Brief Encounters with a Random Key Graph

Random key graphs, also called uniform random intersection graphs, have been used for modeling a variety of different applications since their introduction in wireless sensor networks. In this paper, we review some of their recent applications and suggest several new ones, under the full visibility assumption; i.e., under the assumption that all nodes are within communication range of each other. We also suggest further research in determining the connectivity properties of random key graphs when limited visibility is more realistic; e.g., graph nodes can communicate only with a subset of other nodes due to range restrictions or link failures. The notion of the random key graph (a.k.a. “uniform random intersection graph” [1]) was introduced for probabilistic key pre-distribution in wireless sensor networks [2]. Recently, these graphs have been used for a variety of different applications, such as clustering analysis [3], recommender systems using collaborative filtering [4], and cryptanalysis of hash functions [5]. While random key graphs are not Erdös-Renyi random graphs (e.g., the probabilities that graph edges exist are not necessarily independent), they have similar connectivity properties under the assumption of “full visibility”; i.e., they obey a similar “zero-one” law as Erdös-Renyi graphs for some scaling of their parameters, whenever each node is within communication range of other nodes [6, 7]. Recommender Systems Using Collaborative Filtering. Marbach [4] illustrates an application of random-key-graph connectivity for the modeling of recommender systems with collaborative filtering. In his model, there are N users and a pool of PN objects. Each of the N users picks a set of KN distinct objects uniformly and randomly from the pool, and ranks each object, where KN < PN . A recommender class C is defined as a set of users that gives the same ranking to any particular object that is ranked by at least one user of the set. This implies that (1) every subset of users in class C that rank a common object assigns the same ranking to that object, and (2) not every user in class C ranks all objects ranked by others in C; i.e., there exist users in class C that have not ranked objects ranked by others in class C. A recommender system with collaborative filtering predicts the ranking a user would give an object that the user has not ranked, but that has been ranked by some other users in class C. However, to enable such predictions, the membership