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
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].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F02746391-8719-4497-9e3d-b4944703787f%2Fa9585085-24f1-45d7-ba24-95177e32d697%2Fjbnc0xc_processed.jpeg&w=3840&q=75)
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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