Zero-One Law for Connectivity in Superposition of Random Key Graphs on Random Geometric Graphs

Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2015, Article ID 982094, 9 pages http://dx.doi.org/10.1155/2015/982094 Research Article Zero-One Law for Connectivity in Superposition of Random Key Graphs on Random Geometric Graphs Y. Tang and Q. L. Li College of Mathematics and Computer Science, Hunan Normal University, Changsha 410081, China Correspondence should be addressed to Q. L. Li; Received 24 June 2015; Revised 11 October 2015; Accepted 18 October 2015 Academic Editor: Filippo Cacace Copyright © 2015 Y. Tang and Q. L. Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study connectivity property in the superposition of random key graph on random geometric graph. For this class of random graphs, we establish a new version of a conjectured zero-one law for graph connectivity as the number of nodes becomes unboundedly large. The results reported here strengthen recent work by the Krishnan et al. 1. Introduction Random key graph (RKG), also known as uniform random intersection graph, is a random graph defined below. Consider a set with 𝑛 nodes and another key pool with 𝑃𝑛 keys; we assume each node randomly chooses 𝐾𝑛 distinct keys for its key ring; two nodes can establish a secure link between them if they share at least one common key in their key rings. The random key graph is naturally associated with the random key predistribution scheme of Eschenauer and Gligor [1] for wireless sensor networks (WSNs). A WSN is a collection of distributed sensor devices that are able to communicated wirelessly and supports wide range of applications such as health and environment monitoring, imaging, tracking, and biomedical research; see [2]. These applications require all nodes in the network to be within communication range and to be connected with high probability. Some partial results concerning the connectivity of RKGs were given in [3–5]. In [6], Rybarczyk gave asymptotic tight bounds for the thresholds of the connectivity, phase transition, and diameter of the largest connected component in RKGs for all ranges of 𝐾𝑛 . With the advent of ad hoc sensor networks, an interesting class of random graphs, namely, random geometric graphs (RGGs), has gained new importance and its properties have been the subject of much study. Here the nodes are randomly distributed in a finite Euclidean space and there is an edge between two nodes if the Euclidean distance between them is below a specified threshold. Much work has been done on graph theoretic properties of such graph, comprehensively summarized in the monograph of [7]. Recently, there is interest in random graphs in which an edge is determined by more than one random property, that is, superposition of different random graphs. The superposition of ER random graphs over RGGs has been of interest for quite some time now. Recent work on such random graphs is in [8, 9] where connectivity properties and the distribution of isolated nodes are analyzed. And the superposition of ER random graphs on RKGs is considered in [10]. Such a graph is constructed as follows: a RKG is first formed based on the key distribution and each edge in this graph is deleted with a specified probability. The superposition of RKGs on RGGs is first studied in [11]. The 𝑛 nodes are distributed in a finite Euclidean space and each node is assigned a key ring of 𝐾𝑛 distinct