Sensor Network Localization from Local Connectivity : Performance Analysis for the MDS-MAP Algorithm

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1 Sensor Network Localization from Local Connectivity : Performance Analysis for the MDS-MAP Algorithm Sewoong Oh an Anrea Montanari Electrical Engineering an Statistics Department Stanfor University, Stanfor, CA swoh@stanfor.eu, montanari@stanfor.eu Amin Karbasi LTHC an LCAV EPFL, Lausanne, Switzerlan amin.karbasi@epfl.ch Abstract Sensor localization from only connectivity information is a highly challenging problem. To this en, our result for the first time establishes an analytic boun on the performance of the popular MDS-MAP algorithm base on multiimensional scaling. For a network consisting of n sensors positione ranomly on a unit square an a given raio range r = o(1), we show that resulting error is boune, ecreasing at a rate that is inversely proportional to r, when only connectivity information is given. The same boun hols for the range-base moel, when we have an approximate measurements for the istances, an the same algorithm can be applie without any moification. I. INTRODUCTION Localization plays an important role in wireless sensor networks when the positions of the noes are not provie in avance. One way to acquire the positions is by equipping all the sensors with a global positioning system. Clearly, this as consierable cost to the system. As an alternative, many research efforts have focuse on creating algorithms that can erive positions of sensors base on basic local information such as proximity (which noes are within communication range of each other) or local istances (pairwise istances between neighboring sensors). The problem is solvable, meaning that it has a unique set of coorinates satisfying the given local information, only if there are enough constraints. Note that base on local information any solution is unique only up to rigi transformations (rotations, reflections an translations). The simplest of such algorithms assumes that all pairwise istances are known. However, it is well known that in practical scenarios such information is unavailable an far away sensors (the ones outsie the communication range) cannot obtain their pairwise istances. Many algorithms have been propose to resolve this issue by using heuristic approximations to the missing istances where the success of them has mostly been measure experimentally. Of the same interest an complementary in nature, however, are the theoretical guarantees associate with the performance of the existing methos. Such analytical bouns on the performance of localization algorithms can provie answers to practical questions: for example, how large shoul the raio range be in orer to get the error within a threshol? With this motivation in min, our work takes a forwar step in this irection. In particular, our analysis focuses on proviing a boun on the performance of the popular MDS-MAP [1] algorithm when applie to sensor localization from only connectivity information, which is a highly challenging problem. We prove for the first time that using MDS-MAP, we are able to localize sensors up to a boune error in a connecte network where most of pairwise istances are missing an only local connectivity information is given. A. Relate work A great eal of work over the past few years has focuse on the sensor localization problem. A etaile survey of this area is provie in [2]. In the case when all pairwise istances are known, the coorinates can be erive by using a classical metho known as multiimensional scaling (MDS) [3], [4]. The unerlying principle of the MDS is to convert istances into an inner prouct matrix, whose eigenvectors are the unknown coorinates. In practice, MDS tolerates errors gracefully ue to the overetermine nature of the solution. However, when most pairwise istances are missing, the problem of fining the unknown coorinates becomes more challenging. Two types of practical solutions to this problem have been propose. The first group consists of algorithms who try first to estimate the missing entries of the istance matrix an then apply MDS to the reconstructe istance matrix to fin the coorinates of the sensors. MDS-MAP introuce in [1] can be mentione as a well known example of this class where it computes the shortest paths between all pairs of noes in orer to approximate the missing entries of the istance matrix. The algorithms in the secon group mainly consier the sensor localization as a non-convex optimization problem an irectly estimate the coorinates of sensors. A famous example of this type is a relaxation to semiefinite programming (SDP)[5]. The performance of these practical algorithms are invariably measure through simulations an little is known about the theoretical analysis supporting their results. One exception is the work one in [6] where they use matrix completion methos as a mean to reconstruct the istance matrix. The

