ExBCG-TC: Extended Borel Cayley Graph Topology Control for Ad Hoc Networks

Transcription

1 Ex: Extended Borel Cayley Graph Topology Control for Ad Hoc Netorks Dongsoo Kim and Jaeook Yu Department of ECE Stony Brook University (SUNY) Stony Brook, NY {dongsoo.kim, {dkim, jyu} Eric Noel 200 Laurel Avenue AT&T Labs Middleton, NJ K. Wendy Tang Department of ECE Stony Brook University (SUNY) Stony Brook, NY tang Abstract In our previous ork, e demonstrated that the Borel Cayley Graph topology control () is an efficient topology control algorithm for producing an underlying netork topology resembling a 4-regular Borel Cayley Graph and assuring small diameter, short average path length and superb poer efficiency. But the netorks are unlikely to be connected belo a certain radio range threshold due to isolated nodes or isolated subnetorks. In this paper, e propose a topology control algorithm, EXTENDED (EX), that effectively improves netork connectivity of netork topology generated by. Our EX orks by injecting a small number of nodes into the netork and connecting them to nodes already in the netork. Throughout the efficient rules for establishing connection (namely the Request Criterion and the Accept Criterion), the proportion of isolated nodes or isolated subnetorks is significantly reduced or totally eliminated. The resulting netork topologies are more likely to be fully connected ith radio range much smaller than the radio range threshold of the hile preserving the outstanding netork topological properties and poer efficiency of netork topologies. I. INTRODUCTION Recent development of small and energy-efficient ireless sensors that can communicate ith each other via radio transceivers has resulted in the rapid groth of ireless sensor netorks (WSNs) for a ide variety of applications. Among the various research issues ithin WSNs, topology control has been considered as one of the most important issue [1]. By controlling the topology of the graph representing the communication links beteen nodes, topology control algorithms reduce energy consumption and radio interference and thus preserve netork lifetime of WSNs. Desirable features of the resultant communication graph include fundamental properties such as a small nodal degree (to minimize interference and to allo efficient bandidth utilization), a small diameter, a short average path length and graph connectivity [2] [3]. Among the topology control algorithms introduced so far, the k-neigh algorithm is concerned ith (1) reducing radio interference by confining the maximum number of physical neighbors and (2) generating a connected netork ith high probability [2]. Hoever, the k-neigh algorithm requires more than 10 physical neighbors to guarantee a fully connected netork consisting of a large number of nodes uniformly and randomly distributed in the target area. In the Cone Based Topology Control (CBTC) algorithm, a node searches for its physical neighbor ith the minimum transmission poer in every cone of a certain angle θ centered at [3]. Wattenhofer demonstrates that θ 2π/3 and at most 6 logical neighbors are needed for a connected netork. But, the CBTC needs directional information to find a physical neighbor, hich is required to estimate a direction of another node ithout its position information. This directional information can add more complexity to the topology control algorithm. With a groing interest in topology control, the importance of the underlying netork topology has been rising. In [4], Neman emphasizes the importance of understanding the netork structure that affects the performance of a netork system. In [5], e reported that Borel Cayley Graphs (BCGs) sho attractive topological netork properties as an underlying netork topology. The original BCGs have small diameter, short average path length, and provide fast information distribution over a ide range of netork sizes. Hoever, the original BCGs assume that all nodes are ithin a one-hop communication range. Namely, to adopt the original BCG as an underlying netork topology for ireless netorks, all nodes must be ithin the transmission range of each other, hich can entail a large poer consumption for communication. To relieve this deficiency, e proposed a topology control algorithm for BCG () [6]. generates a netork topology close to a 4-regular original BCG hile assuming that each node has a limited transmission range. The resulting netork topology has better topological netork properties and smaller energy consumption in comparison to other benchmark topology controls. Hoever, the still requires a relatively long transmission range to obtain a fully connected netork. In this paper, e propose an EXTENDED (EX) to reduce required transmission range to generate a fully connected netork hile preserving attractive topological property and poer efficiency of the netork generated by. In the EX, a small number of nodes is randomly injected into the netork generated by. Then these injected nodes are connected to the nodes already in a netork according to a Request Criterion and an Accept Criterion. For performance evaluation, e investigated netork topological properties and poer consumption in information distribution of the formulated netork by EX. For comparison, k-neigh and k-random are used as benchmark. In k-random, a node u randomly selects its logical neighbors among its physical neighbors. The target netork is composed of 1081 nodes uniformly and randomly distributed in an area of m 2. The result shos that the netork topology generated by EX has the best performance among the netork topologies investigated. This paper is organized as follos: In Section II, e revie BCGs, BCG random expansion algorithm and algorithm. In Section III, e present the proposed EX and in Section IV e go over the simulation result. In Section V, e provide a conclusion and future ork.

