Fig 1.
Schematic diagram showing the dynamics of our agent-based modelling.
Here we show a typical case when a selected agent (shown in black) interacts with randomly chosen 5 other agents, i.e., q = 5. Case (a) shows conformity, when the q-panel of selected agent is comprised of all agents with positive opinion (shown in blue). Similarly, case (b) also represents conformity, the q-panel has only agents with negative opinion (shown in red). In case (c), the selected agent is surrounded by 3 agents with positive opinion and 2 agents with negative opinion, and in case (d) it is surrounded by 4 agents with positive opinion and 1 negative agent. In both these cases, the selected agent chooses positive opinion with probability pq+ and negative opinion with probability pq−. The expressions for pq+ and pq− are given by Eqs (1) and (2) respectively.
Fig 2.
Variation of fraction f+ of agents with positive opinion as a function of time t.
Plots are for different values of p & q, and x = 0.51. Simulated (sim) results are shown by solid circles, and analytical (ana) results are shown by solid lines. Simulations are performed using L = 1024 averaging over 100 configurations. The agreement between simulated and analytical results are excellent for q = 2 and 3. For cases with q ≥ 4 the agreement becomes more reasonable as q increases.
Fig 3.
Fraction f+ of agents with positive opinion as a function of time t.
The plots are for x = 0.51 for several values of q and two typical values of p, viz. (a) p = 0.4 and (b) 0.8. Simulations are done for L = 1024 on a complete graph. As q increases, f+ becomes q independent. The black curves are data fittings done using Eq (22).
Fig 4.
Exit probability E(x) as a function of initial fraction x of agents with positive opinion.
Plots are shown for several values of p and two values of q. Simulation were done for L = 64 on a complete graph. The results are qualitatively similar across the values of q shown here.
Fig 5.
Variation of exit probability E(x) as a function of initial fraction x of up spins.
Plots are shown for (a) several system sizes with p = 0.5, q = 10 and for (b) various values of q with p = 0.5, L = 1024. It seems that E(x) maintains its linear behaviour even in the thermodynamic limit and in large q limit.
Fig 6.
Exit probability E(x) as a function of initial fraction x of agents with positive opinion.
Plots are shown for several values of q from 2 to 50 and for (a) L = 256, p = 0.6 and (b) L = 512, p = 0.45. The results converge as q grows larger.
Fig 7.
Data collapse of exit probability E(x).
Plots are shown for several values of system size L from 128 to 1024 for (a) q = 5, p = 0.51 and (b) q = 10, p = 0.46. For p > 0.5 the scaling argument is xL0.95 and for p < 0.5 it is (1 − x)L0.95. Insets show the unscaled data. It is evident that in the thermodynamic limit L → ∞ the exit probability would become a step functions at x = 0 for p > 0.5 and at x = 1 for p < 0.5.
Fig 8.
(a) Variation of xc as a function of p for several values of q as obtained by mean field theory. The solid line in black denotes the results from Monte Carlo simulation for L → ∞. In large q limit, the mean field results converge to Monte Carlo results. Inset shows the behavior of exit probability as expected from the mean field estimations. (b) Generalised phase portrait for our model, where the green circles indicate stable fixed points at 0 & 1, and the red circle denotes the unstable fix point at xc. The values of xc depend on q and p. The arrows indicate the directions of flow, such that for f+(t = 0) < xc, we would have f+(t → ∞) = 0 and similarly f+(t → ∞) = 1 for f+(t = 0) > xc.
Fig 9.
Exponential fittings of f+(t) curves.
(a) shows the obtained values of β and β′ as a function of q for 2 typical values of p keeping x fixed. In (b) we show the obtained values of β and β′ as a function of p. The data is fitted according to Eqs (23) and (24). (c) and (d) shows the fitting of the f+(t) curves for q = 50 according to Eq (22) for 2 typical values of x, viz. x = 0.01 in the region p < 0.5 and x = 0.99 in the region p > 0.5 respectively.
Fig 10.
Consensus time τ as a function of system size L.
Plots are shown for several values of p for (a) q = 2 and (b) q = 3. Insets show the data for p = 0.5. The simulations were done for x = 0.5. We can see that τ ∼ L for p = 0.5, but as p deviates from 0.5 the variation takes a logarithmic form. In (c) we show the variation of τ as function of q for system size L = 1024. τ decreases for lower values of q, however it does not exhibit a systematic dependence as q is made larger.