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Sensor Deployment for Distributed Estimation in Heterogeneous Wireless Sensor Networks SHANYING ZHU, CAILIAN CHEN AND XINPING GUAN Department of Automation, Shanghai Jiao Tong University, and Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai 200240, P.R.China E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] Received: May 16, 2011. Accepted: June 18, 2011.

This paper deals with sensor deployment problem for distributed estimation of unknown signals in heterogeneous wireless sensor networks (HWSNs). We consider a network including two types of sensors with different computational processing abilities: type-I sensors with more powerful ability than type-II sensors. Based on this computational heterogeneity, we present an estimator model for distributed sensor fusion in HWSNs and propose a sensor deployment scheme supporting the proposed distributed estimation. It is discovered that the network topology is closely related to the properties of the estimation algorithm. To satisfy the performance of the estimation algorithm, two sensor deployment algorithms are given. The first is concerned with the connectivity of network topology and the second illustrates a greedy approach to further optimize the network topology and the parameters of the estimators. Simulation results are provided to demonstrate the performance and effectiveness of the proposed estimators and sensor deployment algorithms for distributed estimation in HWSNs. Keywords: Sensor deployment, distributed estimation, heterogeneous sensor network, connectivity

1 INTRODUCTION Wireless sensor networks (WSNs) are massively distributed systems for sensing and processing of spatially dense data, which are composed of large

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number of sensors densely deployed in resource-limited and harsh environments to execute some tasks including industrial process monitoring and civilian applications [1, 7, 30]. The sensor deployment is one of the fundamental problems for many applications in WSNs (see [14] and [31] for a comprehensive survey). A significant amount of studies of sensor deployment have focused on achieving network coverage and connectivity [2,13,17,32], prolonging the network lifetime [29] and guaranteeing fault tolerance of sensor failures[5, 6]. There are also studies on sensor deployment concerned with sensor localization, detection and field estimation. In [9], Jourdan and Roy considered the case of deploying a sensor network that provides range measurements to a mobile agent for localization. The authors used the position error bound (PEB) as a measure of the quality of the sensor configuration and presented an iterative algorithm to place the sensors to minimize PEB. A related problem of sensor deployment for localization is the detection of targets entering guarded areas. Wettergren and Costa [26] studied the sensor deployment for surveillance of mobile targets. An optimization problem of maximizing the probability of successful detection against the targets is developed for placement of sensors. As for the sensor deployment for field estimation, it is usually referred to as how to estimate the quantity of interest at uncovered locations by using observations at locations with sensors [4]. There are two frequently used metrics: entropy and mutual information to measure the quality of the sensor configuration. In [11], Ko et al. studied the problem of selecting a subset of observations to minimize the uncertainty in spatial sampling networks, which is known as the subset selection problem. They showed that this problem is NP-hard. Thus heuristic algorithms are widely used to solve these optimization problems. Different from [11], Krause et al.[12] considered the mutual information between the observed locations and those uncovered. The aim is to maximize this mutual information, which is proved to be NP-complete. On the other hand„ not only the sensor configuration but also estimation algorithms should be carefully designed to fulfill detection and estimation problems over networks. Two prevailing schemes for the estimation are used in the literature, i. e. fusion-center-aided processing [27] and in-network processing [3, 16, 19–24, 28]. In the former scheme, all measurements should be transmitted to the fusion center leads to problems such as sensitivity to congestion around the fusion center and failure of the fusion center itself. Moreover, sensors that are closest to the fusion center deplete their energy budget much faster than other sensors, such that the network life is shortened. Fortunately, the limitations can be overcome by the distributed schemes with which the network itself is responsible for processing the collected data. No fusion center is needed and each sensor only carries out local estimation and adjusts its estimate using its measurement and information from its neighbors.

