Transient Behavior with Bounded Confidence

As noted above, a further reduction of the Markov chain is possible if the dy- namics are considered from the perspective of »party« b. The corresponding partition reads

Y_rb = [

k+m=N −r

X_hk,r,mi, r = 0, 1, . . . , N (3.43)

and the transition probabilities are

P (Xhk,r,mi, Yr+1b ) =

r(N − r)

Chapter III ₆₁

As discussed in Sec. 3.3.5, the probability to converge to Y_Nb = {(bb . . . b)} is 1/δ = 1/3 and the probability to end up in one of the configurations in Yb

0 is 2/3, provided that the model is initialized with an equal number of a,

b and c agents. Notice that contrary to the unbounded confidence case the process is really in its final state whenever r = 0 as all configurations in Y₀b are absorbing.

The convergence times (see Fig. 3.15 for a small system) are composed of the (relatively short) time needed to end up in the class of states Y₀b in- cluding those of opinion polarization and the (relatively long) time needed to converge to Y_Nb. In the following, we use a transformation proposed in Kemeny and Snell (1976), 64/65, in order to assess the two times indepen- dently. Notice that all the results obtained in this section are also applicable to the binary VM (with Xk≡ Ykb) to study the effects of initial opinion bias.

The basic idea is to »compute all probabilities relative to the hypothesis that the process ends up in the given absorbing state« (ibid, 64). This leads to a new absorbing chain with the specified state as the single absorbing state. In fact, for our purpose, it is not necessary to completely determine the transition matrix for that new chain as the fundamental matrix of the original process (F) can be used directly to compute the fundamental matrix of the new chain ( ˜F). Let B = FR where F is the fundamental matrix of the binary chain (3.22) and R the respective 2 × N submatrix of the canonical form (3.19). The elements b1j (b2j) of B correspond to the exit probabilities

of the process started in j to end up in X0 ≡ Y0b (XN ≡ YNb). Recall

that b1j = (N − j)/N and b2j = j/N (see Sec. 3.3.3). Now let D0 be a

diagonal matrix with djj = b1j and respectively define DN as djj = b2j.

Then, according to Kemeny and Snell (1976), 65, the fundamental matrices of the new chains with absorbing state X0 ≡ Y0b and XN ≡ YNb respectively

is given by

˜

F0 = D0−1FD0

˜

FN = D−1N FDN. (3.45)

In our case with the fundamental matrix given in Eq. 3.22 we obtain ( ˜F0)ij = b1j Fij b1i =    iN (N −j) j(N −i) : i ≤ j N : i > j (3.46) and ( ˜FN)ij = b2j Fij b2i =    N : i ≤ j jN (N −i) i(N −j) : i > j . (3.47)

The fundamental matrices ˜F0 and ˜FN allow for a very good understand-

62 Agent–Based Models as Markov Chains steps that the realizations which eventually converge to Y₀b pass through any state Yb

r, and ˜FN informs us about the mean behavior of realizations that

end up in Yb

N. For instance, we can compute the mean convergence time to

each absorbing state independently. For convergence to Y₀b from the initial state Y_rb we have ˜ τ0(r) = rN + N X j=r+1 rN (N − j) j(N − r) (3.48)

and for convergence to uniformity corresponding to Y_Nb ˜ τN(r) = (N − r)N + r−1 X j=1 jN (N − r) r(N − j) (3.49)

For a system of 100 agents these times are shown in Fig. 3.16. It becomes clear that the mean convergence times to Yb

0 and YNb are equal if the initial

situation is unbiased, that is, if there are r = N/2 agents with attribute b and N/2 agents in the other two states a or c. However, with an increasing initial bias, there is an increasing gap between average convergence time to one or the other absorbing state. For the system with three possible attributes a, b and c and random initial conditions the initial number of b agents is around N/3 ≈ 33. This is illustrated by the dashed vertical line. In that case, the mean convergence time for realizations that end up in possible polarized con- figurations with only a and c agents becomes considerably smaller compared to the configuration with all agents in state b.

0 20 40 60 80 100 2000 4000 6000 8000 10 000 k Τ

Figure 3.16: Mean convergence times to X0 ≡ Y0b (red) and XN ≡ YNb (blue)

independently. The vertical dashed line represents the initial bias for the model with δ = 3.

˜

F0and ˜FN enable moreover to study the transient behavior of the model

Chapter III ₆₃ æææææææææææææææææææææææææææææææææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ ææ ææ ææ æææ æ ææææææææææææ ææææ ææææææææææææ 0 20 40 60 80 100 0 20 40 60 80 100 H33,jL F Ž 0 æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææ 0 20 40 60 80 100 0 20 40 60 80 100 H33,jL F Ž N

Figure 3.17: Mean number of steps a process that eventually converges to Yb 0

(l.h.s.) and Y_Nb (r.h.s.) is in the transient states for a system of 100 agents and an initial number of r = 33 agent in state b.

the mean number of steps that a process ending up in Yb

0 (l.h.s.) and YNb

(r.h.s.) is in the transient states provided the model is initialized with 33 agents in b and 67 agents in a or c. This information is encoded in the 33rd row of ˜F0 and ˜FN respectively. We first comment on the l.h.s. showing the

mean behavior of realizations ending up in an absorbing configuration where only a and c agents remain (Y₀b). The figure shows that, in average, all the transient states that are closer to Yb

0 than the initial configuration in Y33b

are met N times. Naturally, the states »to the right« are encountered less frequently. It should be clear that the entries of ˜F0should not be read as the

mean behavior of every single realization, but rather as the average behavior over a large series of realizations. For instance, the mean number of steps to Y₉₉b, which is very close to the opposite absorbing state, is approximately 1/2. However, this does not mean that every second realization approaches the opposite absorbing state so closely. It rather means that there are rare realizations that take that way, and once they are at the opposite extreme, these realizations have a high chance to stay there for some while. In fact, the fundamental matrix ˜F0 tells us that, once a realization reached Y99b, the

mean number of returns to that state is N − 1. The interpretation of the r.h.s. ( ˜FN) goes in the same way.

Finally, the probability distribution of convergence times to one or the other absorbing state can be computed easily for a given N . In this compu- tation, we first compute the respective matrices ˜Q0 as ( ˜Q0)ij = (b1j/b1i)Qij

and ˜QN as ( ˜QN)ij = (b2j/b2i)Qij. This is in complete analogy to the com-

putation of the independent fundamental matrices and follows the work of (Kemeny and Snell, 1976, 64-65). The computation of the probability distri- bution is then based on the evaluation of powers of ˜Q as done in Sec. 3.3.2. The result is shown in Fig. 3.18. For comparison, the distribution of conver- gence times to either absorbing state (dashed, red) is shown for r = 33. All in all, this shows how the general convergence behavior is a composite of the two different convergence trends obtained by considering the two absorbing states independently.

64 Agent–Based Models as Markov Chains 0 5000 10000 15000 20000 0 0.2 0.4 0.6 0.8 1 τ cdf( τ ) τ τ₀ τ_N 0 5000 10000 15000 20000 0 0.5 1 1.5 2x 10 −3 τ Pr[ τ ] τ τ₀ τ_N

Figure 3.18: Probability distribution of convergence times for a system of 100 agents when started with r = 33. Convergence to Yb

0, τ0, is considerably

faster than convergence to Y_Nb, τN.