The random key graph is a random graph naturally associated with the random key predistribution scheme introduced by Eschenauer and Gligor in the context of wireless sensor networks (WSNs). 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 complement and strengthen recent work on this conjecture by Blackburn and Gerke. In particular, the results are given under conditions which are more realistic for applications to WSNs.
The zero-law in ) has recently been established independently by Blackburn and Gerke @cite_17 , and by Ya g an and Makowski @cite_20 . In both papers, it was shown that [ n K (n; (K_n,P_n) ) contains no isolated node = 0 ] whenever @math in ), a result which clearly implies the conjectured zero-law.
@cite_20 @cite_17
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We consider the random graph induced by the random key predistribution scheme of Eschenauer and Gligor under the assumption of full visibility. We show the existence of a zero-one law for the absence of isolated nodes, and complement it by a Poisson convergence for the number of isolated nodes. Leveraging earlier results and analogies with Erdos-Renyi graphs, we explore similar results for the property of graph connectivity. In this paper, we consider a wireless network with unreliable links and investigate its minimum node degree and k- connectivity. In such network, n nodes are uniformly distributed in a region, and edges are established for any two nodes within a certain distance and with a probabilistically active link in between. On a torus topology, we present a zero-one law for the property of minimum degree being at least k, leading to a zero-law for k-connectivity and an intermediate result towards a one-law. On a square with boundary effect, we establish a one- law for minimum degree. Our results are derived from rigorous analysis and also confirmed by the simulation, and they provide guidelines for the design of wireless networks. Index Terms—Connectivity, Erdos-Renyi graph, geometric graph, graph intersection, minimum node degree, random graph, unreliable link, wireless network, zero-one law. Let P be a Poisson process of intensity one in a square Sn of area n. For a fixed integer k, join every point of P to its k nearest neighbours, creating an undirected random geometric graph Gn,k. We prove that there exists a critical constant ccrit such that for c ccrit, Gn,⌊clog n⌋ is connected with probability tending to 1 as n → ∞. This answers a question posed by the authors in [1]. Let P be a Poisson process of intensity one in a square Sn of area n. For a fixed integer k, we join every point of P to its k nearest neighbours, creating an undirected random geometric graph GSn,k = Gn,k in which every vertex has degree at least k. The connectivity of these graphs was studied by the present authors in [1]. It is not hard to see that Gn,k becomes connected around k = �(log n), and we proved in [1] that if k(n) ≤ 0.3043log n then the probability that Gn,k(n) is connected tends to zero as n → ∞, while if k(n) ≥ 0.5139log n then the probability that Gn,k(n) is connected tends to one as n → ∞. However, we were unable to prove the natural conjecture that there exists a critical constant ccrit such that for c ccrit, P(Gn,⌊clog n⌋ is connected) → 1 as n → ∞. In this paper we prove this conjecture. Zero knowledge plays a central role in cryptography and complexity. The seminal work of Ben- (STOC 1988) shows that zero knowledge can be achieved unconditionally for any language in NEXP, as long as one is willing to make a suitable *physical assumption*: if the provers are spatially isolated, then they can be assumed to be playing independent strategies. Quantum mechanics, however, tells us that this assumption is unrealistic, because spatially-isolated provers could share a quantum entangled state and realize a non-local correlated strategy. The MIP^* model captures this setting. In this work we study the following question: does spatial isolation still suffice to unconditionally achieve zero knowledge even in the presence of quantum entanglement? We answer this question in the affirmative: we prove that every language in NEXP has a 2-prover *zero knowledge* interactive proof that is sound against entangled provers; that is, NEXP ⊆ ZK-MIP^*. Our proof consists of constructing a zero knowledge interactive PCP with a strong algebraic structure, and then lifting it to the MIP^* model. This lifting relies on a new framework that builds on recent advances in low-degree testing against entangled strategies, and clearly separates classical and quantum tools. Our main technical contribution is the development of algebraic techniques for obtaining unconditional zero knowledge; this includes a zero knowledge variant of the celebrated sumcheck protocol, a key building block in many probabilistic proof systems. A core component of our sumcheck protocol is a new algebraic commitment scheme, whose analysis relies on algebraic complexity theory.