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DISTRIBUTED ESTIMATION

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In [19], Olfati-Saber and Shamma proposed a distributed filter that drives the sensors to track the input signal, which is one form of the so-called dynamic average consensus [23, 24]. An alternative approach was addressed in [3,22], where the estimation problem was treated as a form of synchronization among coupled dynamical systems. It has been shown that all the sensors could reach a globally optimum maximum likelihood estimate through selfsynchronization in the case of nonlinear coupling [3] and in the presence of propagation delays and asymmetric channels [22]. Optimization approach has also been used to solve distributed estimation problems. Ref. [20, 21] showed that the distributed implementations can guarantee the convergence to their centralized counterparts when communication links between sensors are ideal. Speranzom et al. [25] formulated an optimization problem to minimize the variance of the estimation error for distributed tracking problem. An upper bound of the error variance depending on the number of neighbors was derived. It indicates that sensors with more neighbors would obtain smaller error variance. The main limitation of the estimator models mentioned above is their homogeneity, in other words, all the sensors are identical with same energy, transmit power and computational ability, etc. Recently, it has been recognized that heterogeneous wireless sensor networks (HWSNs) are more practical in real deployments because of their potential ability to increase network lifetime and reliability without significantly increasing the total cost [30]. Based on these considerations, we proposed an estimation method in [33] for HWSNs where two types of sensors coexist. Type-I sensors have high processing ability, while type-II sensors are low-end ones. A new distributed estimator model is proposed to account for the computational heterogeneity. Preliminary results [33] show that the proposed estimator can effectively solve the distributed estimation problem in HWSNs. However, the sensors’ configuration heavily impacts the quality of service of the network as mentioned previously. Consequently, careful consideration should be given to the sensor deployment problem in order to meet the desired performance goals of the distributed estimation in HWSNs with more than one type of sensor. In this paper, we study the sensor deployment problem to ensure the performance of the distributed estimation in HWSNs. The sensor deployment problem in this paper is similar to the relay sensors deployment considered in [2,5,6,17]. Typically, a common objective in such studies is to place a minimum number of relay sensors so that 1) the original or the whole network is k-connected (k ≥ 1) and/or 2) coverage of the monitoring area is guaranteed. However, what is investigated in this paper is to guarantee the distributed estimation performance together with connectivity. Thus both the objective and methodologies in this paper are quite different from the existing results. This paper is an extended and enhanced version of [33]. The contributions are summarized as follows.

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• Based on the distributed estimator model in [33] for HWSNs with two types of sensors of different computational capabilities, we properly construct a Markov chain with an absorbing state (Proposition 2) to discover that the type-II sensors act as relays in the network. It is revealed that the placement of sensors plays an important role in the distributed estimation (Corollary 1). • In order to ensure the performance of the proposed estimation algorithm, we propose a sensor deployment scheme to meet two requirements, i.e. network connectivity and number of neighbors of each type-I sensor (Condition (11)). The scheme is divided into two phases. The first phase (Algorithm 1) deals with the connectivity requirement. In order to reduce the redundant sensors with the resulting network topology in the first phase, Algorithm 2 is proposed to effectively delete some edges and adjust the estimator parameters to guarantee the performance of the estimation algorithm. The remainder of this paper is outlined as follows. Section 2 presents the network model of HWSN, the estimator model allowing computational heterogeneity of two types of sensors and the sensor deployment problem formulation. In Section 3, some preliminaries about the estimation algorithm are given. Moreover, its convergence analysis is addressed under some conditions. These conditions coupled with the connectivity requirement of the network are used in Section 4 to derive a novel estimation-oriented sensor deployment scheme. The effectiveness of the proposed sensor deployment scheme adapting to distributed estimation in HWSN is demonstrated by simulations in Section 5. Finally, we conclude this paper in Section 6.

2 PROBLEM FORMULATION This section presents the network model of HWSN, some basic notations from graph theory, distributed estimator model and the standard estimation-oriented sensor deployment problem statement. 2.1 Model of HWSN The system considered in this paper is composed of a static sensor network in the environment with an unknown ambient signal. The sensors are used to cooperatively trace the behavior of the signal. We consider a HWSN composed of N sensors with two types of sensors: type-I sensors are high-quality sensors and type-II sensors are low-end ones. Since there are no fusion centers, the type-I sensors do most of the data fusion task. All the type-I sensors can sense and observe the unknown signal ζ (t) ∈ with noisy and distorted measurements,

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yi (t) = bi ζ (t) + wi (t),

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(1)

where bi ≥ 0 and wi is independent white Gaussian noise with zero mean and variance σi2 . This linear model (1) is widely used to describe several different types of sensors, such as range bearing sensors and range-only sensors [15]. As for type-II sensors, they do not make observations at all, i. e. they can only receive the local estimates from their neighbors and do some simple computations. Afterward, they then forward the local estimates to their neighbors. We assume that all the sensors have communication range r > 0. Then any two sensors i and j can communicate with each other if and only if the distance dij between them is no longer than r. And all the sensors are equipped with radio transceivers that can transmit and receive signals in all directions. We make a further assumption that the channels are symmetric, which means that if sensor i can receive data from sensor j, then it can also be able to transmit to sensor j with the same channel gain. Moreover, we assume that all the sensors are synchronized so that their estimate updates can be concurrently performed. Based on these assumptions, we can model the communication network as an undirected graph G = (V, E) with the node set V = I ∪ I c = {1, 2, . . . , N} representing the sensors and the edge set E ⊂ V × V referring to the communication links between sensors. I and I c are sets of type-I and type-II sensors, respectively. The set of sensors I˜ ⊂ I that have direct communications with sensors in I c is called the boundary of I. For any two sensors i and j, there is an edge (i, j) ∈ E if and only if dij is no longer than r. All the sensors that are within distance r from sensor i are called its neighbors denoted by Ni = {j ∈ V : (i, j) ∈ E, j = i}, and |Ni | stands for the number of sensors in Ni . For each sensor i, we denote its type-I and type-II neighbors as Ni+ = Ni ∩ I and Ni− = Ni ∩ I c , respectively. We illustrate these notations using the example shown in Figure 2(a). In this example, there are 12 sensors V = {1, 2, . . . , 12} in G, the set of type-I sensors is I = {1, 2, 3, 4, 6, 7, 9}, the set of type-II sensors is I c = {5, 8, 10, 11, 12} and the boundary I˜ = I. Take sensor 1 as an example, it has |N1 | = 4 neighbors N1 = {3, 8, 9, 10}, of which the type-I neighbors are N1+ = {3, 9} and type-II neighbors are N1− = {8, 10}. 2.2 Distributed estimation in HWSN In order to perform distributed estimation for all sensors in noisy environments, one important aspect is to design coordination strategies. In this paper, we adopt the following estimator on each sensor to account for computational heterogeneity of type-I and type-II senors [33]. For type-I sensors, x˙ i (t) = αi (yi (t) − xi (t)) + βi xi (t) +

