Analysis of the high-dimensional naming game with committed minorities

The naming game has become an archetype for linguistic evolution and mathematical social behavioral analysis. In the model presented here, there are $N$ individuals and $K$ words. Our contribution is developing a robust method that handles the case when $K=O\left(N\right)$. The initial condition plays a crucial role in the ordering of the system. We find that the system with high Shannon entropy has a higher consensus time and a lower critical fraction of zealots compared to low-entropy states. We also show that the critical number of committed agents decreases with the number of opinions and grows with the community size for each word. These results complement earlier conclusions that diversity of opinion is essential for evolution; without it, the system stagnates in the status quo [S. A. Marvel et al., Phys. Rev. Lett. 109, 118702 (2012)]. In contrast, our results suggest that committed minorities can more easily conquer highly diverse systems, showing them to be inherently unstable.

Figure 1

Plots of the consensus time (top) and the critical number of zealots (bottom) against the Shannon entropy of various initial conditions. A committed minority of word $A_1$ is introduced only in the bottom figure. Data are shown for $N=200$ and $K=20 (\circ )$ and $N=400$ and $K=40 \left(▵\right)$.

Figure 2

Semilogarithmic plot of the number of words in the system as a function of scaled time $t$. Notice that it takes longer for words to be eliminated when the system is in the outer region. Also shown is a line of best fit for the inner region, which confirms that the states over time tend to an exponential in $t$. The naming game is averaged over 100 runs with $N=K=100$.

Figure 3

Plot of the consensus time averaged over 50 runs of the naming game for various $N$ with $K=N$. Also plotted is the estimate given in Eq. (38) fitted to the data. The best fit yields $c_1=1.18$ and $c_2=2.84$ in the context of Eq. (38). The figure shows good agreement with the theory of Sec. 4b.

Figure 4

Critical fraction of committed agents plotted against $C/N$ for $N=1000 (\circ$ and $▵)$ and $N=500 (+$ and $\times )$. Data for a committed minority of a single word are shown $(\circ$ and $+)$ and the best fit of Eq. (42) in blue. Also shown is the case when there are committed minorities of every word $(▵$ and $\times$) with the fit of Eq. (43) in red.