Clustering coefficient of random intersection graphs with infinite degree variance

Abstract

For a random intersection graph with a power law degree sequence having a finite mean and an infinite variance we show that the global clustering coefficient admits a tunable asymptotic distribution.

Correlation Between Clustering and Degree in Affiliation Networks

“…Such a neighbor belongs to the set W and it is called a witness of the edge v ∼ v . We note that for n, m → +∞ satisfying m/n → β for some β > 0, the random intersection graph G admits a tunable global clustering coefficient and power law degree distribution [3], [4]. Next we introduce network characteristics studied in this paper.…”

Normal and stable approximation to subgraph counts in superpositions of Bernoulli random graphs

“…Since the global and local clustering coefficients are expressed in terms of counts of triangles and wedges, a rigorous asymptotic analysis of clustering coefficients reduces to that of the triangle counts and wedge counts. In particular, the bivariate asymptotic normality for triangle and wedge counts in a related sparse random intersection graph was shown in [4], and related α-stable limits were established in [5]. Another line of research pursued in [11,16] addresses the concentration of subgraph counts in G [n,m] .…”

“…Given F, for any positive sequences {a n } and {b n } we write a n b n (respectively a n ≺ b n ) whenever, for sufficiently large n, we have [5,3] and overlay graph G [5,3] . FIGURE 2.…”

“…Figure 1 depicts an instance of the overlay graph G [5,3] and respective multigraph G [5,3] = G 5,1 ∪ G 5,2 ∪ G 5,3 , where G 5,i has vertex set V 5,i = {i, i + 1, i + 2} and edges labelled (coloured) i. G [5,3] has three polychromatic and two monochromatic copies of K 3 (Figure 2), while G [5,3] has two polychromatic copies of K 3 (induced by {1, 2, 3} and {2, 3, 4}) and one monochromatic copy of K 3 (induced by {3, 4, 5}).…”