σ aij [xj (t) − xi (t)], i ∈ I, ci j∈N i

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(2)

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and for type-II sensors xi (t) =

γij xj (t), i ∈ I c ,

(3)

j∈Ni

where xi ∈ is the estimate of the unknown signal ζ (t) generated by sensor i, αi > 0 and βi > 0 are parameters governing the update rate of information during the estimation process, σ > 0 is the estimator gain, ci > 0 quantifies the confidence of sensor i’s own estimate, aij represents the amplitude of η

the signal received by sensor i, aij = pT |hij |2 /dij , i, j ∈ V, where pT is the transmit power of each sensor, hij = hji is the fading coefficient of the symmetric channel, η is the path loss exponent (typically, 2 ≤ η ≤ 4) and γij are nonnegative weights satisfying 0 < γij ≤ 1 and Nj=1 γij = 1, where we define γij = 0, ∀j ∈ Ni , for convenience. Note that the estimators (2) and (3) are different from the existing models [3, 16, 19, 22–24]. The fact that the estimator must take both filtering and consensus into consideration for all type-I sensors motivates the new estimator models (2) and (3). In fact, the first two terms of (2) are the filtering terms mainly accounting for denoising. It comes from the classic Kalman filter, which is widely used in the context of signal processing, navigation, target tracking and so on [10]. The third term is the consensus term steering the sensors to achieve agreement on the estimates of the unknown signal ζ (t) (see [18] for more details on consensus theory in multi-agent systems). While for type-II sensors, they fuse the local estimate by simply weighting all the incoming data as shown in (3). This distributed estimation in WSNs is schematically shown in Figure 1 . The ability of the HWSN to estimate unknown signals depends on the area coverage and connectivity of the network and the estimation algorithm. Area coverage refers to the union of the area monitored by the sensors. As for

FIGURE 1 A schematic representation of distributed estimation in WSNs.

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the estimation algorithm, convergence is the most important aspect, which has been discovered to be closely related with the network topology. Consequently, in order to guarantee the performance of the estimation, sensor deployment should be carefully considered. Given the above discussions, the estimation-oriented sensor deployment problem can now be stated as follows: Problem (Estimation-orientated sensor deployment). Consider the scenario of estimating the unknown ambient signal ζ (t) by using a HWSN G composed of two types of sensors with different computational capabilities. We seek to place type-I sensors over the monitoring area with coverage requirement and additional type-II sensors to ensure the connectivity of the network of type-I sensors and the performance of the estimation algorithms (2) and (3). In the next two sections, we first explore the properties of the estimator proposed in this section and study its convergence analysis. We can then present a novel sensor deployment algorithm to ensure the performance of the estimator.

3 ANALYSIS OF THE PROPOSED ESTIMATOR MODELS 3.1 Properties of the estimator models (2) and (3) ˘ which is crucial in deriving convergence We first introduce a graph G˜s [I], analysis, following four steps: ˘ of G induced by I˘ I˜ ∪ I c via 1). Induction: Derive the subgraph G[I] ˘ dropping the nodes outside I and the associated edges in graph G. ˜ 2). Deletion: Delete the edges (i, j) ∈ E of which i, j are both in I. − 3). Split: Split each node i ∈ I˜ into |Ni | different nodes possessing the same state, i. e. xij = xi for all j = 1, 2, . . . , |Ni− |, such that each node ij has only one neighbor in I c . Denote the set of all such nodes by I˜ s ˘ In this way, all of the nodes in I˜ s are and the obtained graph by G˜s [I]. pendant ones. ˘ into connected components (CCs), let 4). Partition: Partition graph G˜s [I] us say, G1 = (V1 , E1 ), …, GK = (VK , EK ), where K is the number of CCs. Observe that if graph G is connected, then each Vk , k = 1, 2, . . . , K contains at least one node of I˜ s . We still use the example shown in Figure 2(a) to illustrate the above four steps. Note that here I˜ = I, then I˘ = V. From Step 3), I˜ s can be expressed as I˜ s = {1 , 1

