Random key graphs can have many more triangles than Erdős-Rényi graphs

Random key graphs are random graphs induced by the random key predistribution scheme of Eschenauer and Gligor (EG) under the assumption of full visibility. For this class of random graphs we establish a zero-one law for the existence of triangles, and identify the corresponding critical scaling. This zero-one law exhibits significant differences with the corresponding result in Erdős-Rényi graphs. We also compute the clustering coefficient of random key graphs, and compare them with that of Erdős-Rényi graphs in the many node regime when the expected average degrees are asymptotically equivalent. On the parameter range of practical relevance in wireless sensor networks (WSNs), random key graphs are found to be much more clustered than the corresponding ErdősRényi graphs. These results clearly show the inadequacy of Erdős-Rényi graphs to capture some key properties of the EG scheme in realistic WSN scenarios. The suitability of random key graphs as small worlds in the sense of Watts and Strogatz is also discussed.