2 II. RELATED WORK In this section, e provide a brief summary of the definition and connection rule of BCGs, the BCG random expansion algorithm and the algorithm. A. Borel Cayley Graphs The BCGs are vertex-transitive, symmetric and regular pseudo random graphs derived from the Borel subgroup. The Borel subgroup is a set of triangular and nonsingular 2 2 matrices formulated as ( x y 0 1 ), here x = at (mod-p), a Z p \{0,1}, hile y Z p and t Z k. k is defined as the smallest positive integer satisfying a k (mod-p) = 1. Based on the Borel subgroup, a BCG is defined as: Definition 1 (Borel Cayley Graph [7]). Assume that V and G V \{I} are a Borel subgroup and a generator set, respectively. C is a BCG consisting of vertices represented by 2 2 matrices V and directed edges satisfying u = v g, here u v V,g G and is a modulo-p multiplication chosen as the group operation. The generator set G excludes the identity element I to avoid self-loops. As noted from the definition, a vertex in BCG is represented by a 2 2 matrix. This matrix element needs to be mapped to a unique non-negative integer number for easy computation during netork modeling (e.g. node ID assignment and routing protocol). [5] provides a function that maps 2 2 matrix element of Cayley graphs to a unique integer value by grouping the matrix elements intok classes. From the mapping function, non-negative integer number ranging from 0 to n 1 are generated here n is the number of nodes of BCG. In terms of the connection rule in BCG, arbitrary to nodes u,v represented by 2 2 matrices are logically and bidirectionally connected from u to v by multiplication of one end node v and a certain poer of generator g chosen from a generator set G. This relationship can be expressed as u = v g... g. Since the generator set G is closed under inversion (g 1 G), the connection in opposite direction is obtained by v = u g 1...g 1. Effectively, a symmetric (undirected) connection is established in BCG. Furthermore, selecting a generator g gives each node of BCG to edges (in g,g 1 directions). Therefore, choice of to generators g 1,g 2 and their inverses g1 1,g 1 2 generates a 4-regular undirected BCG. B. BCG random expansion algorithm From the definition of BCG, the netork size of the BCG is determined by parameters p and k (N = p k). This resulting size inflexibility is one of the most critical issues of BCGs. In [8], the BCG random expansion algorithm is introduced to relax this size inflexibility of BCGs ithout performance degradation. The BCG random expansion algorithm expands the original BCG by injecting a node beteen to distinct edges that are randomly selected from the original BCG to maintain the 4-regular property. If there exists any common node on the selected edges, the ne edges become multiple edges hich should be avoided. Once the to edges are selected, they are unired and then the end nodes of the selected to edges are reired to the nely added node. This reiring preserves the 4-regular property of the BCGs. In [8], the topological properties and information distribution performance of the BCG random expansion algorithm are discussed further. C. BCG topology control algorithm aims at generating a netork topology close to a 4-regular BCG ith a more realistic constraint on the transmission range for each node. From the definition of BCGs, a 4-regular undirected BCG is created by defining to generators and their inverses (i.e. generators g 1, g 2, g1 1 and ). Hoever, BCG has an assumption that all nodes are in a one-hop communication range. In [6], to overcome this limitation, e presented algorithm that considers a limited transmission range of netork to generate a netork topology resembling a 4-regular BCG. The assumes all nodes in a netork are aare of their IDs and the IDs of their logical neighbors. After deployment, during Phase I of, a node searches for its logical neighbors among its physical neighbors. If the resulting node from the Phase I has less than 4 neighbors, the node performs Phase II that utilizes the Cut-Through Reiring (CTR) algorithm [5]. Using the CTR algorithm, the node finds the next available neighbor in the same direction in hich the node didn t find a neighbor during the Phase I. As a result, the algorithm produces a netork topology resembling the 4-regular original BCG. g2 1 III. EXTENDED Even though generates netork topologies close to the original BCG topology, it places a stringent constraint on the node a radio range to guarantee the resulting netork topology forms a fully connected graph. In [5], ith 1081 nodes uniformly and randomly distributed in area m 2, it as found that the radio range must be at least 100m to guarantee a fully connected netork. This large transmission