2 main contribution of their paper is that they are able to provably localize the sensors up to a boune error. However, their analysis is base on a number of strong assumptions. First, they assume that even far away sensors have non-zero probability of etecting their istances. Secon, the algorithm explicitly requires the knowlege of etection probabilities between all pairs. Thir, their theorem only works when the average egree of the network (i.e., the average number of noes etecte by each sensor) grows linearly with the number of sensors in the network. Our work has a similar flavor as [6], namely we provie theoretical guarantee that backs up experimental results. We use shortest paths (the same way it is use in MDS-MAP) as our primary guess for the missing entries in the istance matrix an apply MDS to fin the topology of the network. In contrast to [6], we require weaker assumptions for our result. More specifically, we assume that only neighboring sensors have information about each other an that only connectivity information is known. Furthermore, the knowlege of etection probabilities plays no role in our analysis or the algorithm. An last, in our analysis we assume that the average egree grows logarithmically with the number of sensors not linearly, which results in neeing much less reveale entries in the istance matrix. In particular, the last conition is quite realistic: if the average egree grows any slower then the network is not even connecte. Due to the fact that the shortest paths algorithm works for both rage-free an rangeaware cases, our analysis inclues both an provies the first error bouns on the performance of MDS-MAP. The organization of this paper is as follows. Section II escribes the state of the art algorithm known as MDS-MAP an states the main result on the performance of MDS-MAP. In Section III, we provie etaile proof of the main theorem. II. LOCALIZATION USING MULTIDIMENSIONAL SCALING In this section, we first efine the basic mathematical moel for the sensors an the measurements an state preliminary facts on MDS, a technique to erive sensor configuration from pairwise istances [1]. We will then escribe the algorithm an give a boun on the error between X an X, the estimate returne by the algorithm. A. Moel efinition For the sensor localization problem, we consier the ranom geometric graph moel G(n, r) = (V, E, P ) where V is a set of n vertices that correspon to n sensor noes which are istribute uniformly at ranom within the -imensional hypercube [0, 1], E V V is a set of unirecte eges that connect pairs of sensors which are close to each other, an P : E R + is a non-negative real-value function. We consier the function P as a mapping from a pair of connecte noes (i, j) to the approximate measurement for the istance between i an j, which we call the proximity measurement. Let be the Eucliean norm on R an let r be some positive parameter. Given a set of ranom positions of n sensors X = {x 1,..., x n }, where x i is the position of sensor i in -imensional space, a common moel for the ranom geometric graph is the isc moel where noe i an j are connecte if the Eucliean istance i,j = x i x j is less than r. More formally, (i, j) E i,j r. To each ege (i, j) E, we associate the proximity measurement P i,j between sensors i an j, which is a function of the istance i,j an ranom noise. In an ieal case when our measurements are exact, we have P i,j = i,j. On the other extreme, when we are given only network connectivity information an no istance information, we have P i,j = r for all (i, j) E. The sensor localization algorithms can be classifie into two ifferent categories. For the connectivity-base moel, which is alternatively also known as the range-free moel, only the connectivity information is available. Formally, P i,j = r if (i, j) E. In the range-base moel, which is also known as the rangeaware moel, the istance measurement is available but may be corrupte by noise or have limite accuracy. P i,j = [ i,j + z i,j ] + if (i, j) E, where z i,j moels the measurement noise (in the noiseless case z i,j = 0), possibly a function of the istance i,j, an [a] + max{0, a}. Common examples are the aitive Gaussian noise moel, where the z i,j s are i.i.. Gaussian ranom variables, an the multiplicative noise moel, where P i,j = [(1 + N i,j ) i,j ] +, for inepenent Gaussian ranom variables N i,j s. In the following, let D enote the n n square istance matrix where D ij = 2 i,j an X enote the n position matrix where the ith row correspons to x i, the -imensional position vector of sensor i. Given the graph G(n, r) = (V, E, P ) with associate proximity measurements for each eges in E, the goal of sensor network localization is to fin an embeing of the noes that best fits the measure proximity between all the pairs in E. The notion of embeing, or configuration, will be mae clear in the following sections. The main contribution of this paper is that we provie, for the first time, the performance boun for the MDS- MAP algorithm when only the connectivity information is available as in the connectivity-base moel. However, the algorithm can be reaily applie to the range-base moel without any moification. Further, since we are given extra information about the approximate istance measurements in the range-base moel, the performance is at least as goo as if we only ha the connectivity information. Note that given G(V, E, P ) from the range-base moel we can easily prouce G (V, E, P ) where P i,j = r for all (i, j) E. B. Multiimensional scaling (MDS) MDS is a technique use in fining the configuration of objects in a low imensional space such that the measure pairwise istances are preserve. If all the pairwise istances