keys drawn randomly from a pool of 𝑃𝑛 keys. Two nodes have an edge if and only if they share at least one common key in their key rings and their Euclidean distance is at most 𝑟𝑛 . Pietro et al. [11] have shown that under the scaling 𝜋𝑟𝑛2 𝐾𝑛2 /𝑃𝑛 = 𝑐(log 𝑛/𝑛), the one-law that this class of random graphs is connected follows if 𝑟𝑛 > 0 and 𝑐 > 20𝜋. Another notable work is due by Krzywdziński and Rybarczyk [12], where the authors have improved these results and established the one-law for 𝑐 > 8 without any constraint on 𝑟𝑛 . Recently, Krishnan et al. [13] 2 Discrete Dynamics in Nature and Society have shown that for large 𝑛, this class of random graphs will be connected if 𝐾𝑛 ≥ 2, 𝑟𝑛 and 𝑃𝑛 are selected such that 𝐾𝑛 , 𝑃𝑛 󳨀→ ∞, 𝐾𝑛2 󳨀→ 0, 𝑃𝑛 𝑃𝑛 ≥ 2𝐾𝑛 , (1) 𝑃𝑛 ≥ 𝜎𝑛𝑟𝑛2 , 𝜋𝑟𝑛2 𝐾𝑛2 2𝜋 log 𝑛 > , 𝑃𝑛 1−𝛾 𝑛 Theorem 1. Let 𝐾𝑛 ≥ 2, 𝐾𝑛 , 𝑃𝑛 → ∞, 𝐾𝑛2 /𝑃𝑛 → 0, 𝑃𝑛 ≥ 𝑛. Then for any 𝜎 > 0 and 0 < 𝛾 < 1. They also observed that for large 𝑛 and 0 < 𝑐1 < ∞, the probability that this class of random graphs is disconnected is at least 𝑒−𝑐1 /4 if the scaling satisfies 𝜋𝑟𝑛2 𝐾𝑛2 log 𝑛 + 𝑐1 = . 𝑃𝑛 𝑛 (2) The connectivity in the superposition of RKGs on RGGs is still studied in this paper. Assuming that 𝑃𝑛 ≥ 𝑛, we show that given 𝑛𝜋𝑟𝑛2 𝐾𝑛2 /𝑃𝑛 = log 𝑛+𝑐𝑛 , this class of random graphs is disconnected if 𝑐𝑛 → −∞, and for 𝑐𝑛 → ∞, this class of random graphs is connected. In this paper, we use standard, asymptotic notations: 𝑎𝑛 = Θ(𝑏𝑛 ), 𝑎𝑛 = 𝜔(𝑏𝑛 ), 𝑎𝑛 = 𝑜(𝑏𝑛 ), and 𝑎𝑛 ∼ 𝑏𝑛 for ∃𝑐,𝐶>0 𝑐𝑏𝑛 ≤ 𝑎𝑛 ≤ 𝐶𝑏𝑛 , 𝑎𝑛 /𝑏𝑛 → ∞, 𝑎𝑛 /𝑏𝑛 → 0, and 𝑎𝑛 /𝑏𝑛 → 1, respectively, all limits are taken as 𝑛 → ∞. The phrase “with high probability” (abbreviated whp) means with probability tending to one as 𝑛 tends to infinity. The rest of the paper is organized as follows. Our main result is presented in Section 2. Namely, the theorem concerning zero-one law for graph connectivity is presented. Section 3 contains technical proof of Theorem 1. Finally, Section 4 discusses prospects of establishing tighter connectivity thresholds in the superposition of RKGs on RGGs. 2. Main Result The 𝑛 nodes are uniformly and independently distributed in R = [0, 1]2 . Let 𝑥𝑖 ∈ R be the location of point 𝑖. A key pool with 𝑃𝑛 cryptographic keys is designated for the network of 𝑛 nodes. Node 𝑖 randomly chooses a subset 𝑆𝑖 of keys from the key pool with |𝑆𝑖 | = 𝐾𝑛 . Our interest is in the random graph 𝐺(𝑃𝑛 , 𝐾𝑛 , 𝑟𝑛 ) with 𝑛 nodes and edges formed as follows. An edge (𝑖, 𝑗), 1 ≤ 𝑖 < 𝑗 ≤ 𝑛, is present in 𝐺(𝑃𝑛 , 𝐾𝑛 , 𝑟𝑛 ) if both of the following two conditions are satisfied: 󵄩 󵄩 𝐸1 : 󵄩󵄩󵄩󵄩𝑥𝑖 − 𝑥𝑗 󵄩󵄩󵄩󵄩 ≤ 𝑟𝑛 , 𝐸2 : 𝑆𝑖 ∩ 𝑆𝑗 ≠ 0, at least one common key. Thus 𝐺(𝑃𝑛 , 𝐾𝑛 , 𝑟𝑛 ) is a superposition of RKG on RGG. In the following, to avoid technicalities which obscure the main ideas, we will neglect edge effects resulting due to the fact that 𝑛 nodes are distributed uniformly and independently over a folded unit square R = [0, 1]2 with continuous boundary conditions and a node is close to the boundary of R. Throughout the paper, we set 𝑛𝜋𝑟𝑛2 = 𝑑𝑛 , where 𝑑𝑛 = 𝜔(log 𝑛) and 𝑑𝑛 = 𝑜(𝑛(1−𝛿)/4 ) for any small 0 < 𝛿 < 1. The following theorem gives zero-one law for the connectivity of a superposition of RKG on RGG. (3) where ‖ ⋅ ‖ represents the Euclidean norm. Condition 𝐸1 produces a random geometric graph with the transmission range 𝑟𝑛 . Imposing condition 𝐸2 on 𝐸1 retains the edges of the random geometric graph for which the two nodes share (i) if 𝜋𝑟𝑛2 𝐾𝑛2 /𝑃𝑛 = (l (...truncated)