range is not practical and causes undesirably large poer consumption in communication. To relieve the radio range constraint of hile maintaining a small constant degree (k = 4) and other attractive properties of BCG, e propose an EXTENDED (EX). A. Protocol The EX is aimed at eliminating isolated nodes and isolated subnetorks hen the radio range is smaller than the threshold that guarantees a full connectivity of netork topology. The isolated nodes and isolated subnetorks are eliminated by injecting a small number of nodes and connecting them to the existing nodes. For example, in [8], Request Criterion and Accept Criterion are provided for the injected node to select logical neighbors among its physical neighbors. In EX, e define Request Criterion and Accept Criterion as: Request Criterion: All physical neighbors receiving HELLO send REQ to an injected node. This is to prevent an isolation of injected nodes hen there are no physical neighbors qualified to send REQ. Accept Criterion: The physical neighbors ith a degree smaller than4have a higher priority to be a logical neighbor of the injected node. Thus, an isolated physical neighbor has the highest priority to become a logical neighbor of injected nodes. To limit a maximal nodal degree to 4, the injected node can select at most 4 logical neighbors. Note that these criteria are applied to the netork topology. The Request Criterion and the Accept Criterion defined by the EX make it possible for the netork topology to become fully connected ith a smaller transmission range than that of the.

3 r (a) Broadcast HELLO (b) Receive REQ (c) Send ACK-I (d) Receive ACK-II Fig. 1: Illustration of 4 steps of EXTENDED. The EX is composed of four steps: (a) identification of physical neighbors (broadcasting HELLO and receiving REQ), (b) selection of logical neighbors, (c) transmission of connection requests (sending ACK-I) and (d) connection confirmation (receiving ACK-II). In the EX, e assume that each injected node is assigned an ID prior to deployment that is consistent ith nodes ID. When more than 2 nodes are injected, they perform the EX in sequence by increasing order of their IDs. Each injected node is allocated a fixed time period to perform the EX. Also note that all nodes together ith the injected nodes have the same transmission range. Thus, if to nodes are logically connected, a bidirectional connection beteen the to nodes is created. Algorithm 1 EX Require: Π = { 1, 2,..., i}, i := injected node Require: r:= transmission range of i Require: V > 0 1: procedure EXTENDED(Π) 2: for each Π do 3: V V := a set of vertex 4: V V := a set of vertex 5: /* collect physical neighbors */ 6: for each v ithin r do 7: V V {v} 8: /* collect vertex ith degree smaller than 4 */ 9: if d v < 4 then 10: V V {v} 11: end if 12: end for 13: /* select neighbors and connect */ 14: if V = 0 then Case. 1 15: BCG Random Expansion Section III-C 16: else if V = 1 V = 2 then Case. 2 17: E(V) E(V) {e(,v)}, v V 18: choose random e(x,y) E(V \V ) 19: E(V) E(V)\{e(x,y)} 20: E(V) E(V) {e(,x),e(,y)} 21: else if V = 3 V = 4 then Case. 3 22: E(V) E(V) {e(,v)}, v V 23: else if V > 4 then Case. 4 24: sort V by degree 25: V first 4 vertices in V 26: E(V) E(V) {e(,v)}, v V 27: end if 28: end for 29: end procedure In the EX algorithm, the transmission range is denoted by r for the rest of this paper and Π denotes an injected node in the netork generated by. V represents the set of physical neighbors of, hile V is the set of nodes ith a degree smaller than 4 out of V. We use d v to represent the degree of node v. In the next section, e discuss the operation of EX. B. EX mode of operation The EX is used to establish a connection beteen the injected nodes and their physical neighbors hile limiting the maximum nodal degree to 4. In Figure 1, the area surrounded by the dashed line corresponds to the transmission range of the injected node,, and the hite nodes represent existing nodes. Once is injected, it advertises its existence by broadcasting HELLO ith its ID and starts the timer, ReqTimer, to manage reception of REQ (see Figure 1(a)). In Figure 1(b), all physical neighbors send REQ ith their IDs, degree values and neighbor lists folloing the Request Criterion of EX. When the ReqTimer expires, starts the procedure to select logical neighbors from the collected REQs based on the the Accept Criterion of the EX. Note that is assumed to receive at least one REQ since e regard the target netork as a very dense ireless netork and thus there is at least one physical neighbor. ill face one of folloing four cases: Case1:All physical neighbors sending REQ have degree 4 ( V = 0). For this case, the injected node performs the BCG random expansion discussed in