3 are measure without error then a naive application of MDS exactly recovers the configuration of sensors [1]. Algorithm : Classical Metric MDS [1] Input: imension, estimate istance matrix M 1: Compute ( 1/2)LM L, where L = I n (1/n)1 n 1 T n ; 2: Compute the best rank- approximation U Σ U T of ( 1/2)LM L; 3: Return MDS (M) U Σ 1/2. There are many types of MDS techniques, but, throughout this paper, whenever we say MDS we refer to the classical metric MDS, which is efine as follows. Let L be an n n symmetric matrix such that L = I n (1/n)1 n 1 T n, where 1 n R n is the all ones vector an I n is the n n ientity matrix. Let MDS (D) enote the n matrix returne by MDS when applie to the square istance matrix D. Then, in formula, given the singular value ecomposition (SVD) of a symmetric an positive efinite matrix ( 1/2)LDL as ( 1/2)LDL = UΣU T, MDS (D) U Σ 1/2, where U enotes the n left singular matrix corresponing to the largest singular values an Σ enotes the iagonal matrix with largest singular values in the iagonal. This is also known as the MDSLOCALIZE algorithm in [6]. Note that since the columns of U are orthogonal to 1 n by construction, it follow that L MDS (D) = MDS (D). It can be easily shown that when MDS is applie to the correct square istance matrix without noise, the configuration of sensors are exactly recovere [6]. This follows from the following equality (1/2)LDL = LXX T L. (1) Note that we only get the configuration an not the absolute positions, in the sense that MDS (D) is one version of infinitely many solutions that matches the istance measurements D. Intuitively, it is clear that the pairwise istances are invariant to a rigi transformation (a combination of rotation, reflection an translation) of the positions X, an therefore there are multiple incients of X that result in the same D. For future use, we introuce a formal efinition of rigi transformation an relate terms. Denote by O() the orthogonal group of matrices. A set of sensor positions Y R n is a rigi transform of X, if there exists a -imensional shift vector s an an orthogonal matrix Q O() such that Y = XQ + 1 n s T. Y shoul be interprete as a result of first rotating (an/or reflecting) sensors in position X by Q an then aing a shift by s. Similarly, when we say two position matrices X an Y are equal up to a rigi transformation, we mean that there exists a rotation Q an a shift s such that Y = XQ + 1 n s T. Also, we say a function f(x) is invariant uner rigi transformation if an only if for all X an Y that are equal up to a rigi transformation we have f(x) = f(y ). Uner these efinitions, it is clear that D is invariant uner rigi transformation, since for all (i, j), D ij = x i x j 2 = (x i Q + s T ) (x j Q + s T ) 2, for any Q O() an s R. Note that when solving a localization problem, there are two possible outputs : a relative map an an absolute map. The task of fining a relative map is to fin a configuration of sensors that have the same neighbor relationships as the unerlying graph G. In the following we use the terms configuration, embeing, an relative map interchangeably. The task of fining an absolute map is to etermine the absolute geographic coorinates of all sensors. In this paper, we are intereste in presenting an algorithm MDS-MAP, that fins a configuration that best fits the proximity measurements, an proviing an analytical boun on the error between the estimate configuration an the correct configuration. While MDS works perfectly when D is available, in practice not all proximity measurements are available because of the limite raio range r. The next section escribes the MDS- MAP algorithm introuce in [1], where we apply the shortest paths algorithm on the graph G = (V, E, P ) to get a goo estimate of the unknown proximity measurements. C. Algorithm Base on MDS, MDS-MAP consists of two steps : Algorithm : MDS-MAP [1] Input: imension, graph G = (V, E, P ) 1: Compute the shortest paths, an let D be the square shortest paths matrix; 2: Apply MDS to D, an let X be the output. The original MDS-MAP algorithm introuce in [1] has aitional post-processing step to fit the given configuration to an absolute map using a small number (typically + 1) of special noes with known positions, which are calle anchors. However, the focus of this paper is to give a boun on the error between the relative map foun by the algorithm an the correct configuration. Hence, the last step, which oes not improve the performance, is omitte here. Shortest paths. The shortest path between noe i an j in graph G = (V, E, P ) is efine as a path between two noes such that the sum of the proximity measures of its constituent eges is minimize. Let ˆi,j be the compute shortest path between noe i an j. Then, the square shortest paths matrix D R n n is efine as D ij = ˆ 2 i,j for i j, an 0 for i = j. Note that D is well efine only if the graph G is connecte. If G is not connecte, there are multiple configurations resulting in the same observe proximity measures an global localization is not possible. In the unit square, assuming sensor positions are rawn uniformly at ranom as efine in the previous section, the graph is connecte, with high probability, if πr 2 > (log n + c n )/n for c n [7]. A similar conition can be erive for generic -imensions as C r > (log n + c n )/n, where C π is a constant that epens on. Hence, we focus in the regime where r = (α log n/n) 1/ for some positive constant α.

