3.3 Isolated Node (10 points)
For the Erdos-Renyi random graph model G(n, p), let p be very large.
Also assume that the graph is n = 1.
Compute the probability that an arbitrary node has degree 0 (i.e., v is isolated) in the random graph (2 points).
Using the value that you found above, now find an upper bound on the probability that the graph contains at least one isolated node (3 points).
[Hint: Use simple union bound: P(U∪A) < E[P(A)]]
Using the above two parts, argue that at p = 2√(ln(n))/n, G(n, p) has no isolated nodes (5 points).
[Hint: e = exp(1)]