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3. In the Erdos-Renyi random graph above, let W be a subset of the vertex set of size k. It is called a clique if for every two vertices v₁, v, in W the edge e = (v₁, vj) is present in the graph. Compute the probability that W is a clique. For each such subset W, let Xw = 1 if W is a clique and Xw = 0 if W is not a clique. Are the random variables Xw and Xw, independent if W and W' are two such subsets? Show that Nk = Σ Χw WC{1,...,n} |W|=k is the number of cliques of size k in the graph. Compute E[N].

3. In the Erdos-Renyi random graph above, let W be a subset of the vertex set of size k. It is called a clique if for every two vertices v₁, v, in W the edge e = (v₁, vj) is present in the graph. Compute the probability that W is a clique. For each such subset W, let Xw = 1 if W is a clique and Xw = 0 if W is not a clique. Are the random variables Xw and Xw, independent if W and W' are two such subsets? Show that Nk = Σ Χw WC{1,...,n} |W|=k is the number of cliques of size k in the graph. Compute E[N].

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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3. In the Erdos-Renyi random graph above, let W be a subset of the vertex set of size k. It is called a clique if for every two vertices v₁, v, in W the edge e = (v₁, vj) is present in the graph. Compute the probability that W is a clique. For each such subset W, let Xw = 1 if W is a clique and Xw = 0 if W is not a clique. Are the random variables Xw and Xw, independent if W and W' are two such subsets? Show that Nk = Σ Xw WC{1,...,n} |W|=k is the number of cliques of size k in the graph. Compute E[N].
Transcribed Image Text:3. In the Erdos-Renyi random graph above, let W be a subset of the vertex set of size k. It is called a clique if for every two vertices v₁, v, in W the edge e = (v₁, vj) is present in the graph. Compute the probability that W is a clique. For each such subset W, let Xw = 1 if W is a clique and Xw = 0 if W is not a clique. Are the random variables Xw and Xw, independent if W and W' are two such subsets? Show that Nk = Σ Xw WC{1,...,n} |W|=k is the number of cliques of size k in the graph. Compute E[N].
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