Consider a network given by the adjacency matrix a_ij = a_ji = 1 with probability p and a_ij = a_ji = 0 with probability 1 − p.
(a) In the limit of large N and fixed pN, compute the degree distribution and calculate the connectivity.
(b) Now exclude the nodes of degree zero and neglect degree correlations. Use the distribution derived in (a) in the limit of large N and Np fixed to express the average return time r_i for the nodes of non-zero degree in terms of Ei(pN), where Ei(pN) denotes the exponential integral.

(c) Compare the expression in (b) with the estimate one obtains by the definition of r_i in Eq. (8.99), replacing k_i by the average degree and L by the average number of links. Consider the limit Np → 0 and N ≫ 1.

Eq. (8.99)

Transcribed Image Text:

2L ki