4 As will be shown in Lemma III.1, the key observation of the shortest paths step is that the estimation is guarantee to be arbitrarily close to the correct istance for large enough raio range r an large enough n. Moreover, the all-pairs shortest paths problem has an efficient algorithm whose complexity is O(n 2 log n + n E ) [8]. For r = (α log n/n) 1/ with constant α, the graph is sparse with E = O(n log n), whence the complexity is O(n 2 log n). Multiimensional scaling. In step 2, we apply the MDS to D to get a goo estimate of X. As explaine in II-B, we compute X = MDS ( D). The main step in MDS is singular value ecomposition of a ense matrix D, which has complexity of O(n 3 ). D. Main results Our main result establishes that MDS-MAP achieves an arbitrarily small error from a raio range r = (α log n/n) 1/ with a large enough constant α, when we have only the connectivity information as in the case of the connectivity-base moel. The same boun hols immeiately for the range-base moel, when we have an approximate measurements for the istances, an the same algorithm can be applie without any moification. Let X enote an n estimation for X with estimate position for noe i in the ith row. Then, we nee to efine a metric for the istance between the original position matrix X an the estimation X which is invariant uner rigi transformation of X or X. Define L I n (1/n)1 n 1 T n as in the MDS algorithm. L is an n n rank n 1 symmetric matrix which eliminates the contributions of the translation, in the sense that LX = L(X + 1s T ) for all s R. Note that L has the following nice properties : (1) LXX T L is invariant uner rigi transformation. (2) LXX T L = L X X T L implies that X an X are equal up to a rigi transformation. This naturally efines following istance between X an X. 1 (X, X) = (1/n) LXX T L L X X T L F, (2) where F enotes the Frobenius norm. Notice that the factor (1/n) correspons to the usual normalization by the number of entries in the summation, an (1/ ) correspons to the maximum istance between any two noes in the [0, 1] hypercube. Inee this istance is invariant to rigi transformation of X an X. Furthermore, 1 (X, X) = 0 implies that X an X are equal up to a rigi transformation. Define ( ) log n r 0. (3) 10 n Theorem II.1. Assume n sensors istribute uniformly at ranom in the [0, 1] hypercube, for a boune imension = O(1). For a given raio range r, a corresponing graph G(V, E, P ) is efine accoring to the connectivitybase moel. Then, with high probability, the istance between the estimate X given by MDS-MAP an the correct position matrix X is boune by 1 (X, X) r 0 r + O(r), (4) for r > r 0, where 1 ( ) is efine in Eq. (2) an r 0 in Eq. (3). The proof is provie in Section III. As escribe in the previous section, we are intereste in the regime where r = (α log n/n) 1/ for some constant α. Given a small positive constant δ, this implies that MDS-MAP is guarantee to prouce estimate positions that satisfy 1 (X, X) δ with a large enough constant α an large enough n. Perhaps a more interesting an challenging moel to look at is a probabilistic connectivity base moel, where a sensor etects another sensor whithin istance r with a fixe probability p. This moel, which is a natural generalization of the moel consiere in this paper, is a subject for future research, an we believe that a similar boun as in Theorem II.1 can be prove to hol with a larger constant in the efinition of r 0 that epens on the value of p. In the following, whenever we write that a property A hols with high probability (w.h.p.), we mean that P(A) approaches 1 as the number of sensors n goes to infinity. Given a matrix A R m n, the spectral norm of A is enote by A 2, an the Frobenius norm is enote by A F. For a vector a R n, a enotes the Eucliean norm. Finally, we use C to enote generic constants that o not epen on n. III. PROOF OF THEOREM II.1 We start by bouning the istance 1 (X, X), as efine in Eq. (2), in terms of D an D. Since L(XX T X X T )L has rank at most 2, it follows that L(XX T X X T )L F 2 L(XX T X X T )L 2, (5) where we use the fact that, for any rank 2 matrix A 2 with singular values {σ 1,..., σ 2 }, A F = i=1 σ2 i 2 A 2. The spectral norm can be boune in terms of D an D. Let M = (1/2)L DL. Then, L(XX T X X T )L 2 LXX T L M 2 + M X X T 2 (1/2) L( D + D)L 2 + (1/2) L( D + D)L 2 D D 2, (6) where in the first inequality we use the triangular inequality an the fact that X = L X as shown in Section II-B. In the secon inequality we use Eq. (1) an the fact that M X X T 2 M A 2, (7) for any rank- matrix A. Since the rank of (1/2)LDL is, the secon term follows by setting A = (1/2)LDL. The inequality in (6) follows trivially from the observation that L 2 = 1.

