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.

Experimental results show that in large complex networks such as Internet or biological networks, there is a tendency to connect elements which have a common neighbor. This tendency in theoretical random graph models is depicted by the asymptotically constant clustering coefficient. Moreover complex networks have power law degree distribution and small diameter (small world phenomena), thus these are desirable features of random graphs used for modeling real life networks. We survey various variants of random intersection graph models, which are important for networks modeling.

Experimental results show that in large complex networks (such as internet, social or biological networks) there exists a tendency to connect elements which have a common neighbor. In theoretical random graph models, this tendency is described by the clustering coefficient being bounded away from zero. Complex networks also have power-law degree distributions and short average distances (small world phenomena). These are desirable features of random graphs used for modeling real life networks. We survey recent results concerning various random intersection graph models showing that they have tunable clustering coefficient, a rich class of degree distributions including power-laws, and short average distances.

The classical Erdős-Pósa theorem states that for each positive integer k there is an f(k) such that, in each graph G which does not have k + 1 disjoint cycles, there is a blocker of size at most f(k); that is, a set B of at most f(k) vertices such that G − B has no cycles. We show that, amongst all such graphs on vertex set {1, . . . , n}, all but an exponentially small proportion have a blocker of size k. We also give further properties of a random graph sampled uniformly from this class, concerning uniqueness of the blocker, connectivity, chromatic number and clique number.A key step in the proof of the main theorem is to show that there must be a blocker as in the Erdős-Pósa theorem with the extra 'redundancy' property that B − v is still a blocker for all but at most k vertices v ∈ B.

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.