, 4 , 4

, 4

, 7 , 7

}, where x1 = x1

= x1 , x4 = x4

= x4

= x4 and x7 = x7

= x7 . By operating the aforementioned steps, the resultant ˘ with 3 CCs is shown in Figure 2(b). Since graph G is connected, graph G˜s [I] i. e. any two nodes can reached from each other via multi-hop edges, each CC

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FIGURE 2 (a) A sensor network G with 12 sensors consisting of 7 type-I sensors and 5 type-II sensors. (b) ˘ with 3 CCs, each one is enclosed in an ellipse. The corresponding graph G˜s [I]

contains at least one node of I˜ s , namely, {1

} ⊂ CC G1 , {1 , 4

, 7

} ⊂ CC G2 and {4 , 4

, 7 } ⊂ CC G3 . ˘ has the following important properties. The graph G˜s [I] Proposition 1. If graph G is connected, then the state of each node in Vk \I˜ s ˜ can be expressed as a convex combination of the sates of nodes in Vk ∩ I, k = 1, 2, . . . , K, i. e. xi =

γ˜ij xj ,

(4)

j∈Vk ∩I˜

for all i ∈ Vk \I˜ s , where 0 ≤ γ˜ij ≤ 1 and

j∈Vk ∩I˜

γ˜ij = 1.

Proof. See Appendix 1. Proposition 1 reveals that the estimate of each type-II sensor is locally determined by those of type-I sensors in the same CC. This is clear by carefully examining the estimator models (2) and (3). This suggests that we only need to deal with the estimator of type-I sensors to solve the distributed estimation problem, once the weights γij ’s are given. Interestingly, γ˜ij are not at all meaningless. Actually, we can interpret them from the viewpoint of Markov chain. To this end, we introduce an absorbing Markov chain through the following operations. Let xi , i ∈ I˜ s ∪ I c correspond to the states of a homogeneous Markov chain MC. To define the one-step transition probability pij from state xj to state xi , five cases need to be considered, namely, i) i = j, ii) i = j, xi , xj ∈ I˜ s ,

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iii) i = j, xi ∈ I˜ s and xj ∈ I c , iv) i = j, xi ∈ I c and xj ∈ I˜ s , and v) i = j, xi , xj ∈ I c . For the first three cases i)-iii), we set pij = 0. Now consider case iv), in this case, there exist two integers 1 ≤ k1 , k2 ≤ K such that xi ∈ I c ∩ Vk1 and xj ∈ I˜ s ∩ Vk2 , now we can define pij = γij , if k1 = k2 , and 0 otherwise. Likewise, for case v), we set pij = γij , if i and j both belong to some CC Gk , k = 1, 2, . . . , K, and 0 otherwise. In addition, an absorbing state x0 is introduced such that p00 = 1 and k∈I˜ s ∪I c pkj + p0j = 1 for each state xj , j ∈ I˜ s ∪ I c . Therefore, the state space of the Markov chain MC is S = {xi , x0 , i ∈ I˜ s ∪ I c }. Figure 3 depicts the state transition diagram of MC, where SkI = {xi , i ∈ Vk ∩ I˜ s }, SkII = {xi , i ∈ Vk \I˜ s }, k = 1, 2, . . . , K, and PT ←S represents the transition probability matrix from subset S ⊂ S to subset T ⊂ S. Now we obtain the following proposition. − ˜ we have γ˜ij = |Ni | γîjh , where jh is Proposition 2. For all i ∈ I c , j ∈ I, h=1 defined in Step 3) and γîjh is the ever visiting probability from state xjh to state xi of the Markov chain MC on |I˜ s | + |I c | + 1 states {xi , x0 , i ∈ I˜ s ∪ I c }. Proof. See Appendix 2. Observing from the above proposition, type-II sensors can be regarded as relays, who collect the information of their neighbors and then rebroadcast them. The existence of type-II sensors enables type-I sensors communicate with each other by multi-hops even they are far away. Moreover, the cost of HWSNs can be reduced. However, the longer the distance between two type-I sensors is, the fewer information can be successfully received, which is determined by the multiplication of the γ˜ij along the links. This state transition diagram for the network in Figure 2 can be shown in Figure 4.

FIGURE 3 Schematic view of the state transition diagram of Markov chain MC with an absorbing state x0 .

