On Topological Properties of Wireless Sensor Networks under the q-Composite Key Predistribution Scheme with On/Off Channels

The q-composite key predistribution scheme [1] is used prevalently for secure communications in large-scale wireless sensor networks (WSNs). Prior work [2]-[4] explores topological properties of WSNs employing the q-composite scheme for q = 1 with unreliable communication links modeled as independent on/off channels. In this paper, we investigate topological properties related to the node degree in WSNs operating under the q-composite scheme and the on/off channel model. Our results apply to general q and are stronger than those reported for the node degree in prior work even for the case of q being 1. Specifically, we show that the number of nodes with certain degree asymptotically converges in distribution to a Poisson random variable, present the asymptotic probability distribution for the minimum degree of the network, and establish the asymptotically exact probability for the property that the minimum degree is at least an arbitrary value. Numerical experiments confirm the validity of our analytical findings.Comment: Best Student Paper Finalist in IEEE International Symposium on Information Theory (ISIT) 201

A classification of isomorphism-invariant random digraphs

We classify isomorphism-invariant random digraphs \linebreak (IIRDs) according to where randomness lies, namely, on arcs, vertices, vertices and arcs together as arc random digraphs (ARD), vertex random digraphs (VRD), vertex-arc random digraphs (VARD) as an extension of the classification of isomorphism-invariant random graphs (IIRGs) \cite{beer:2011}, and introduce randomness in direction (together with arcs, vertices, etc.) also which in turn yield direction random digraphs (DRDs) and its variants, respectively. We demonstrate that for the number of vertices $n\ge 4$, ARDs and VRDs are mutually exclusive and are both proper subsets of VARDs, and also demonstrate the existence of VARDs which are neither ARDs nor VRDs, and the existence of IIRDs that are not VARDs (e.g., random nearest neighbor digraphs(RNNDs)). We demonstrate that to obtain a DRD as an IIRD, one has to start with an IIRG and insert directions randomly. Depending on the type of IIRG, we obtain direction-edge random digraphs (DERDs), direction-vertex random digraphs (DVRDs), and direction-vertex-edge random digraphs (DVERDs), and demonstrate that DERDs and DVRDs have an overlap but are mutually exclusive for $n \ge 4$, and both are proper subsets of DVERDs which is a proper subset of DRDs and also the complement of DRDs in IIRDs is nonempty (e.g., RNNDs). We also study the relation of DRDs with VARDs, VRDs, and ARDs and show that for $n\ge 4$, the intersection of DERDs and VARDs is ARDs; we provide some results and open problems and conjectures. For example, the relation of DVRDs and DVERDs with the VARDs (hence with ARDs and VRDs) are still open problems for $n \ge 4$. We also show positive dependence between the arcs of a VARD whose tails are same which implies the asymptotic distribution of the arc density of VRDs and ARDs has nonnegative variance

ON VERTEX, EDGE, AND VERTEX-EDGE RANDOM GRAPHS (EXTENDED ABSTRACT)

We consider three classes of random graphs: edge random graphs, vertex random graphs, and vertex-edge random graphs. Edge random graphs are Erdős-Rényi random graphs [9, 10], vertex random graphs are generalizations of geometric random graphs [21], and vertex-edge random graphs generalize both. The names of these three types of random graphs describe where the randomness in the models lies: in the edges, in the vertices, or in both. We show that vertex-edge random graphs, ostensibly the most general of the three models, can be approximated arbitrarily closely by vertex random graphs, but that the two categories are distinct