Effects of mixing in threshold models of social behavior

Andrei R. Akhmetzhanov, Lee Worden, and Jonathan Dushoff

Phys. Rev. E 88, 012816 – Published 25 July 2013

Abstract

We consider the dynamics of an extension of the influential Granovetter model of social behavior, where individuals are affected by their personal preferences and observation of the neighbors’ behavior. Individuals are arranged in a network (usually the square lattice), and each has a state and a fixed threshold for behavior changes. We simulate the system asynchronously by picking a random individual and we either update its state or exchange it with another randomly chosen individual (mixing). We describe the dynamics analytically in the fast-mixing limit by using the mean-field approximation and investigate it mainly numerically in the case of finite mixing. We show that the dynamics converge to a manifold in state space, which determines the possible equilibria, and show how to estimate the projection of this manifold by using simulated trajectories, emitted from different initial points. We show that the effects of considering the network can be decomposed into finite-neighborhood effects, and finite-mixing-rate effects, which have qualitatively similar effects. Both of these effects increase the tendency of the system to move from a less-desired equilibrium to the “ground state.” Our findings can be used to probe shifts in behavioral norms and have implications for the role of information flow in determining when social norms that have become unpopular in particular communities (such as foot binding or female genital cutting) persist or vanish.

Received 18 April 2013

DOI:https://doi.org/10.1103/PhysRevE.88.012816

Authors & Affiliations

Andrei R. Akhmetzhanov^*, Lee Worden^†, and Jonathan Dushoff^‡

Theoretical Biology Laboratory, Department of Biology, McMaster University, Hamilton, Ontario L8S4K1, Canada

^*aakhmetz@um2.fr; currently at Institute of Evolutionary Sciences, University of Montpellier II, Montpellier 34095, France.
^†worden.lee@gmail.com; also at the University of California at Berkeley, Berkeley, California, USA, and San Francisco Art Institute, San Francisco, California, USA.
^‡Corresponding author: dushoff@mcmaster.ca

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Issue

Vol. 88, Iss. 1 — July 2013

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Images

Figure 1
Simulations of the threshold model (TM) on a two-dimensional lattice of size $100^{2}$ with no mixing among individuals and different neighborhood sizes. The thresholds are normally distributed with the mean $0.45$ and standard deviation $0.3$ . The initial pattern of thresholds and initial states is the same for all simulations shown. Activated individuals are shown in orange (dark). The $\infty$ symbol denotes an equilibrium, which is reached in the TM.Reuse & Permissions
Figure 2
The mean-field dynamics of the TM in the phase plane $(y, d y / d t)$ for different neighborhood sizes: $n = 4$ [magenta (light)], 12 [red (medium)], and 24 [green (dark)]. The dashed curve corresponds to the case of infinitely large neighborhood size. The corresponding potential functions $V_{n} (y)$ are shown in the inset. The thresholds' PDF is Gaussian with the mean $0.6$ and standard deviation $0.225$ . The crosses show the results of numerical simulations of the TM on a two-dimensional lattice of the size $100^{2}$ at mixing rate $μ = 4$ and $y (0) = 1$ .Reuse & Permissions
Figure 3
The probability $p_{+}$ that the TM will approach the upper equilibrium as a function of $μ$ or scaled $μ / L$ (the inset). The TM is on a two-dimensional lattice of size $N = L^{2}$ with the neighborhood 8. The thresholds' PDF is Gaussian with mean 0.55 and standard deviation $0.225$ . Different colors (shades) stand for different initial values $y (0)$ , while the symbols stand for different values of $L$ (see the legend in the bottom right corner). In the main figure the points with $y (0) = 0.71$ are shown in cyan (light), with $y (0) = 0.69$ in blue (dark). In the inset overlapping points in the middle correspond to $y (0) = y_{*}^{\infty}$ , the ones above them to $y (0) = 0.69$ , and the ones below them to $y (0) = 0.66$ . The bifurcation value $\tilde{μ} (y (0) = 0.69) \approx 1.556$ is indicated. To estimate the probability, we performed $10^{5}$ simulations with different random initial individual states and thresholds and update order.Reuse & Permissions
Figure 4
The dependence of $y_{\times}^{μ}$ on $μ$ . The TM is posed on a two-dimensional lattice of size $200^{2}$ with neighborhood 8 and Gaussian distribution of thresholds with the mean $0.55$ and standard deviation $0.225$ . The extrapolation curve, shown by the dashed line, gives $\bar{μ} = μ (y_{\times}^{μ} = 1.0) \approx 0.168$ .Reuse & Permissions
Figure 5
The projection curve $F_{n}^{μ}$ on a two-dimensional lattice of size $800^{2}$ with neighborhood 8 and the Gaussian distribution of thresholds (the mean 0.55 and standard deviation $0.225$ and mixing rate equals $0.278$ ). The trajectories, shown in red (thin) lines, start from $y (0) = i / 10$ ( $i = 1, ..., 9$ ), while the green (solid) curve consists of the trajectories initiated at $y (0) = 0$ , $y (0) = 1$ and $y (0) = y_{\times}^{μ} = 0.76$ . Initially, the states and thresholds are distributed randomly among individuals, hence all initial points fall along the curve $F_{n} (y) \equiv F_{n}^{\infty} (y)$ (dashed curve).Reuse & Permissions
Figure 6
The mean transition time $〈 T_{R} 〉$ , necessary for the TM to evolve from complete activation $y (0) = 1.0$ to a given intermediate value $y (T_{R}) = y_{*}^{\infty} \approx 0.6752$ . The TM is posed on a two-dimensional lattice of size $N = L^{2}$ with neighborhood 8 and Gaussian distribution of the thresholds with the mean 0.55 and standard deviation 0.225. The inset illustrates the distribution of $T_{R}$ at $μ = 0.1$ for $L = 100$ [green (light)], 200 [magenta (medium)], and 400 [blue (dark)].Reuse & Permissions

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