Clustering function: another view on clustering coefficient

Abstract

Assuming that actors u and v have r common neighbors in a social network we are interested in how likely is that u and v are adjacent. This question is addressed by studying the collection of conditional probabilities, denoted cl(r), r = 0, 1, 2,. .. , that two randomly chosen actors of the social network are adjacent, given that they have r common neighbors. The function r → cl(r) describes clustering properties of the network and extends the global clustering coefficient. Our empirical study shows that the function r → cl(r) exhibits a typical sigmoid pattern. In order to better understand this pattern we establish the large scale asymptotics of cl(•) for two related random intersection graph models of affiliation networks admitting a non-vanishing global clustering coefficient.

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

“…The proof (8) is similar to that of (7). It makes use the observation that the typical adjacency relation is witnessed by a single attribute.…”

“…Now, the statistical dependence of neighboring adjacency relations persists as n, m → +∞, and the asymptotic bivariate degree-degree distribution is not a product of marginal asymptotic degree distributions, see (7). One can show that in this case the degrees of adjacent vertices are positively correlated and G admits a non-vanishing positive Newman's assortativity coefficient, provided that the vertex degree distribution has a finite third moment (cf.…”

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

“…We note that the moment conditions EX 2 1 < ∞ and EY 1 < ∞ of Theorem 2 are the minimal ones as the numbers a 2 = EX 2 1 and b 1 = EY 1 enter (implicitly) both formulas (7) and (8).…”

“…Proof of (i). The intuition behind formula (7) is that with a high probability the adjacency relation v 1 ∼ v 2 as well as all common neighbors of v 1 and v 2 are witnessed, by the same common attribute (all attributes having equal chances). Furthermore, conditionally on the event that this attribute is w 1 , and given Y 1 , Y 2 , X 1 , we have that the random variables…”

Moment-Based Parameter Estimation in Binomial Random Intersection Graph Models

“…In this case the limiting model can be parameterised by its mean degree λ = βγ 2 and attribute intensity µ = βγ. By extending the model by introducing random node weights, we obtain a statistical network model which is rich enough to admit heavy tails and nontrivial clustering properties [4,5,8,10]. Such models can also be generalised to the directed case [6].…”