Engineering, 18.10.2019 18:20 ahicks2004

Consider a set of n people who are members of an online social network. suppose that each pair of people are linked as "friends" independently with probability 1/2. we can think of their relationships as a graph with n nodes (one for each person), and an undirected edge between each pair that are friends. a clique is a fully connected subset of the graph, or equivalently a subset of people for which all pairs are friends.

a) a clique of size 2 is simply a pair of nodes that are linked by an edge. find the expected number of edges as a function of the number of nodes, n. what is the expected number of friend relationships among n = 10 people?

b) a clique of size 3 is a triplet of nodes within which all three pairs are linked by an edge. find the expected number of 3-cliques as a function of the number of nodes, n. what is the expected number of 3-cliques among n = 10 people?

c) larger cliques may occur involving groups of nodes of any size k. find the expected number of cliques of size k ≥ 3 as a function of the number of nodes, n. what is the expected number of cliques of size k ≥ 3 with n = 10 people?

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