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FIGURE 4 Part of state transition diagram of the Markov chain MC corresponding to graph G shown in Figure 2, here only 6 states x4 , x4

, x7 , x11 , x12 and x0 are given with x0 being an absorbing state.

As far as the previous example in Figure 2 is concerned, we can easily derive from (3) together with (4) that γ˜12,7 =

γ12,7 . 1 − γ12,11 γ11,12

On the other hand, simple calculation based on Figure 4 yields that the transition probability from states x7 to state x12 is γ˜12,7 + γ˜12,7

= γ12,7

∞

j γ11,12 γ12,11 = γ˜12,7 .

j=0

This verifies the assertion of Proposition 2 for γ˜12,7 . For more general cases, similar claims can still be obtained. Now, substituting (4) to (2), we can obtain that, for type-I sensor i ∈ I, x˙ i = αi (yi − xi ) + βi xi +

σ aij (xj − xi ) ci j∈Ni+



 σ aij  γ˜jk xk − xi  + ci j∈Ni−

k∈I˜

= (−αi + βi ) xi +

σ aij (xj − xi ) + αi yi ci j∈Ni+

+

σ aij γ˜jk (xk − xi ). ci j∈Ni−

˜ k∈I\{i}

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Without loss of generality, we assume that the sensors in I are the first M ones in V. Concatenating xi , yi , i ∈ I in x = [x1 , x2 , . . . , xM ]T , y = [y1 , y2 , . . . , yM ]T , respectively, we can rewrite (2) into a compact form as follows

x˙ (t) = − σ Dc−1 Lˆ x(t) + y(t), (5) where = diag {β1 − α1 , . . . , βM − αM }, = diag {α1 , α2 , . . . , αM }, Dc = diag {c1 , c2 , . . . , cM } and Lˆ = [Lˆ ij ] ∈ M×M with entries  aij + aij γ˜jk ,   ˜ j∈Ni+ j∈Ni− ˆLij = k∈I\{i} aik γ˜kj , −aij −   −

j = i, j = i.

k∈Ni

With the definitions above, we have the following lemma. Lemma 1. If graph G is connected, then zero is a simple eigenvalue of Lˆ and the corresponding left eigenvector [ξ1 , ξ2 , . . . , ξM ]T is positive. Proof. Since graph G is connected, from the structure of Lˆ and Proposition 2, we can see that Lˆ is irreducible. Therefore the claim can be obtained based on [22, Corollary 3].

Let = diag{ξ1 , ξ2 , . . . , ξM }. Then it’s obvious that 21 Lˆ + Lˆ T is a symmetric matrix with zero row sums, and thus is positive semidefinite. This property will be utilized to analyze the convergence of the estimation algorithm in the next subsection. 3.2 Convergence analysis on -consensus In this subsection, under some conditions, we shall establish that all the typeI sensors can reach the -consensus asymptotically, which means under the distributed estimators (2) and (3) each type-I sensor can approximately track the time-varying signal ζ (t) with the aid of type-II sensors disregarding an -ball in the long run. First, let us define the error variables ei (t) = xi (t) − ζ (t), i ∈ I.

(6)

Then (6) can be collected into the following dynamics

e˙ (t) = − σ Dc−1 Lˆ e(t) + y(t) + ζ (t)1 − ζ˙ (t)1, where 1 is the column vector with all ones.

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(7)

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Now we are ready to state the stability result about the performance of the distributed estimators (2) and (3). The following theorem summarizes the convergence result for the noise-free case, i. e. wi = 0, for all i ∈ V. Theorem 1. Consider the connected network G composed of two types of sensors. Suppose that there exist ν ≥ 0, µ ≥ 0 such that the signal ζ (t) satisfies |ζ (t)| ≤ µ, |ζ˙ (t)| ≤ ν and

σ

− Dc +

Lˆ + Lˆ T > 0, (8) 2 then e(t) asymptotically converges to an -ball with radius = λmin ( )

η¯ , η

as t → ∞, where η¯ = max{ξi ci , i = 1, 2, . . . , M}, η = min{ξi ci , i = 1, 2, . . . , M} and  M √ = η¯ ν M + µ (βi + αi (bi − 1))2  . 