5 Eq. (7) can be prove as follows. From the efinition of X = MDS ( D) in Section II-B, we know that X XT is the best rank- approximation to M. Hence, X XT minimizes M A 2 for any rank- matrix A, an this proves Eq. (7). To boun D D 2, we use the following key result on the shortest paths. The main iea is that, for sensors with uniformly ranom positions, shortest paths provie arbitrarily close estimation to the correct istances for large enough raio range r. Define r 0 (β) 24 ((1 + β) log n/n) 1/. (8) For simplicity we enote r 0 (0) by r 0. Lemma III.1 (Boun on the shortest paths). Uner the hypothesis of Theorem II.1, w.h.p., the shortest paths between all pairs of noes in the graph G(V, E, P ) are uniformly boune by [3] Y. Shang, W. Ruml, Y. Zhang, an M. P. J. Fromherz, Localization from connectivity in sensor networks, IEEE Trans. Parallel Distrib. Syst., vol. 15, no. 11, pp , [4] X. Ji an H. Zha, Sensor positioning in wireless a-hoc sensor networks using multiimensional scaling, in Proc. IEEE INFOCOM, March [5] P. Biswas an Y. Ye, Semiefinite programming for a hoc wireless sensor network localization, in IPSN 04: Proceeings of the 3r international symposium on Information processing in sensor networks. New York, NY, USA: ACM, 2004, pp [6] P. Drineas, A. Jave, M. Magon-Ismail, G. Panurangant, R. Virrankoski, an A. Savvies, Distance matrix reconstruction from incomplete istance information for sensor network localization, in Proceeings of Sensor an A-Hoc Communications an Networks Conference (SECON), vol. 2, Sept. 2006, pp [7] P. Gupta an P. Kumar, Critical power for asymptotic connectivity, in Proceeings of the 37th IEEE Conference on Decision an Control, vol. 1, 1998, pp vol.1. [8] D. B. Johnson, Efficient algorithms for shortest paths in sparse networks, J. ACM, vol. 24, no. 1, pp. 1 13, ˆ 2 i,j ( 1 + ( r 0 /r) ) 2 i,j + O(r), for r > r 0, where r 0 is efine as in Eq. (8). The proof of this lemma is provie in a longer version of this paper. Define an error matrix Z = D D. Then by Lemma III.1, Z is element-wise boune by 0 Z ( r 0 /r)d + O(r). Since all the entries are non-negative, we can boun the spectral norm of Z. Let u an v be the left an right singular vectors of Z respectively. Then by Perron-Frobenius theorem, u an v are non-negative. It follow that D D 2 = u T Zv ( r 0 /r)u T Dv + O(rn) ( r 0 /r) D 2 + O(rn). (9) The first inequality follows from the element-wise boun on Z an the non-negativity of u an v. It is simple to show that D 2 = Θ(n). Typically, we are intereste in the region where r = o(1), which implies that the first term in Eq. (9) ominates the error. Now we are left with the task of bouning D 2. using the following lemma, an the proof is provie in a longer version of this paper. Lemma III.2. D 2 ( / 20 ) n + o(n), w.h.p.. Applying Lemma III.2 to Eq. (9) an substituting D D 2 in Eq. (6), we get the esire boun, an this finishes the proof of Theorem II.1. ACKNOWLEDGMENT We thank Rüiger Urbanke for stimulating iscussions on the subject of this paper. REFERENCES [1] Y. Shang, W. Ruml, Y. Zhang, an M. P. J. Fromherz, Localization from mere connectivity, in MobiHoc 03: Proceeings of the 4th ACM international symposium on Mobile a hoc networking & computing. New York, NY, USA: ACM, 2003, pp [2] G. B. J. Hightower, Locations systems for ubiquitous computing, IEEE Computer, vol. 34, pp , 2001.