Degree-Degree Distribution in a Power Law Random Intersection Graph with Clustering

Abstract

The bivariate distribution of degrees of adjacent vertices, degree-degree distribution, is an important network characteristic defining the statistical dependencies between degrees of adjacent vertices. We show the asymptotic degree-degree distribution of a sparse inhomogeneous random intersection graph and discuss its relation to the clustering and power law properties of the graph.

Degree-degree distribution in a power law random intersection graph with clustering

“…Connections between Newman's assortativity coefficient and the clustering coefficient in related random graph models have been discussed in [6]. The present paper complements, revises and extends the results of [3], presented at the 12th Workshop on Algorithms and Models for the Web Graph, WAW 2015. In particular, the factor b 1 is included in (11).…”

“…Here τ i , τ i , i ≥ 1 are independent and identically distributed random variables, which are independent of Y 1 , Y 2 , X 1 and have distribution (3). Define the events…”

“…Define the random vectors L(2) and L(3) in the same way as L (2) and L (3) above, but with ξ 2i and ξ 3i replaced by ξ2i = ξ i + ∆ i and ξ3i = ξ i + ∆ i respectively. We note that given Y 1 , Y 2 , X 1 the random vector L(2) has the same conditional distribution as L (2) , and L(3) has the same conditional distribution as L (3) . Hence in order to prove (54) for h = 2 it suffices to show that for any realized values Y 1 , Y 2 , X 1 we have…”

“…Furthermore, the tail P(d * > r) of a randomly stopped sum d * is as heavy as the heavier one of Y 1 and τ 1 , see, e.g., [1]. Hence, choosing a power law weights X and Y we obtain a power law asymptotic degree distributions, namely, the distributions of d * and Λ 3 .…”

“…In particular, the factor b 1 is included in (11). It was missing in the respective formula (7) of [3].…”