Clustering coefficients of large networks

Let G be a network with n nodes and eigenvalues lambda(1) >= lambda(2) >= center dot center dot center dot >= lambda(n) Then G is called an (n, d, lambda)-network if it is d-regular and lambda = max{vertical bar lambda(2)vertical bar, vertical bar lambda(3)vertical bar, ... , vertical bar lambda(n)vertical bar}. It is shown that if G is an (n, d, lambda)-network and lambda = O(root d), the average clustering coefficient (c) over bar (G) of G satisfies (c) over bar (G) similar to d/n for large d. We show that this description also holds for strongly regular graphs and Erdos-Renyi graphs. Although most real-world networks are not constructed theoretically, we find that many of them have (c) over bar (G) close to (d) over bar /n and many close to 1 - (mu) over bar (2)(n-(d) over bar -1)/(d) over bar((d) over bar -1), where (d) over bar is the average degree of G and (mu) over bar (2) is the average of the numbers of common neighbors over all non-adjacent pairs of nodes. (C) 2016 Elsevier Inc. All rights reserved.