On Connectivity Thresholds in Superposition of Random Key Graphs on Random Geometric Graphs

In a random key graph (RKG) of n nodes each node is randomly assigned a key ring of K-n cryptographic keys from a pool of P-n keys. Two nodes can communicate directly if they have at least one common key in their key rings. We assume that the n nodes are distributed uniformly in [0, 1](2). In addition to the common key requirement, we require two nodes to also be within r(n) of each other to be able to have a direct edge. Thus we have a random graph in which the RKG is superposed on the familiar random geometric graph (RGG). For such a random graph, we obtain tight bounds on the relation between K-n, P-n and r(n) for the graph to be asymptotically almost surely connected.