Connectivity in secure wireless sensor networks under transmission constraints

Abstract

In wireless sensor networks (WSNs), the Eschenauer-Gligor (EG) key pre-distribution scheme is a widely recognized way to secure communications. Although the connectivity properties of secure WSNs with the EG scheme have been extensively investigated, few results address physical transmission constraints. These constraints reflect real-world implementations of WSNs in which two sensors have to be within a certain distance from each other to communicate. In this paper, we present the first zero-one laws for connectivity in WSNs employing the EG scheme under transmission constraints. These laws improve recent results [10, 11] significantly, are sharp, and help specify the critical transmission ranges for connectivity. Our analytical findings, which are also confirmed via numerical experiments, provide precise guidelines for the design of secure WSNs in practice. The application of our theoretical results to frequency hopping of wireless networks is discussed in some detail.

<inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-Connectivity in Random Key Graphs With Unreliable Links

“…One possible candidate is the so-called disk model [27] where two nodes have to be within a certain distance to each other to have a link in between; this induces a random geometric graph. Intersection of random key graphs with random geometric graphs has already received some attention [22], [23], but the model is proven to be difficult to analyze with results obtained thus far for its connectivity [22], [23], [41], not for k-connectivity.…”

“…Therefore, (38) will follow immediately (withK n =K n andP n =P n ) if (41) holds. We now give the coupling argument that leads to (41). As seen from (3), G on is the intersection of a random key graph G(n, K n , P n ) and an Erdős-Rényi graph G(n, p n ).…”

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

“…Denote the circle by ; let = 1 be the event that node is isolated in , and let (V ) be the intersection of and the disk centered at positionV ∈ with radius , where node is at positionV . Similar to the discussion in [16], the number of nodes within (V ) follows a Poisson distribution with mean (V ); and to have an edge with , a node not only has to be within a (V ) but also has to share at least a key with node , so the number of nodes neighboring to follows a Poisson distribution with mean (V ). IntegratingV over , the probability that node is isolated in is given by…”

“…The upper bound on ( − 1) ( 1 2 ) that nodes 1 and 2 are isolated if < ≤ 2 and 0 ≤ ≤ is obtained together using (16) and (17). So 3 ( − 1) Pr( 1 2 = 1 | ≤ ≤ 2 ) + ( − 1) Pr( 1 2 = 1 | 0 ≤ ≤ ) is upper-bounded as follows:…”

Secure Connectivity of Wireless Sensor Networks Under Key Predistribution with on/off Channels

“…Then given (36) (i.e., K n ≥ K n for each n), from the definitions of graphs G q (n, K n , P n ) and G q (n, K n , P n ), we can construct them on the same probability space such that G q (n, K n , P n ) is a spanning subgraph of G q (n, K n , P n ). Given (38) and (39) (i.e., p n ≤ p n ≤ 1 for each n), p n is indeed a probability, and we can define Erdős-Rényi graphs G(n, p n ) and G(n, p n ) on the same probability space such that G(n, p n ) is a spanning subgraph of G(n, p n ). Summarizing the above, we can define G q (n, K n , P n ) ∩ G(n, p n ) and G q (n, K n , P n ) ∩ G(n, p n ) on the same probability space such that G q (n, K n , P n ) ∩ G(n, p n ) is a spanning subgraph of G q (n, K n , P n ) ∩ G(n, p n ).…”