On existence of triangles, clustering, and small-world property of random key graphs

Random key graphs have been introduced about a decade ago in the context of key predistribution schemes for securing wireless sensor networks (WSNs). They have received much attention recently with applications spanning the areas of recommender systems, epidemics in social networks, cryptanalysis, and clustering and classification analysis. This paper is devoted to analyzing various properties of random key graphs including containment and number of triangles, clustering coefficient, and small-world behavior. In particular, we establish a zero-one law for the existence of triangles in random key graphs, and identify the corresponding critical scaling. This zero-one law exhibits significant differences with the corresponding result in Erdős-Rényi (ER) graphs. We also compute the clustering coefficient of random key graphs, and compare them with that of ER graphs in the many node regime when the expected average degrees are asymptotically equivalent. We show that on the parameter range of practical relevance in both wireless sensor network and social network applications, random key graphs are much more clustered than the corresponding ER graph. The suitability of random key graphs as small worlds in the sense of Watts and Strogatz is also demonstrated.