Propensity and stickiness in the naming game: Tipping fractions of minorities

Agent-based models of the binary naming game are generalized here to represent a family of models parameterized by the introduction of two continuous parameters. These parameters define varying listener-speaker interactions on the individual level with one parameter controlling the speaker and the other controlling the listener of each interaction. The major finding presented here is that the generalized naming game preserves the existence of critical thresholds for the size of committed minorities. Above such threshold, a committed minority causes a fast (in time logarithmic in size of the network) convergence to consensus, even when there are other parameters influencing the system. Below such threshold, reaching consensus requires time exponential in the size of the network. Moreover, the two introduced parameters cause bifurcations in the stabilities of the system's fixed points and may lead to changes in the system's consensus.

Figure 1

Trajectories of propensity models.

Figure 2

Average density of the opinion $B$ at consensus time $T_c$ as a function of propensity $p$, initialized with a uniform distribution of single opinions with the size of the network $N$.

Figure 3

Trajectories of stickiness model.

Figure 4

Average ratio of time to consensus $T_c$ to size of the network $N$ as a function of stickiness to mixed opinion $s$, initialized with a uniform distribution of single opinions.

Figure 5

Trajectories of two-parameter naming game.

Figure 6

Average ratio of consensus time, $T_c$, to the network size, $N$, as a surface of propensity, $p$, and stickiness, $s$. Lighter color implies longer consensus time.

Figure 7

Weighted average ratio of consensus time, $T_c$, to the network size, $N$, as a contour of propensity, $p$, and stickiness, $s$. Darker color implies consensus toward $B$, and lighter color implies consensus toward $A$.

Figure 8

Trajectories of stickiness model with committed agents computed using Runge-Kutta.

Figure 9

Global stability in $c_A-s$ plane.

Figure 10

Average ratio of time to consensus, $T_c$, to the network size, $N$, as a function of stickiness, $s$, and the fraction, $c_A$, of nodes committed to opinion $A$. Lighter color implies longer consensus times.