1. Introduction Approximation by the Poisson distribution arises naturally in the theory of random graphs, as in many other fields, when counting the number of occurrences of individually rare and unrelated events within a large ensemble. For example, one may be concerned with the number of times that a particular small configuration is repeated in a large graph, such questions being considered...

We study a random graph model which combines properties of the edge percolation model on Z d and a classical random graph G(n, c/n). We show that this model, being a homogeneous random graph, has a natural relation to the so-called " rank 1 case " of inhomogeneous random graphs. This allows us to use the newly developed theory of inhomogeneous random graphs to describe completely the phase diag...

For a general random intersection graph we show an approximation of the vertex degree distribution by a Poisson mixture. key words: intersection graph, random graph, degree distribution

We study the applicability of random graph theory in modeling secure connectivity of wireless sensor networks. Specifically, our work focuses on the highly influential random key predistribution scheme by Eschenauer and Gligor to examine the appropriateness of the modeling in finding system parameters for desired connectivity. We use extensive simulation and theoretical results to identify rang...

We consider the combinatorial properties of the trace of a random walk on the complete graph and on the random graph G(n, p). In particular, we study the appearance of a fixed subgraph in the trace. We prove that for a subgraph containing a cycle, the threshold for its appearance in the trace of a random walk of length m is essentially equal to the threshold for its appearance in the random gra...