? Giant component of the random graph. Consider a random graph ensemble G(N,p) formed by all…

? Giant component of the random graph. Consider a random graph ensemble G(N,p) formed by all…
с Giant component of the random graph. Consider a random graph ensemble G(N,p) formed by all networks of N nodes with each pair of nodes connected with probability p. Take p N – 1 with c > 0 indicating the average degree of the network. Let S indicate the probability that a node is in the giant component. A node i is not in the giant component of a random graph if for every other node ; of the graph either one of the following events occurs: (i) i is not connected to j by a link, (ii) i is linked to j but j doesn't belong to the giant component. [8] [5] (a) Show that in the large network limit N » 1, the probability S satisfies the equation S=1-e-cs, where c is assumed to be independent of the network size N. (b) Using graphical arguments, justify that in the large network limit N »1, the network will have a non-null giant component if and only if ( > 1. (c) Consider the case in which p is given by p = 0.5. Does the random graph G(N,p) have a giant component in the limit N +00? (Why?) (d) Consider the case in which p is given by p = 0.5/N2.5. Does the random graph G(N,p) have a giant component in the large network limit N +00? (Why?) (e) Show that the average degree c that ensures that the random network is connected, i.e. such that there are no isolated nodes, is approximately given by c~In(N). [2] [2] [8]   Apr 08 2022 01:51 PM

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