(9)

i=1

Proof. Since Dc is positive definite from Lemma 1, it is natural to define the Lyapunov functional candidate as V = 21 eT Dc e. The time derivative of V along the trajectory of the error dynamics (7) is given by 1 T V˙ = e Dc e˙ + e˙ T Dc e 2 σ = eT Dc − ( Lˆ + Lˆ T ) e + eT Dc ζ (t)(b + 1) − ζ˙ (t)1 , 2 (10) where b = [b1 , b2 , . . . , bM ]. In view of the hypothesis (8), Dc − σ2 ( Lˆ + Lˆ T ) is negative definite. Then employing Schwarz inequality yields V˙ ≤ −λmin ( )e2 + e 2 2 = −λmin ( ) e − + , 2λmin ( ) 4λmin ( ) where is defined in (9). Let R = {e : V (e) ≤ R} be the level-set of Lyapunov functional candidate 2 η¯ V with R = 21 2 , then for every e ∈ M \R , it is easy to see that V˙ < 0. λmin ( )

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This means that if e is outside the set R , then there is a attractive force to pull it to R until it lies in R . As a result, for arbitrary t0 ∈ , there exists t1 > t0 such that e(t) ∈ R , ∀t ≥ t1 , i. e. R is a trapping region for error dynamics (7). Consequently, for sufficiently large t ∈ , we can conclude that 1 2 η¯ 1 ηe2 ≤ V (e) ≤ R = . 2 2 λ2min ( ) Therefore, e(t) asymptotically converges to an -ball with the radius = λmin ( )

η¯ . η

This completes the proof. The following corollary follows Theorem 1 to present a necessary condition. It reveals the relation between sensor deployment and distributed estimation in HWSNs. Corollary 1. For type-I sensors, if βi > αi , then it is necessary that the following condition j∈Ni

aij −

aij γ˜ji >

j∈Ni−

ci (β − αi ) σ i

(11)

must hold in order to guarantee condition (8). Proof. The idea of the proof is based on the fact that the diagonal entries of a positive definite matrix must be positive as well. The i-th diagonal entry of is given by σ ξi Lˆ ii + ξi (αi + i − βi ), i = 1, 2, . . . , M. It thus follows from (8) that the necessary condition is σ Lˆ ii + αi + i − βi > 0, i = 1, 2, . . . , M, since ξi is a positive scalar according to Lemma 1. Combined with the structure ˆ it infers that of L,    σ aij + aij γ˜jk  > ci (βi − αi ). 

j∈Ni+

j∈Ni−

˜ k∈I\{i}

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(12)

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Note that γ˜jk = 0, for all j ∈ I c and k ∈ I˜ which is an immediate result from ˘ in the subsection 3.1, then we have the construction of graph G˜s [I]

γ˜jk =

k∈I˜

γ˜jk .

k∈I

In consequence, j∈Ni−

aij

γ˜jk =

aij ,

j∈Ni−

k∈I˜

where use was made of Proposition 1. Thus substitution of the above identity into (12) yields    aij − aij γ˜ji  > ci (βi − αi ). σ 

j∈Ni−

j∈Ni

This completes the proof. ˜ j ∈ I c , then the following relations hold Note that 0 ≤ γ˜ji ≤ 1, ∀i ∈ I, j∈Ni

aij −

aij γ˜ji =

j∈Ni−

j∈Ni+

aij +

aij (1 − γ˜ji )

(13)

j∈Ni−

≥ min aij |Ni+ |. j∈Ni+

(14)

As a result, if |Ni+ | >

ci (β − αi ) σ minj∈N + aij i

(15)

i

is satisfied, then condition (11) must hold. This means that condition (15), and thus condition (11) to some extent, enforces a requirement on the number of type-I neighbors of each type-I sensor for distributed estimation in HWSNs, i. e. the number of type-I neighbors of each type-I sensor should be greater than a threshold related to its own confidence, estimator gain and the amplitude of signals sent from their neighbors. In other words, only the deployment of type-I sensors matters for distributed estimation using the proposed estimators (2) and (3). At first sight, this is doubtful, since the deployment of type-II sensors seems to be irrelevant to the distributed estimation. However, after a second thought, this is reasonable by recalling that the type-I sensors are more powerful than type-II ones. The estimation accuracy is primarily determined

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15

by configuration of type-I sensors. Therefore, more attention should be paid to the deployment of type-I sensors, if the number of type-I sensors is large enough. On the other hand, we can not place arbitrarily many type-I sensors to make condition (15) enforced in view of the overall cost of the network. Thus condition (15) might be much restrictive in some scenarios. Consequently, in the next section, we use condition (11) instead of (15) coupled with the connectivity requirement needed in Theorem 1 to present the sensor deployment algorithms.

