Preferential attachment models have been widely studied in complex networks, because they can explain the formation of many networks like social networks, citation networks, power grids, and biological networks, to name a few. Motivated by the application of key predistribution in wireless sensor networks (WSN), we initiate the study of preferential attachment with degree bound. Our paper has two important contributions to two different areas. The first is a contribution in the study of complex networks. We propose preferential attachment model with degree bound for the first time. In the normal preferential attachment model, the degree distribution follows a power law, with many nodes of low degree and a few nodes of high degree. In our scheme, the nodes can have a maximum degree @math , where @math is an integer chosen according to the application. The second is in the security of wireless sensor networks. We propose a new key predistribution scheme based on the above model. The important features of this model are that the network is fully connected, it has fewer keys, has larger size of the giant component and lower average path length compared with traditional key predistribution schemes and comparable resilience to random node attacks. We argue that in many networks like key predistribution and Internet of Things, having nodes of very high degree will be a bottle-neck in communication. Thus, studying preferential attachment model with degree bound will open up new directions in the study of complex networks, and will have many applications in real world scenarios.
Yagan @cite_14 @cite_20 @cite_2 studied the key graphs formed by @cite_25 and @cite_18 under full and partial visibility. They showed that graphs defined in @cite_18 are connected with high probability. They also studied the security under the ON-OFF secure channel, which means that the links between nodes might or might not be active at a given instant.
@cite_18 @cite_14 @cite_2 @cite_25 @cite_20
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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. Random key graphs form a class of random intersection graphs that are naturally induced by the random key predistribution scheme of Eschenauer and Gligor for securing wireless sensor network (WSN) communications. Random key graphs have received much attention recently, owing in part to their wide applicability in various domains, including recommender systems, social networks, secure sensor networks, clustering and classification analysis, and cryptanalysis to name a few. In this paper, we study connectivity properties of random key graphs in the presence of unreliable links. Unreliability of graph links is captured by independent Bernoulli random variables, rendering them to be on or off independently from each other. The resulting model is an intersection of a random key graph and an Erdős–Renyi graph, and is expected to be useful in capturing various real-world networks; e.g., with secure WSN applications in mind, link unreliability can be attributed to harsh environmental conditions severely impairing transmissions. We present conditions on how to scale this model’s parameters so that: 1) the minimum node degree in the graph is at least @math and 2) the graph is @math -connected, both with high probability as the number of nodes becomes large. The results are given in the form of zero-one laws with critical thresholds identified and shown to coincide for both graph properties. These findings improve the previous results by Rybarczyk on @math -connectivity of random key graphs (with reliable links), as well as the zero-one laws by Yagan on one-connectivity of random key graphs with unreliable links. We investigate the secure connectivity of wireless sensor networks under the random key distribution scheme of Eschenauer and Gligor. Unlike recent work which was carried out under the assumption of full visibility, here we assume a (simplified) communication model where unreliable wireless links are represented as on off channels. We present conditions on how to scale the model parameters so that the network: 1) has no secure node which is isolated and 2) is securely connected, both with high probability when the number of sensor nodes becomes large. The results are given in the form of full zero-one laws, and constitute the first complete analysis of the EG scheme under non-full visibility. Through simulations, these zero-one laws are shown to be valid also under a more realistic communication model (i.e., the disk model). The relations to the Gupta and Kumar's conjecture on the connectivity of geometric random graphs with randomly deleted edges are also discussed. To be considered for an IEEE Jack Keil Wolf ISIT Student Paper Award. We study the secure and reliable connectivity of wireless sensor networks. Security is assumed to be ensured by the random pairwise key predistribution scheme of Chan, Perrig, and Song, and unreliable wireless links are represented by independent on off channels. Modeling the network by an intersection of a random K-out graph and an Erdős-Renyi graph, we present scaling conditions (on the number of nodes, the scheme parameter K, and the probability of a wireless channel being on) such that the resulting graph contains no nodes with degree less than k with high probability, when the number of nodes gets large. Results are given in the form of zero-one laws and are shown to improve the previous results by Yagan and Makowski on the absence of isolated nodes (i.e., absence of nodes with degree zero). Via simulations, the established zero-one laws are shown to hold also for the property of k-connectivity; i.e., the property that graph remains connected despite the deletion of any k − 1 nodes or edges.