Consensus from group interactions: An adaptive voter model on hypergraphs

We study the effect of group interactions on the emergence of consensus in a spin system. Agents with discrete opinions $\{0,1\}$ form groups. They can change their opinion based on their group's influence (voter dynamics), but groups can also split and merge (adaptation). In a hypergraph, these groups are represented by hyperedges of different sizes. The heterogeneity of group sizes is controlled by a parameter $\beta$. To study the impact of $\beta$ on reaching consensus, we provide extensive computer simulations and compare them with an analytic approach for the dynamics of the average magnetization. We find that group interactions amplify small initial opinion biases, accelerate the formation of consensus, and lead to a drift of the average magnetization. The conservation of the initial magnetization, known for basic voter models, is no longer obtained.

Figure 1

Dynamics of the magnetization for 50 simulations starting from the same initial configuration with $M\left(0\right)=0.08\ne 0$. Parameters: $N=100, p=0.5, \beta =1.5, \gamma =0.3$.

Figure 2

Final average magnetization vs $\beta$ for various values of $N$ for $\langle M\left(0\right)\rangle =0.1$. Each data point is averaged over 20 trajectories and 10 initial configurations.

Figure 3

Time to equilibrium (log scale) vs (a) $\beta$ and (b) $\gamma$. Parameters: $N=100, r=0.5, \mathbb{E}\left[m_0\right]=0.1$. (a) $\gamma =0.3$, (b) $\beta =3$. The points and the shaded area are the average and the standard deviation of the time of convergence for 100 trajectories of 50 initial configurations for a given $\gamma$.

Figure 4

Switching rate $S$ vs initial magnetization $\langle M\left(0\right)\rangle$. Parameters: $N=100, r=0.5, \beta =1.5, \gamma =0.3$. The red points and the shaded area are the mean and the standard deviation of the switching rate averaged over 50 trajectories and 30 initial configurations. The black line is the best fit ($R^2=0.88$) of $S$, Eq. (21), with $k_{B}T=0.010\pm 0.007$ and $\lambda =0.45\pm 0.02.$

Figure 5

(a) $ln\left(kT\right)$ and (b) $ln\left(\lambda \right)$ vs $ln\left(\beta \right)$ for seven values of $N$ for $r=0.5$ and $\gamma =0.3$. The $R^2$ of the fitting lines are on average 0/85 and at least 0.78 (a) and 0.75 (b).

Figure 6

Average magnetization, Eq. (24) (black) and from simulations (red) over time. (a) $S=0$, (b) $S$ from Eq. (21). Parameters: $N=200, \langle M\left(0\right)\rangle =0.1, \beta =1.5, \gamma =0.3, r=0.55$. We run 100 simulations starting from 50 different initial configurations.

Figure 7

Final average magnetization, Eq. (24) for $t\to \infty$ (black dots), vs (a) $N$ and (b) $\beta$. Parameters: $\langle M\left(0\right)\rangle =0.1, \gamma =0.3, r=0.55$. (a) $\beta =1.5$, (b) $N=100$. Red dots and error bars from 100 simulations starting from 50 different initial configurations.

Figure 8

Dynamics of the average magnetization obtained from Eq. (A1) (black line) and from simulations. Parameters: $N=500, \beta =3, \langle M\left(0\right)\rangle =0.3, r=0.5, \gamma =0.3$. The red points and the shaded area are the average and standard deviation of the average final magnetization for 20 initial configurations, respectively. For each configuration, we simulate 50 trajectories.