Consequence of reputation in an open-ended naming game

We study a modified version of the naming game, a recently introduced model which describes how shared vocabulary can emerge spontaneously in a population without any central control. In particular, we introduce a mechanism that allows a continuous interchange with the external inventory of words. A playing strategy, influenced by the hierarchical structure that individuals’ reputation defines in the community, is implemented. We analyze how these features influence the convergence times, the cognitive efforts of the agents, and the scaling behavior in memory and time.

Figure 1

Top: temporal evolution for the total number of words $\left[N_{\mathit{tot}}\left(t\right)\right]$ and for the number of different words $\left[N_{\mathit{dif}}\left(t\right)\right]$. Bottom: the success rate $\left[S\left(t\right)\right]$. Data are averaged over 100 simulations with $P=500$. The agents’ initial reputations follow a Gaussian distribution with standard deviation 5. In the inset we present the distribution of the convergence time ($P=100$, $\sigma =5$). Data are fitted by a log-normal distribution.

Figure 2

Temporal evolution for the number of different words for a population with a reputation obtained from Gaussian distributions with different standard deviations $\left(\sigma \right)$. The inset shows the very slow convergence towards the absorbing state, characterized by the presence of long-standing quasistationary states. Data averaged over 100 simulations with $P=500$.

Figure 3

Temporal evolution for the number of different words for populations with an initial asymmetric distributions in $R$. The population with 10% of agents having $R=+5$ (teachers) and the others having $R=-5$ shows the minor memory requirement. The population with 90% of teachers needs the largest amount of memory. Data averaged over 100 simulations with $P=500$.

Figure 4

Maximum number of total words and, in the inset, maximum number of different words for different population sizes. The lines have slope $3∕2$. In all simulations the agents’ initial reputations follow a Gaussian distribution. These results are robust with respect to the choice of different $\sigma$ values.

Figure 5

(Color online) Convergence time ($T_{\mathit{con}}$, data on the top) and time position of the peak that corresponds to the maximum number of total words ($T_{\mathit{max}}$, data on the bottom) for different population sizes $P$. $T_{\mathit{con}}$ displays a crossover: for $P<10000$, $T_{\mathit{con}}\propto P^{1.23\pm 0.01}$; for $P>10000$, $T_{\mathit{con}}\propto P^{1.49\pm 0.03}$. $T_{\mathit{max}}$ is well described by a power law with exponent $1.54\pm 0.01$. The inset shows the power-law dependence of the alignment time $T_{\mathit{con}}-T_{\mathit{max}}$ (the fitted exponent is $1.14\pm 0.02$). We present simulations where the agents’ initial reputations follow a Gaussian distribution with different $\sigma$ values.