In the book and movie Divergent, society is divided into five factions: Abnegati

Question

In the book and movie Divergent, society is divided into five factions: Abnegation, Amity, Candor, Dauntless, and Erudite. Upon reaching the age of 16, each individual is allowed to choose which faction to belong to as an adult. The vast majority chooses the faction they grew up in, but a few decide on a different faction. For this assignment, we will be investigating the population of each of faction as time goes on. W e will let u_0 = (u_1, u_2, u_3, u_4, u_5) be the initial state vector where u_i is the percentage of the population in each faction at the start of our investigation. (u_1 = Abnegation, u_2 = Amity, u_3 = Candor, u_4 = Dauntless, u_5 = Erudite.). The migration matrix A determines how the population changes from one generation to the next, where A = [.80 .12 .02 .02 .01 .08 .76 .04 .02 .04 .06 .04 .82 .06 .05 .02 .02 .04 .84 .02 .04 .06 .08 .06 .88] To determine the state vector u_1, the percentage of the population in each faction one generation after the start of our investigation, we use the equation u_1 = Au_0. We can continue this process to find the percentages for any subsequent generation: u_2=Au_1 =A(Au_0) = A^2u_0 u_3 = Au_2=A(A^2 u_0) = A^3 u_0 and so on. In general, we get that u_n = A^n u_0. Suppose that the population starts out uniformly across the five factions (equal percentages for each). What would the state vector be one generation later? What about 5 generations later? 100 generations? How much change is there between the percentages for the 100th generation compared to the 101st generation? Does the original state vector have much of an effect on the percentages many generations later? Experiment with various initial state vectors and see what happens after 5, 10, 100 generations with each. (Remember, the percentages in the state vector must total 100%.) What effect did the initial state vector have on later generations? What do the columns of A represent? What do the rows of A represent? The diagonal entries of A are much larger than the other entries. How does this relate to the movement of individuals between factions? What would it indicate if those values were lower? Now, in reality, there is a 6th group of individuals, the factionless. Since nobody chooses to be factionless at their choosing ceremony, the choosing matrix will be the same as before, except with another row and column to account for the factionless: C = [.80 .12 .02 .02 .01 0 .08 .76 .04 .02 .04 0 .06 .04 .82 .06 .05 0 .02 .02 .04 .84 .02 0 .04 .06 .08 .06 .88 0 0 0 0 0 0 1] So, after the choosing ceremony, the percentages in each faction would be given as Cu_0. However, before the next choosing ceremony, each faction loses a certain percentage according to the factionless matrix: F = [.96 0 0 0 0 0 0 .93 0 0 0 0 0 0 .90 0 0 0 0 0 0 .75 0 0 0 0 0 0 .85 0 .04 .07 .10 .25 .15 1] So u_1 = FCu_0. What effect does this have on the percentages after many generations? After how many generations will the factionless be the largest group? After how many generations will the factionless be more than 50% of the population? What about 90%?

Explanation / Answer

C = [0.8 0.12 .02 .02 .01 0;0.08 0.76 .04 .02 .04 0 ; .06 .04 .82 .06 .05 0; .02 .02 .04 .84 .02 0; .04 .06 .08 .06 .88 0;0 0 0 0 0 1];

F=[0.96 0 0 0 0 0;0 .93 0 0 0 0;0 0 .9 0 0 0;0 0 0 .75 0 0; 0 0 0 0 .85 0 ;.04 .07 .10 .25 .15 1];
A =FC

u0 =[0.2; 0.2; 0.2; 0.2; 0.2; 0];

a) u1 = FC u0

similarly

un = (FC)n u0

we see if n is large ,say 100

then only factionless individual are left ,that is

un = [0.0000
0.0000
0.0000
0.0000
0.0000
1.0000 ];

b) u2 = [0.1713
0.1540
0.1694
0.1027
0.1775
0.2252]; so they become largest after 2 genration only.

c) u6 = [0.1154
0.0964
0.1108
0.0396
0.1215
0.5163] ;

u5 = [ 0.1280
0.1080
0.1238
0.0482
0.1347
0.4573];

so after 5 generation,they are more than 50 %

u21  = [0.0222
0.0180
0.0205
0.0059
0.0231
0.9103];

u20 = [0.0248
0.0202
0.0230
0.0066
0.0258
0.8997];

,hence after 21 generations ,they are more than 90 % .

Related Questions

Navigate

• Browse (All)
• Browse I
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.