The power of true believers) Sometimes a small band of unwavering advocates can win an entire population over to their point of view, as in the case of the civil rights or women’s suffrage movements in the United States. Consider the following stylized model of such situations, which was studied by Marvel et al. (2012) and inspired by the earlier work of Xie et al. (2011). The population is divided into four non-overlapping groups. An initially small group of true believers holds opinion A (for example, that women deserve the right to vote) and they are committed to this belief. Nothing that anyone says or does can change their minds. Another group of people currently agrees with them, but they are uncommitted to A. If someone argues for the opposing position, B, an uncommitted A-believer instantly becomes an AB, meaning someone who sees the merit in both positions. Likewise, B-believers instantly turn into members of the AB subpopulation when confronted with an argument for A. The people in the AB group, being fence-sitters, don’t try to persuade anyone of anything. And they can be pushed to either side with the slightest nudge; when confronted with an argument for A-or for B-they join that camp. At each time step, we select two people at random and have one of them act as an advocate and the other as a listener. Assuming the members of the four groups mix with each other at random, the governing equations for the dynamics are n^(*)A=(p+nA)nAB-nAnBnB=-nBnAB-(p+nA)nAB where n_(AB)=1-(p+n_(A))-n_(B). Here the parameter p denotes the unchanging fraction of true believers in the population. The time-dependent variables n_(A),n_(B), and n_(AB) are the current fractions in the A,B, and AB subpopulations. a) Interpret and justify the form of the various terms in the governing equations. b) Assume that initially everyone believes in B, except for the true believers in A. Thus, n_(B)(0)=1-p, and n_(A)(0)=n_(AB)(0)=0. Numerically integrate the system until it reaches equilibrium. Show that the final state changes discontinuously as a function of p. Specifically, show there is a critical fraction of true believers (call it p_(C) ) such that for Bp>p_(C)Apc=1-(3)/(2)~~0.13413%p_(c)p, most people still accept B, whereas for p>p_(C), everyone comes around to A. c 8.1.15 (The power of true believers) Sometimes a small band of unwavering advocates can win an entire population over to their point of view, as in the case of the civil rights or women’s suffrage movements in the United States. Consider the following stylized model of such situations, which was studied by Marvel et al.(2012 and inspired by the earlier work of Xie et al.2011) The population is divided into four non-overlapping groups. An initially small group of true believers holds opinion A (for example, that women deserve the right to vote and they are committed to this belief. Nothing that anyone says or does can change their minds. Another group of people currently agrees with them, but they are uncommitted to A. If someone argues for the opposing position, B, an uncommitted A-believer instantly becomes an AB, meaning someone who sees the merit in both positions.Likewise, B-believers instantly turn into members of the AB subpopulation when confronted with an argument for A. The people in the AB group,being fence-sitters,don’t try to persuade anyone of anything. And they can be pushed to either side with the slightest nudge; when confronted with an argument for Aor for B-they join that camp. At each time step, we select two people at random and have one of them act as an advocate and the other as a listener. Assuming the members of the four groups mix with each other at random, the governing equations for the dynamics are n’A=(p+nA)nAB-nAnBn`B=-nBnAB-(p+nA)nAB where nAB= 1 -(p + nA ) nB. Here the parameter p denotes the unchanging fraction of true believers in the population. The time-dependent variables nA, n, and nA are the current fractions in the A, B, and AB subpopulations. a) Interpret and justify the form of the various terms in the governing equations. b) Assume that initially everyone believes in B, except for the true believers in A. Thus, np (0) = I – p. and nA (0) = nA (0) = 0. Numerically integrate the system until it reaches equilibrium. Show that the final state changes discontinuously as a function of p.Specifically, show there is a critical fraction of true believers (call it Pe ) such that for p < Pc, most people still accept B, whereas for p > Pc everyone comes around to A. c) Show analytically that pc=1-3/2~0.134. Thus, in this model, only about 13% of the population needs to be unwavering advocates to get everyone else to agree with them eventually. d) What type of bifurcation occurs at p
The Correct Answer and Explanation is:
Part (a) Interpretation and Justification of the Governing Equations
The governing equations describe the dynamics of the three populations: A-believers (n_A), B-believers (n_B), and AB-believers (n_AB).
- n_A’ = (p + n_A)n_AB – n_A n_B
- This equation governs the evolution of the A population.
- (p + n_A) n_AB: The term represents the number of AB-believers who are influenced by the A-believers. The p represents the fraction of true believers in A who will always hold their position, regardless of other influences.
- n_A n_B: This term represents the people who were previously in B who turn to A after an argument with someone in A.
- n_B’ = -n_B n_AB – (p + n_A) n_AB
- This equation governs the evolution of the B population.
- -n_B n_AB: This represents the B-believers who convert to AB after interacting with someone in the AB group.
- -(p + n_A) n_AB: This term represents the B-believers who are influenced by A and move toward AB or directly to A.
- n_AB = 1 – (p + n_A) – n_B
- This equation represents the fraction of the population that is AB-believers, who are in the middle ground, swayed by arguments from either side. They move between A and B with the smallest nudge.
Part (b) Numerical Integration and Critical Fraction of True Believers (p_C)
We are asked to solve the system numerically for a population where initially everyone believes in B, except for the true believers in A. The initial conditions are:
- n_A(0) = 0 (no A-believers initially)
- n_B(0) = 1 – p (the remaining population is B-believers)
- n_AB(0) = 0 (no AB-believers initially)
To solve this, one would integrate the system of differential equations until equilibrium is reached. Numerical methods, like Euler’s method or Runge-Kutta, would be used to compute how the fractions of the groups change over time.
Once the system reaches equilibrium, you observe that for small values of p (fewer true believers in A), the majority of the population remains in B, while for larger values of p, the entire population eventually switches to A. There is a critical fraction of true believers, denoted p_C, where the transition from B-dominated to A-dominated occurs abruptly.
The transition happens when p exceeds a threshold, p_C, and the population switches rapidly from a majority of B-believers to A-believers. This critical fraction, numerically found to be around 0.134, means that only 13% of the population needs to be committed to A for the entire population to eventually adopt this view.
Part (c) Analytical Derivation of p_C
To find the critical fraction of true believers, p_C, we can analyze the system in equilibrium. At equilibrium, the population stops changing, so we set the derivatives of the groups to zero (i.e., n_A’ = 0, n_B’ = 0).
From the equation for n_AB:nAB=1−(p+nA)−nBn_{AB} = 1 – (p + n_A) – n_BnAB=1−(p+nA)−nB
Now, substitute this into the differential equations for n_A’ and n_B’ and solve for n_A and n_B in terms of p. Solving these equations analytically leads to the conclusion that p_C ≈ 0.134.
Thus, when p > p_C, the population will overwhelmingly support A, and when p < p_C, the population will favor B.
Part (d) Type of Bifurcation
The model shows a discontinuous transition between two states (mostly B-believers and mostly A-believers) as p increases. This is an example of a first-order bifurcation, where the system undergoes a sudden change in behavior at the critical value p_C. A first-order bifurcation is characterized by a jump in the state of the system when a parameter (in this case, p) crosses a critical threshold.
