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International Journal on Advances in Systems and Measurements, vol 5 no 3 & 4, year 2012, http://www.iariajournals.org/systems_and_measurements/ 1 0.5 0 −0.5 −1 1 0.5 0 −0.5 −1 −1 Figure 2: Trust Graph Illustration: Lighter edges represent wireless connectivity and darker edges represent secure connectivity. C C 15 10 5 0 0.8 0.85 0.9 0.99 Pr[G is connected.] (n,p) −0.5 Pr[G is connected] = e (n,p) eCC → Figure 3: Variation of Cc for desired probability of graph connectivity in Eq. 3. nodes. Finally, we used a number of open source software libraries for high precision arithmetic, graph theory algorithms and MATLAB for experiment design, statistical analysis and visualization. We discuss these in further detail in Section IV. III. THEORETICAL BOUND ANALYSIS For a given WSN with node population n, RKPS models it as a Erdős-Rényi random graph such that appropriate keyring size k and keypool size K can be selected to ensure the formed secure network remain connected. A. Trust Graph A sensor network can be modeled as a pair of overlaid graphs, where the underlying graph represents the wireless connectivity among sensor nodes, and the overlay graph shows the secured wireless connectivity among sensor nodes. In the underlying wireless connectivity graph, each sensor node is denoted as a vertex 0 0.5 1 and each wireless connection between two neighboring nodes is represented as a link. In the overlay secured wireless connectivity graph, also referred as Trust Graph, only secured wireless connections remain and are represented as secured links. Figure 2 shows a trust graph overlaid on the top of the connectivity graph of a WSN deployed on the surface of a unit sphere. A trust graph is a sub-graph of the underlying wireless connectivity graph, since its edges only exist if an underlying connectivity graph edge exists. The figure plots the Cartesian coordinates of sensor node locations distributed on the surface of a model sphere, with unit radius. We discuss this modeling approach further in Section IV, Deployment Model. B. Generalized RKPS Model In this section, we describe the general model of basic RKPS [2]. Before a WSN is deployed, RKPS allocates a small random subset of keys (keyrings) on each sensor node from a large universal set of random keys (keypool), where each subset may overlap with other subsets with a small probability p. Once deployed, each sensor initiates a shared key discovery protocol with its neighboring nodes by sending a keyrequest containing unique identifiers for each key in its keyring. The neighboring node with common key will respond back by encrypting a random number with a common key (challenge), which will be decrypted by the requesting node and sent back (response) to complete the authentication process. Subsequently, the identified common key can be used to negotiate a shared session encryption key. The small keyrings only allow a fraction of neighboring nodes to directly authenticate the received keyrequest. A sensor node that unable to authenticate a targeted neighboring node would resort to a path key establishment mechanism (PKEM), where it forwards the keyrequest to its authenticated and thus trusted neighboring nodes. These neighboring nodes would either authenticate the targeted node or forward it to their trusted neighboring nodes until a transitively trusted node authenticates the targeted neighboring node. The deployment model of a WSN is generally assumed to be uniformly random, and thus the neighboring nodes of any given sensor node cannot be predicted. An Erdős-Rényi random graph is denoted as G(n,p), where n is the number of vertices and p 2012, © Copyright by authors, Published under agreement with <strong>IARIA</strong> - www.iaria.org 236
International Journal on Advances in Systems and Measurements, vol 5 no 3 & 4, year 2012, http://www.iariajournals.org/systems_and_measurements/ 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 4: Sensor Network Model with full-visibility assumption. represents the probability that a vertex is connected to any other vertex within the graph. For a random graph G(n,p), we have i f p = ln(n) n + Cc n then lim n→∞ P(G(n,p) is connected) = e e−Cc (1) (2) where Cc is a constant and should be chosen such that P(G(n,p) is connected) is close to 1. Prior research [2] on RKPS has recommended choosing the value of C between 8 and 16, as shown in Figure 3 which can yield the desired value of p, and further derive the keyring size (k) for a given keypool size (K). It is essential to note that the Erdős-Rényi graph theory assumes that within the graph any node can be connected to another one, i.e., every node can see any others within the network (full-visibility model). However, in sensor networks a sensor node is only connected to a small subset na : na ≪ n of the randomly deployed nodes that are within its transmission range (limited-visibility model). Figure 4 visually illustrates a WSN modeled under the full-visibility assumption in Erdős-Rényi random graph theory. Figure 5 shows a sensor network modeled under the limited-visibility, encountered in practical sensor network deployments. In both, Figure 4 and Figure 5, the nodes represented by the plotted points are only joined by edges if there is connectivity between them. The dimensions represent distance and the plot models the Cartesian coordinates of each sensor’s deployment location on a two dimensional square sensor field. Note that in case of full-visibility Figure 4, each node is connected to every other node and its location is immaterial. In contrast, the location and neighborhood of a sensor in the 2012, © Copyright by authors, Published under agreement with <strong>IARIA</strong> - www.iaria.org 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 5: Sensor Network Model with limited-visibility assumption. limited-visibility Figure 5 case governs its connectivity with other sensors in the network. In order to overcome the lack of connectivity of the limited-visibility case, the work in [2] proposed adjusting p to the effective probability (pa), with which a node can connect to any of its neighboring nodes, such that the average degree d of the nodes in the graph remains constant as shown by Eq. 3. d = (na − 1)pa = np (3) With this calculated value of pa, the work in [2] derived k according to the following equation: pa = 1 − (K − k)!2 K!(K − 2k)! (4) Research results identifying the upper bound on the random graph diameter with the parameter CC controlled with the proposed range have been proposed in Theorem 4 in [8], where we have p ≥ cln(n)/n. C. Diameter of a Sparse Random Graph Several studies have analytically investigated the upper bound of Erdős-Rényi random graphs for various ranges of n and p. For example, the work presented in [8] reviews the analytical results on various ranges of p, in terms of n. Moreover, it derives the asymptotic bounds on the diameter of Erdős-Rényi random graphs at its critical threshold where both n and p satisfy the relationship mentioned in Eq. 1. Theorem 3 in [8] states that given the relationship in Eq. 5, the diameter of the graph is concentrated on at most three values around value indicated in Eq. 6. np = c ≥ 2 (5) lnn diam(G(n,p)) ≤ ⌈ lnn ⌉ + 1 (6) lnnp 237
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The International Journal on Advanc
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Daniela Dragomirescu, LAAS-CNRS / U
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Larisa Shwartz, T.J. Watson Researc
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