the next section. Case 2 : If the number of the nodes of degree d < 4 is 1 or 2, randomly selects one edge based on the responding nodes IDs and neighbor lists from the REQ received. Note that the selected edge should not have any nodes in common ith the node(s) of degree d < 4. Then sends ACK-I to the node(s) of degree d < 4 and to the to end nodes of the selected edge. As a result, the selected edge is disconnected, and the to end nodes of the selected edge together ith the selected node(s) (of degree d < 4) are connected to. Case 3: When there are 3 or 4 physical neighbors of degree d < 4, sends ACK-I to each of them. Case 4: When more than four physical neighbors of degree d < 4 exist, selects the four physical neighbors ith the smallest degree. Then, sends ACK-I to the selected four nodes. ACK-I includes the ne degree value and neighbor list for each selected logical neighbor. After the injected node sends ACK-I, Ack-II-Timer is initiated to manage a collection of

4 TABLE I: Simulation parameters v v Parameters Values u y x u y x area dimension 100m 100m number of nodes (N) 1081 number of injected nodes (N in ) 10,20,30,40 radio range (r) 10, 20,..., 150m node distribution uniformly and randomly distributed sample size 100 samples per radio range BCG generating parameters p = 47,c = 23,a = 2 BCG generator set t 1 = 3, t 2 = 7 (a) Randomly select to edges in r and send ACK-I both to end nodes, u,v,x and y, of the edges. (b) After receiving ACK-II from u, v, x and y, and construction of connection is completed. Fig. 2: BCG random expansion in EX ACK-II. The physical neighbors receiving ACK-I update their degree and neighbor list and reply ith ACK-II to for confirmation (see Figure 1(d)). Once Ack-II-Timer expires, the next injected node, if any, performs EX. C. BCG random expansion in EX BCG random expansion [8] is an algorithm that connects a nely injected node to nodes already in the original BCG hile preserving a constant maximum nodal degree. When a node is injected into the original BCG, to distinct edges ith no common end node are randomly selected. Each of the selected edges is unired and then reired to the injected node. In [8], since it is assumed that all nodes are ithin a one-hop from one another, selecting to edges is not restricted by the transmission range. Hoever, for the BCG random expansion in EX, to pairs of end nodes of the selected to edges should be in the transmission range of the injected node. Figure 2 shos an example of the BCG random expansion in EX. When all physical neighbors have degree 4, the injected node randomly selects to edges ithin its transmission range based on the REQs received from its physical neighbors. Figure 2(a) corresponds to the case here edges e(u,v) and e(x,y) are randomly selected, and sends ACK-I to to both end nodes of the selected edge. Then, the to edges are unired and is ired to u,v,x and y in Figure 2(b). To summarize e described ho EX and BCG random expansion in EX ork. When the netork generated by includes isolated nodes or isolated subnetorks, EX can eliminate such isolations by randomly injecting nodes and connecting them to their physical neighbors. In EX, an injected node advertises its existence, finds physical neighbors, requests connections, receives acknoledgements and constructs connections. In the next section, e discuss the performance of EX by evaluating the topological properties and poer consumption in information distribution of the resulting netork topology. IV. PERFORMANCE EVALUATION In this section, e evaluate the netork connectivity, netork topological properties and poer consumption of the netork topology generated by EX. We compare the resulting netork topology from our algorithm to the netork topologies of, k-neigh and k-random. We note that the number of injected nodes required by Ex beyond that of is negligible (less than 4% at most) and had little impacts on our benchmark metrics. A. Setup In the simulation, all netorks under consideration are based on 1081 nodes uniformly and randomly distributed in an area of m 2. For each radio range (10m,20m,...,150m), 100 netork samples ere collected. Table I shos the detail of the parameters. The maximum nodal degree of, EX and k-random is constrained to 4. Note that it is impossible to generate a fully connected netork ith maximum nodal degree 4 in k-neigh. Therefore, e increased the nodal degrees ofk-neigh to10 [2]. Ink-RANDOM, a node randomly selects the logical neighbors among its physical neighbors. Also, note that the injected nodes into EX are uniformly and randomly injected into the target area. B. Netork connectivity By connecting isolated nodes to injected nodes, e reduce the probability for a netork to form a disconnected