of silence. To quantify the burstiness effects on the dynamics of social models, we place in competition two groups exhibiting different speakers' waiting-time distributions. These competitions are implemented in the binary naming game and show that the relevant aspect of the waiting-time distribution is the density of the head rather than that of the tail. We show that even with identical mean waiting times, a group with a higher density of short waiting times is favored in competition over the other group. This effect remains in the presence of nodes holding a single opinion that never changes, as the fraction of such committed individuals necessary for achieving consensus decreases dramatically when they have a higher head density than the holders of the competing opinion. Finally, to quantify differences in burstiness, we introduce the expected number of small-time activations and use it to characterize the early-time regime of the system.

Figure 1

PDFs for the nonexponential speakers' waiting-time distributions with various chosen parameters compared to the exponential one: (a) a power law with lower cutoff, (b) a shifted power law, (c) a Weibull distribution, and (d) a uniform distribution.

Figure 2

Fraction of runs (out of 1000 trials) vs system size that the non-Poisson ($A$-opinion) nodes won the opinion competition against the Poisson ($B$-opinion) nodes in the binary NG on a complete graph. As before, the speakers' waiting-time distribution for the non-Poisson nodes is (a) a power law with lower cutoff, (b) a shifted power law, (c) a Weibull distribution, and (d) a uniform distribution.

Figure 3

Time to consensus conditioned on each side winning in the binary NG on a complete graph. Initially, half the nodes have a nonexponential speakers' waiting-time distribution and hold opinion $A$, while the other half follows an exponential distribution and holds opinion $B$. (a) Results for the power law with a lower cutoff and $\gamma =1.7$ in an attempt to show results for a system with similar burstinesses. Also shown is the case of a more bursty nonexponential distribution for (b) the shifted power law with $\gamma =2.9$ and (c) the Weibull distribution with $\alpha =0.7$. (d) The less bursty case for a uniform distribution with $b=1.9$. All simulations were run 10 000 times and then averaged based on their final consensus.

Figure 4

Time to consensus conditioned on each side winning in the binary NG on a complete graph, where half of the nodes are in opinion $A$ initially and follow a Weibull waiting-time distribution and the other half of the nodes initially are in opinion $B$ following an exponential waiting-time distribution. The results of the conditioned simulations are shown for the Weibull distribution with (a) $\alpha =0.7$, (b) $\alpha =0.85$, (c) $\alpha =1$, (d) $\alpha =1.15$, and (e) $\alpha =1.3$.

Figure 5

Average number of speaking events in the binary NG for each type of node per one system time step over the course of the entire simulation. Each simulation is averaged over 1000 runs with $N=1000$ on a complete graph. Shown, as before, are (a) the power law with lower cutoff, (b) the shifted power law, (c) the Weibull distribution, and (d) the uniform distribution.

Figure 6

Effects committed agents with non-Poisson speakers' communication patterns in the binary NG on a complete graph. (a) Critical fraction of committed agents (tipping point) necessary to create consensus for the minority opinion with respect to the various parameters that control the burstiness of the committed agents' waiting-time distributions, averaged over 1000 runs on systems with $N=1000$. It is important to note that the parameters $\gamma , \alpha$, and $b$ are specific to the distribution in which they are used and their impact on the burstiness varies from one distribution to another. As such, they should not be compared directly to each other. (b) Fraction of runs reaching consensus driven by a committed minority by time $t=150$. Committed agents have a power law with the lower cutoff speakers' waiting-time distribution with the parameter $\gamma$. The critical fractions $p_c$ [shown in (a)] were defined as the population at which the system reaches consensus (of the initial minority opinion) in over half of the runs. (c) Finite-size effects of the tipping point for a power law with the lower cutoff and $\gamma =1.5$, indicating no significant shift in the value of $p_c$ as $N\to \infty$.

Figure 7

Comparison of the second-order approximation of the small-time activation densities for each of the nonexponential distributions vs the exponential: (a) the shifted power law, (b) the power law with lower cutoff, (c) the uniform distribution, and (d) the Weibull distribution.

Figure 8

Fraction of runs (out of 1000 trials) vs network size that the non-Poisson ($A$-opinion) nodes won the opinion competition against the Poisson ($B$-opinion) nodes in the voter model on a complete graph. As before, the speakers' waiting-time distribution for the non-Poisson nodes is (a) a power law with lower cutoff, (b) a shifted power law, (c) a Weibull distribution, and (d) a uniform distribution.

Figure 9

(a) Critical populations of committed nodes (tipping points) in the binary NG on a complete graph when each node in the network has identical waiting-time distributions and the system size is $N=1000$. (b) Fraction of runs reaching consensus in 1000 simulations (by time $t=150$) vs the fraction of committed individuals for various system sizes. In this plot, each node has the Weibull waiting-time distribution with $\alpha =1.3$.

Figure 10

(a) Number of speaking events per unit system time relative to the type of waiting-time distribution used in the system for the same simulations as in Fig. 9. As such, the simulations are still done on a system of $N=1000$, over 1000 runs on a complete graph. (b) Average number of speakers' events per time step for a single node updating with a Weibull distributed waiting time over different time intervals. The values are for the updates of a single node averaged over 1000 simulations. The inset shows the data for $\alpha =0.1$ on extended (logarithmic) time scales.

Figure 11

Fraction of runs (out of 1000 trials) that reached consensus on opinion $A$ in ER networks with $N=1000$ nodes and various values of the average degree $\langle k\rangle$. Half of the nodes follow a nonexponential waiting-time distribution and initially have opinion $A$. The other half follow the exponential waiting-time distribution and initially have opinion $B$. The nonexponential distributions are (a) the shifted power law, (b) the power law with lower cutoff, (c) the Weibull distribution, and (d) the uniform distribution.

Figure 12

Critical population $p_c$ of committed nodes following a nonexponential wait-time distribution that resulted in half of 1000 trials reaching consensus on opinion $A$ in ER networks with $N=1000$ nodes and various values of the average degree $\langle k\rangle$. In each simulation a minority fraction of the population $p$ is committed to opinion $A$ and follow a nonexponential waiting-time distribution. The rest of the nodes have opinion $B$ and follow the exponential distribution. The nonexponential distributions are (a) the shifted power law, (b) the power law with lower cutoff, (c) the Weibull distribution, and (d) the uniform distribution.