Fig 1.
(a) Illustration showing the effect of mean-field theory on a network of interconnected agents that influence each others decisions. In the mean-field case each agent is influenced by the same information I(n, t) which they all contribute towards. (b) Diagram showing the 1D process that is equivalent to the mean-field case of the binary decision model. The number in each circle denotes the number of right deciding agents, and the expressions above and below the arrows denote the associated probabilities of transitioning between each configuration. (c) Plots showing the correspondence between our analytic solution in Eq (16) and Monte Carlo simulations provided by the SSA. The parameters for this time-evolution are F = 0, J = 10, α = 0, β = 1, γ = 1, N = 100 and P(n, 0) = δn,N/2. The probability distribution from the SSA is calculated from 2.5 × 103 trajectories. As t → ∞ we see the emergence of a single steady-state consisting of two equal modes of height ≈0.5 at m(0) = −1 and m(N) = 1. In the limit N → ∞ this steady-state bimodality corresponds to symmetry breaking behaviour [30]. Note that the SSA simulations in the solid lines show random perturbations due to the stochastic nature of the simulations, whereas the bars showing Eq (16) do not show these deviations.
Fig 2.
Plots showing the lock-in phenomena in the mean-field model.
(a) For F > 0 and β > βc we show the evolution of the probability distribution given by Eq (16) from an initial condition at n = N/2. Beyond the initial condition we see the emergence of two modes of behaviour, the agents either increasingly choose the left or the right technology. The evolution of the probability distribution becomes very slow for t > 10 and the distribution at t = 1010 is indistinguishable from that at t = 10. (b) A plot of the analytic steady-state distribution from Eq (22) shows that in the true steady-state limit almost all the agents will be locked into the right technology. Importantly, this is completely distinct from the time-dependent solution even at large times. Note the presence of a small mode at the unfavoured left-hand technology seen in the inset. (c) Individual SSA trajectories for the system showing the lock-in effect on individual populations. (d) As t ≫ 1 the dynamics of the system becomes metastable and can be approximately mapped to a two state process with very small transition rates. Parameters for plots in this figure are F = 0.1, J = 5, α = 0, β = 1, γ = 1 and N = 50.
Fig 3.
Figure showing the coalescence of the agents onto stable equilibria where the modes are not found at m = ±1 and fluctuations are present, for parameters F = 0.025, J = 1.5, α = 0, β = 1, γ = 1 and N = 50 using the analytic solution from Eq (16).
(a) Plots of P(m, t) at times from near the initial condition at m = 0 to the steady-state. Note that for t ≳ 3 × 103 the time-dependent solution becomes indistinguishable from the steady-state. (b) Plot showing that the metastable modes have the same shape as the modes of the steady-state distribution. We show that the right-hand mode rescaled by a pre-factor (= 0.455 here for t = 103) becomes indistinguishable from the steady-state distribution. (c) New dynamics considered for the calculation of the mean first passage time τlr. Note that a similar, but separate, diagram can be drawn for the calculation of τrl. (d) Plot of the mean first passage time to reach mu given one starts at m(n). It is clear that since F > 0 the mean first passage times to hit mu are greater for m > 0. (e) Plot of the time-dependent distribution at (using our approximation) versus the steady-state distribution.
Fig 4.
Figure showing the calibration procedure on a single trajectory/realisation of data produced by the SSA for the parameter set explored in Fig 3.
(i) Exploring the calibration when only one equilibrium of the two is realised on the data trajectory. i(a) The data which we perform the calibration on, clearly showing that only the right-hand equilibrium is manifest. Red dashed lines show the various times used for calibration. In i(b) and i(c) we explore the various calibration times and plot the error on the calibrated parameters compared to the true values for Etot and f respectively defined in Eqs (41) and (42). Although larger calibration times result in better parameter inference there are still large errors ≫ 1 even for large calibration times. (ii) Exploring the calibration procedure for a trajectory that realises both equilibria. ii(a) The trajectory that we use for calibration that explores both equilibria. ii(b) and ii(c) show that increasing the calibration time such that the exploration of both equilibria occurs results in much improved inference of the parameters where for calibration times ∼ 103 the errors are of order . (iii) Exploring the trajectory in ii(a) for a varied number of time/data points. iii(a) Bar chart showing the different number of time points explored in the calibration. iii(b) and iii(c) show that although having an increased number of time points benefits calibration, it does so with diminishing returns. Note throughout the figure that the optimiser used from BlackBoxOptim generally identifies good local minima of the likelihood function and not the global minimum due to the complexity of the likelihood function [62].
Fig 5.
Figure showing the performance of the calibration procedure when one has access to multiple realisations of a binary decision process.
(a) We simulated 103 trajectories of binary decision data from the SSA for the parameter set used in Fig 3. We emphasise five possible trajectories in the foreground, with the yellow background giving an idea of all the trajectories we used. (b) and (c) show that the model calibration can be significantly improved if one has access to multiple realisations of the same binary decision process. For > 102 realisations we find that the errors are typically ≤ 1, which corresponds to a very accurate calibration.
Fig 6.
For the set of parameters explored in Fig 3 we explore how the metastable relaxation times (using our very accurate approximation) and the equilibrium distribution behave with respect to changing agent altruism.
The left-hand plot shows that as agent altruism increases the metastable waiting times (black line) become exponentially large in α. The blue dots show an exponential approximation to , clearly exhibiting its close-to exponential behaviour. The right-hand plot shows the change in the equilibrium (t → ∞) distributions. For increasing agent altruism populations become more polarised in each behavioural mode (with much reduced fluctuations) even in the metastable regime.