Continuous time limits of the utterance selection model

In this paper we derive alternative continuous time limits of the utterance selection model (USM) for language change [G. J. Baxter et al., Phys. Rev. E 73, 046118 (2006)]. This is motivated by the fact that the Fokker-Planck continuous time limit derived in the original version of the USM is only valid for a small range of parameters. We investigate the consequences of relaxing these constraints on parameters. Using the normal approximation of the multinomial approximation, we derive a continuous time limit of the USM in the form of a weak-noise stochastic differential equation. We argue that this weak noise, not captured by the Kramers-Moyal expansion, cannot be neglected. We then propose a coarse-graining procedure, which takes the form of a stochastic version of the heterogeneous mean field approximation. This approximation groups the behavior of nodes of the same degree, reducing the complexity of the problem. With the help of this approximation, we study in detail two simple families of networks: the regular networks and the star-shaped networks. The analysis reveals and quantifies a finite-size effect of the dynamics. If we increase the size of the network by keeping all the other parameters constant, we transition from a state where conventions emerge to a state where no convention emerges. Furthermore, we show that the degree of a node acts as a time scale. For heterogeneous networks such as star-shaped networks, the time scale difference can become very large, leading to a noisier behavior of highly connected nodes.

Figure 1

Illustration of the two time scales of the problem. Here $t_{\mathrm{int}}$ represents the time of one interaction and $t_{\mathcal{G}}$ represents the time of one network update.

Figure 2

Illustration of the coarsening problem by a continuous time approximation. The discrete models are on the left. Agent-based models are on top; their evolution depends on $t_{\mathrm{int}}$ and on the bottom the population models evolve according to $t_{\mathcal{G}}$. For the USM, it is known how to obtain a continuous time limit at the agent level using the Kramers-Moyal expansion. The other arrows are not clear. In this paper we will take the diagonal approach.

Figure 3

Structure of the USM interaction. On the $\left(ij\right)$ edge, the agents use their internal beliefs ${\mathit{x}}^{\left(i\right)}$ to produce an utterance ${\mathit{u}}^{\left(i\right)}$ through the process $\mathcal{U}$, which depends on the matrix $M$. The utterances are then used to update the internal beliefs depending on a weighting parameter $h^{\left(ij\right)}$.

Figure 4

Comparison of models for the USM and continuous time limits. The red horizontal line represents the initial error $z\left(0\right)$. The dashed black line represents the solution of the deterministic limit. We display the average over 100 simulations and the corresponding variances. The continuous time limits have to be compared with the discrete USM displayed in red stars. The first panel shows the averaged $z$ at time $T=1$, the second panel the averaged $z$ at time $T=100$, the third panel the variance of $z$ at time $T=1$, and the fourth panel the variance of $z$ at time $T=100$.

Figure 5

Illustration of the network reduction for regular networks (left) and star-shaped networks (right). In each panel the left part is the original network and the right part is the reduced network.

Figure 6

Illustration of the critical parameter $q_{*}\left(r\right)$ separating the bell-shaped and the $\mathsf{U}$-shaped domain. The red curve is the approximate behavior. For illustration, we display the positions of the two examples considered in this section (regular network of degree 3 for $N=10$ and 100 agents).

Figure 7

Comparison between the discrete USM (left column) and the SHMF approximation limit of it (right column). For these simulations, the parameters are $h=0.5, \lambda =0.1, V=2, L=2, T={10}^4$, and $q=0.001$ and the number of agents is 10 in the top row and 100 in the bottom row. The regular graph is of degree $k=3$. At the beginning of the simulation, all agents share the convention to use the variant $v=1$.

Figure 8

Comparison between the stationary distribution of the discrete USM on a regular network with the distribution predicted by the SHMF approximation. Shown on the left is the stationary distribution for 10 agents. On the right is the stationary distribution for 100 agents.

Figure 9

Comparison of time series of the discrete USM and of the SHMF approximation limit of it for the star-shaped network. For these simulations, the parameters are $\lambda =0.1, V=2, L=2, T={10}^4$, and $q=0.001$. The top part of each graph displays the behavior of the degree $k=1$ nodes and the bottom part of each graph displays the behavior of the central node. The value of $N$ is (a)–(d) 10 and (e)–(h) 100. The first and third columns display the results of the USM and the second and fourth columns display the results of the SHMF approximation. In the first two columns $h=0.9$ and in the last two columns $h=0.1$. At the beginning of the simulation, all agents share the convention to use the variant $v=1$.

Figure 10

Comparison between the discrete USM and its SHMF approximation limit for the star-shaped network. For these simulations, the parameters are $\lambda =0.1, V=2, L=2, T=4\times {10}^3$, and $q=0.001$. The top part of each graph displays the distribution of the degree $k=1$ nodes and the bottom part of each graph displays the distribution of the central node. In the top row the value of $N$ is 10 and in the second row the value of $N$ is 100. The first and third columns display the results of the USM and the second and fourth columns display the results of the SHMF approximation. In the first two columns $h=0.9$ and in the last two columns $h=0.1$. The red line is the solution of the mean field approximation. It helps to see how the star-shaped network differs from the regular network case. The first two columns and the last two columns have to be compared.