4 SENSOR DEPLOYMENT FOR BOTH CONNECTIVITY AND DISTRIBUTED ESTIMATION Theorem 1 establishes the convergence property of the distributed estimation algorithm presented in Section 2. In this section, we aim at placing sensors to satisfy the conditions needed in Theorem 1. In order to guarantee the desired performance of the estimation algorithm, two main requirements of the sensor configuration are: connectivity requirement and the number of type-I neighbors requirement in (11). Intuitively, the second requirement can be easily solved by placing all the type-I sensors together. Actually, this does not work in our scenario, because area coverage is also necessary to monitor the area and estimate the ambient signal ζ (t) in HWSNs. There is a tradeoff between these two factors. To deploy the sensors such that the requirement of condition (11) is satisfied, we can follow two steps: first, deploy the type-I sensors, then deploy the type-II sensors. The reason behind this handling is that the left hand side of (11), i. e. j∈Ni aij − j∈Ni− aij γ˜ji can be rewritten as j∈Ni+ aij + j∈Ni− aij (1− γ˜ji ). Note that the first term depends only on type-I sensors, while the second term on type-II sensors. This observation enables the two-step implementation of sensor deployment. After the deployment of type-I sensors in the first step, check whether condition (11) is satisfied for each type-I sensor with βi > αi . If not, then deploy some additional type-II sensors until it is satisfied. This can always work, since j∈N − aij (1−γ˜ji ) ≥ 0, i ∀i ∈ I. Based on the two-step implementation of sensor deployment of type-I and type-II sensors, in this paper, we focus on the deployment of type-II sensors, given that the type-I sensors have been deployed to meet the requirement of coverage of the monitoring area. The algorithms in [2, 13, 32] can be used here to deploy the type-I sensors in the first step. Algorithm 1 is concerned with the connectivity requirement of the network of the type-I sensors by placing some additional type-II sensors. In Algorithm 1, we first compute a weighted undirected complete graph, and the weight corresponding to each edge (i, j) is the distance dij . Second, we

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Algorithm 1 Type-II sensor deployment to ensure connectivity in HWSN Input: A set of M type-I sensors denoted by I = {1, 2, . . . , M}, their coordinates p1 , p2 , . . . , pM , the communication range r of both type-I and type-II sensors and positive constants αi , βi , i = 1, 2, . . . , M. 1: Construct an undirected complete graph G

= (I, E

) and associate each edge (i, j) ∈ E

with weight dij pi − pj . 2: Define a subgraph S = (I, E ) initialized as S = (I, ∅). 3: for each edge (i, j) ∈ E

in increasing order of weight do 4: if dij ≤ r then 5: Put edge (i, j) into graph S. 6: end if 7: if S is connected then 8: break 9: else 10: if (βi ≤ αi and βj ≤ αj ) or (βi > αi and βj > αj ) then 11: if r < dij ≤ 2r then 12: Place a type-II sensor at some position with distance r from pi and pj on the perpendicular bisector of line segment [pi , pj ]. 13: else if dij > 2r then 14: Place two type-II sensors at positions p = (1 − κ) pi + κpj and p

= κpi + (1 − κ) pj , respectively, where κ = r . dij dij − 2r 15: Place τ r type-II sensors on the line segment

[p , p ] and the k-th one is at 1 − kκ p + kκ p

, k = 1, 2, . . . , τ , where κ = r

. p − p 16: end if 17: else if βi ≤ αi and βj > αj then 18: if dij > r then 19: Place τ + 2 type-II sensors on the line segment [pi , pj ] and the k-th one is at kκpi + (1 − kκ) pj , k = 1, 2, . . . , τ + 2. 20: end if 21: else if βi > αi and βj ≤ αj then 22: if dij > r then 23: Place τ + 2 type-II sensors on the line segment [pi , pj ] and the k-th one is at (1 − kκ) pi + kκpj , k = 1, 2, . . . , τ + 2. 24: end if 25: end if 26: Put edge (i, j) into graph S. 27: end if 28: end for

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place additional type-II sensors on each edge with weight greater than r to enable the multi-hop communication between the two sensors associated with this edge. This process ends until the graph S is connected. During this process, note that for such type-I sensors with βi > αi , the additional typeII sensors are placed with distance r from them. The purpose is to make the value of the second term of left-hand side of (11) as small as possible, since aij is inversely proportional to dij . This will be further discussed in Algorithm 2. Algorithm 2 illustrates a greedy algorithm to adjust the network topology and parameters of the estimator. This algorithm is divided into two phases. The first phase is used to delete the redundant edges and the resulting isolated type-II sensors from graph S generated by Algorithm 1. But put it back if it is necessary to guarantee the connectivity of graph S. Those edges between closer type-I and type-II sensors are discarded first for the same reason as in Algorithm 1. After the edge pruning, if condition (11) is still not satisfied for some type-I sensor i = 1, 2, . . . , M, then we decrease the value of ci in the second phase until it is finally satisfied.

