Strategic Practice and Homework 8 - Projects at Harvard

142356Group iGroup j141342352Group iGroup jGroup iGroup jFigure 1: Two groups with 0 (upper), 1 (lower left), 2 (lower right) people in common.since 5 distinct pairs of people must know each other to make X i X j equal to 1.There are n 4=6 n pairs of groups {i, j} (i 6= j)with1pairofpeopleincommon4 2 4(choose 4 people out of the n, thenchoosewhich2ofthe4aretheoverlapofthegroups). The remaining pairs of groups have covariance 0. Thus, the variance of thenumber of cliques is◆7 n64✓3✓ n+2· 6 ·4◆164 64✓ = 7 ◆ n+ 3 ◆ n.3 16✓4(c) Suppose that n < 2 (k 2) 1 .Showthatthereisanetworkwithn nodes containingkno cliques of size k or anticliques of size k.Hint: explain why it is enough to show that for a random network with n nodes, theprobability of the desired property is positive; then consider the complement.7