graph hen r is belo the connected graph threshold. Figure 3(a) shos the ratio of average isolated nodes versus r. The netork topologies of the and the EX ith r = 10m have almost the same ratio of average isolated nodes, but the difference beteen the to ratios becomes larger as r increases. Particularly, for r 60m, the netork topologies of the EX ith N in = 10,20,30 and 40 have no isolated node, hile the generates isolated nodes up to r = 90m. We also considered the ratio of fully connected netorks as a function of r. The ratio for each r is obtained by dividing the number of fully connected netorks by the number of netork samples (100). As demonstrated in the figure, the netork topologies resulting from EX are more likely to produce connected netorks than the netork topologies of. Specifically, ith r = 30m, only 4 out of 100 netork samples generated by are fully connected hile 21 fully connected netorks out of 100 netork samples are generated by EX ith N in = 10. When N in = 40 and r = 30m, 99% of netork samples are fully connected. In particular, the number of fully connected netorks of EX ith N in = 40 quickly gros from 0 to 99 hen r increases from 20m to 30m. It is said that by performing EX e obtained at least 5 times more connected netorks than ith of same radio range. C. Netork Topological Properties In this section, e discuss the netork topological properties of the resulting netork topologies (diameter and average path length). The diameter is the greatest number of hops beteen to arbitrary nodes in the netork. Thus, the shorter

5 The ratio of average isolated nodes Ex (N 10-3 in (a) The ratio of average isolated nodes for 100 sample netorks The ratio of fully connected netork (b) The ratio of fully connected netorks for 100 sample netorks Fig. 3: Netork connectivity Diameter (hop) Average Path Length (hop) (a) The diameter for 100 sample netorks (b) The average path length for 100 sample netorks Fig. 4: Netork topological properties diameter allos the more efficient information distribution. Figure 4(a) shos the average diameter of the netork topologies investigated. The netork topologies of thek-neigh have a constant average diameter of 33 over all r. In case of the netork topologies of k-random, the diameter decreases from 25 to 10 in the range of 10 r 40. Hoever, it increases to 19 ith r 50 again. On the other hand, the diameters of netork topologies of and EX are even smaller than the ones of k-neigh independent of r. The netork topologies generated by and EX have almost the same diameter ranging from 14 to 9 as r increases. We also evaluated the average path length of the netork topologies. The average path length is defined as the average number of hops along the shortest paths for all pair of nodes in a netork. This metric is a good indicator of ho efficient information distribution is over the netork. In Figure 4(b), the netork topologies of k-neigh have an invariant average path length of 13.28, hich is larger than the one for the netork topologies of and EX. The average path length of the netork topologies of and EX is equal to or less than 7 over all r here the connected netork is generated. D. Poer Consumption Poer efficiency is one of the most important factors in ireless netork. To evaluate the poer efficiency of the netork topologies introduced in this paper, e rely on the average consensus protocol [9]. The average consensus protocol can be used to measure the information distribution performance of a netork. In the consensus protocol, each node n has a state value k n hich it exchanges ith its neighbors at every step. When the state values of all nodes in the netork are the same ithin a pre-defined threshold, the netork is said to have reached a consensus. The smaller the number of steps required, the better the information distribution performance of the netork. In this paper, e calculate the poer consumption of the netork to reach a consensus using the radio model in [10]. For each exchange of state value beteen nodes, e only consider the poer consumption for transmitting and receiving the state values, P t and P r, respectively. Using the equation in [6], the total poer to reach a consensus denoted by P total. P t, P r and P total can be computed as folloing: P t = S(δ 1 +δ 2 d l (u,v)), (1) P r = Sτ, (2) P total (λ) = λs (δ 1 +δ 2 d l (u,v)+τ), (3) (u,v) E u v Where d(u,v) is a distance beteen node u and v, τ is the number of steps to reach a consensus and E corresponds to

