We establish asymptotic vertex degree distribution and examine its relation
to the clustering coefficient in two popular random intersection graph models
of Godehardt and Jaworski [Electron. Notes Discrete Math. 10 (2001) 129-132].
For sparse graphs with a positive clustering coefficient, we examine
statistical dependence between the (local) clustering coefficient and the
degree. Our results are mathematically rigorous. They are consistent with the
empirical observation of Foudalis et al. [In Algorithms and Models for Web
Graph (2011) Springer] that, "clustering correlates negatively with degree."
Moreover, they explain empirical results on $k^{-1}$ scaling of the local
clustering coefficient of a vertex of degree k reported in Ravasz and Barabasi
[Phys. Rev. E 67 (2003) 026112].Comment: Published in at http://dx.doi.org/10.1214/12-AAP874 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
We study a connectivity property of a secure wireless network that uses random pre-distribution of keys. A network is composed of n sensors. Each sensor is assigned a collection of d different keys drawn uniformly at random from a given set of m keys. Two sensors are joined by a communication link if they share a common key. We show that for large n with high probability the connected component of size (n) emerges in the network when the probability of a link exceeds the threshold 1/n. Similar component evolution is shown for networks where sensors communicate if they share at least s common keys.
We consider sparse random intersection graphs with the property that the clustering coefficient does not vanish as the number of nodes tends to infinity. We find explicit asymptotic expressions for the correlation coefficient of degrees of adjacent nodes (called the assortativity coefficient), the expected number of common neighbours of adjacent nodes, and the expected degree of a neighbour of a node of a given degree k. These expressions are written in terms of the asymptotic degree distribution and, alternatively, in terms of the parameters defining the underlying random graph model.
We present and investigate a general model for inhomogeneous random digraphs with labeled vertices, where the arcs are generated independently, and the probability of inserting an arc depends on the labels of its endpoints and on its orientation. For this model, the critical point for the emergence of a giant component is determined via a branching process approach.
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