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Random key graphs were introduced to study various properties of theEschenauer-Gligor key predistribution scheme for wireless sensor networks(WSNs). Recently this class of random graphs has received much attention incontexts as diverse as recommender systems, social network modeling, andclustering and classification analysis. This paper is devoted to analyzingvarious properties of random key graphs. In particular, we establish a zero-onelaw for the the existence of triangles in random key graphs, and identify thecorresponding critical scaling. This zero-one law exhibits significantdifferences with the corresponding result in Erdos-Renyi (ER) graphs. We alsocompute the clustering coefficient of random key graphs, and compare it to thatof ER graphs in the many node regime when their expected average degrees areasymptotically equivalent. For the parameter range of practical relevance inboth wireless sensor network and social network applications, random key graphsare shown to be much more clustered than the corresponding ER graphs. We alsoexplore the suitability of random key graphs as small world models in the senseof Watts and Strogatz.

Counting triangles, tunable clustering and the small-world property in random key graphs (Extended version)