You can use Matlab for problems in this homework. X2 X1 X1 X1 X1 14 (a) (b) (c) Problem 1. (Opinion Network relooked) For each of the social networks a function f(x) (called the potential function) of z = =1 12 13 24] can be de f(z) := )-1, (i,j) where in the summation, (i, j) ranges over all pairs of individuals that are friends (i) The potential function f(x) can be written as f(x) = EPI for some proper (called the graph Laplacian matrix) PER1x4. For each of the four social I matrix P. Explain why P is always positive semidefinite. (ii) For social networks (a), (b), (c), show that f(x) = 0 if and only if the four reached consensus. Show that this is not necessarily the case for social net (iii) Suppose the opinion x(k] evolves according to the LII dynamics z [k+ 1] = A 1 of HW3. Show that the potential function is non-increasing: f(x[k + 1]) of the social networks. Further, show that for social networks (a), (b), (c), strictly decreasing: f(x[k + 1])