6 Energy used (J) E-3 1E-4 Fig. 5: Average poer consumption of node to reach the consensus an edge set of a netork under consideration. The constants used in (1), (2) and (3) are listed in Table II. TABLE II: Radio model parameters Parameters Values Description l 2 (free space) path loss exponent δ 1 45n J/bit poer dissipated by transmitter module δ 2 45n J/bit/m 2 poer dissipated by transmit amplifier τ 135n J/bit poer dissipated by receiver module S 320 bits data size of one state value In Figure 5, e displayed the average poer consumption for each node to reach a consensus in the netork topologies under test. It is obtained by dividing P total by the number of nodes in the netork. The average nodal poer consumption in the netork topologies generated by k-neigh has a constant value J independent of r. On the other hand, the netork topology of consumes less poer than one of k-neigh over the hole transmission range. Further, e found that the netork of EX requires a slightly different amount of poer from that the needs. More specifically, ith r = 50m each node in the netork of needs J to reach the consensus and one of EX is J ith N in = 10. Note that as the number of injected nodes increases, the needed poer to reach the consensus is reduced. Among the netork topologies of and EX ith N in = 10,20,30 and 40, the netork topology of EX ith N in = 10 requires the least poer to reach a consensus. Overall, e observed that the proposed EX generates more fully connected netorks ith smaller transmission range than netorks. Moreover, EX did not degrade the netork topological properties such as diameter and average path length. When it comes to the poer consumption in information distribution, e found EX to need almost the same or slightly less amount of poer to reach a consensus than. V. CONCLUSION AND FUTURE WORK In conclusion, the proposed EX is a poerful topology control algorithm to improve a connectivity of the netork topology generated by. In EX, a small number of nodes (e.g., 10, 20,..., 40 out of 1081) is randomly injected and connected to nodes already in the netork. Connections of these injected nodes are established based on a set of the Request Criterion and the Accept Criterion in order to reduce or eliminate isolated nodes or isolated subnetorks that cause a disconnected netork. We simulated the EX algorithm starting from the netork topology generated by ith 1081 nodes uniformly and randomly distributed in the area of m 2. Using the EX, e found the resulting netork topologies are more likely to be fully connected than those of and k-neigh hile preserving the superior netork topological properties (small diameter and average path length) and poer efficiency in communication. For the future ork, e plan to provide an analytical model to identify the exact number of injected nodes needed to guarantee a full connectivity of. Further, e ill expand our algorithm to other topology controls such as k-neigh and CBTC. VI. ACKNOWLEDGEMENT The authors are partially supported by the National Science Foundation under Grant No. CNS and IIP Any opinions, findings and conclusions or recommendations expressed in this article are those of the authors and do not necessarily reflect the vies of the National Science Foundation. REFERENCES [1] P. Santi, Topology control in ireless ad hoc and sensor netorks, ACM Computing Surveys (CSUR), vol. 37, no. 2, pp , [2] D. Blough, M. Leoncini, G. Resta, and P. Santi, The k- neighbors approach to interference bounded and symmetric topology control in ad hoc netorks, vol. 5, no. 9, pp , [3] R. Wattenhofer, L. Li, P. Bahl, and Y. Wang, Distributed topology control for poer efficient operation in multihop ireless ad hoc netorks, in IEEE INFOCOM, vol. 3, 2001, pp [4] M. E. J. Neman, The structure and function of complex netorks, SIAM Revie, vol. 45, p. 167, [5] J. Yu, E. Noel, and K. W. Tang, A graph theoretic approach to ultrafast information distribution: Borel Cayley Graph resizing algorithm, Computer Communications, 2010, (accepted for publication). [6], Degree constrained topology control for very dense ireless sensor netorks, in Proc. of the 2010 IEEE Global Communications Conference (GLOBECOM 10), Dec.6 10, 2010, pp. 1 6, (to appear). [7] K. W. Tang and B. W. Arden, Representations of Borel Cayley graphs, SIAM Journal on Discrete Mathematics, vol. 6, no. 4, pp , [8] J. Yu, D. Kim, E. Noel, and K. W. Tang, BCG Random Expansion: Formulation of scalable topology for ultrafast information dissemination in ad hoc netorks, submitted to the 2010 IFIP/IEEE Wireless Days Conference (WD 10). [9] R. Olfati-Saber and R. M. Murray, Consensus problems in netorks of agents ith sitching topology and time-delays, vol. 49, no. 9, pp , Sep [10] M. Marta and M. Cardei, Using sink mobility to increase ireless sensor netorks lifetime, in Proc. of the 2008 International Symposium on a World of Wireless, Mobile and Multimedia Netorks (WoMom 08). Washington, DC, USA: IEEE Computer Society, 2008, pp