Algorithm 2 Network topology optimization and estimator design for distributed estimation in HWSN Input: Graph G = (I ∪ I , E), where I is the set of placed type-II sensors in Algorithm 1, coordinates of all sensors p1 , p2 , . . . and the estimator parameters σ , αi , βi , ci and γ˜ij , i = 1, 2, . . . , M. 1: Associate each edge (i, j) of graph G with weight dij . 2: for each edge (i, j) ∈ I × I in increasing order of weight do 3: if subgraph (I ∪ I , E\{(i, j)}) is connected then 4: Delete edge (i, j) from graph G. 5: end if 6: end for 7: for each edge (i, j) ∈ I × I in decreasing order of weight do 8: if subgraph (I ∪ I , E\{(i, j)}) is connected then 9: Delete edge (i, j) from graph G. 10: end if 11: end for 12: Discard those pendant type-II sensors from graph G. 13: for each type-I sensor i ∈ I do 14: Compute the set of neighbors Ni of sensor i and weight aij , ∀j ∈ Ni . 15: if βi > αi and (11) is violated then 16: Decrease the value of ci such that (11) is satisfied. 17: end if 18: end for

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5 SIMULATIONS In this section, we consider the estimation problem using a HWSN in 2-D plane. The performance of the proposed estimator and sensor deployment algorithms is evaluated by several simulations. In the following simulations, we first randomly place 30 type-I sensors in the monitoring area [ − 40, 40] × [ − 40, 40] as shown in Figure 5 such that the distance between any two type-I sensors is at least 2.5 m. This is setup to meet the requirement of coverage. Each type-I sensor can communicate with others within a disc of radius r = 10 m and can only measure a noise version of the ambient signal yi (t) = ζ (t) + wi (t), where ζ (t) = 3 sin (t) + cos (2t) and wi (t) is zero-mean white noise with intensity σi2 = 0. 125, for all i = 1, 2, . . . , 30. Based on the local measurement and estimates transmitted from its neighbors, each sensor implements a local estimator (2) or (3) to estimate the state of the unknown signal ζ . We apply Algorithm 1 and Algorithm 2 described in Section 4 to determine the positions of the additional type-II sensors such that the whole network is connected and condition (11) is satisfied. They are necessary to ensure the performance of the estimators (2) and (3) at both type-I and type-II sensors. We first use Algorithm 1 to place the type-II sensors until the network is connected. 22 type-II sensors are placed in this phase. Then we employ Algorithm 2 to delete the redundant edges between type-II sensors and the corresponding type-II sensors and adjust the parameters of the estimator to enforce condition (11). Here 1 type-II sensor and 4 edges are discarded with no influence on the network connectivity. After this adjustment, we find that the

FIGURE 5 Topology of a random network with 30 type-I sensors with the distances between them being at least 2.5 m.

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FIGURE 6 The resulting HWSN with 21 type-II sensors placed in the network shown in Figure 5 using Algorithm 1 and Algorithm 2.

sufficient condition (8) is satisfied as well. Figure 6 depicts the corresponding HWSN, where 21 type-II sensors are placed. Although edges pruning is performed in Algorithm 2, it can be easily seen that there is still edge redundance in Figure 6. This is normal in greedy approaches, since greedy approaches (although efficient in most cases) may be arbitrarily away from optimal solutions. In the following simulations, we adopt a simple model for normalized path loss without path fading aij = 1/(1 + dij2 ), where dij is the distance between sensors i and j. The weight γij is set to be γij =

aij

j∈Ni aij

, ∀i ∈ I c , j ∈ V.

Other parameters are selected as follows: σ = 20, αi = 0. 5, βi = 1, ci = 0. 1 if i is odd, and αi = 40, βi = 0. 01, ci = 0. 05 if i is even. The initial estimates of all type-I sensors are randomly chosen from the interval [ − 3, 3]. All the results are presented by averaging 10 independent runs. Figure 7 shows the measurements and estimates of ambient signal ζ (t). We can easily see that all the type-I sensors are able to track the signal ζ (t) with an acceptable error bound determined by Theorem 1. We use two metrics to measure the performance of the estimator. The first one is the mean-square error over all type-I sensors 1 |xi (t) − ζ (t)|2 , 30 30

MSE =

i=1

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FIGURE 7 (a) Measurements and (b) estimates of the ambient signal ζ (t) of all 30 type-I sensors, where the bold red curve is the ambient signal ζ (t).

which measures the estimation accuracy of the algorithm. And the second one characterizing the differences of the estimates among the type-I sensors is the mean disagreement of the estimates defined by 1 |xi (t) − x¯ (t)|2 , 30 30

MSD =

i=1

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FIGURE 8 Mean-square error (MSE) and disagreement of the estimates (MSD) among all type-I sensors.

1 30 xi . The simulation results of MSE and MSD are shown in where x¯ = 30 i=1 Figure 8. It is observed that both MSD and MSE are enveloped by some curve, in other words, the estimator works well, even if the environment is quiet noisy. Furthermore, small MSD means the proposed estimator (2) possesses the ability of enabling all the type-I sensors to cooperatively estimate the signal ζ (t) and reach agreement on the final estimate with a small error. Therefore, the proposed estimator performs in a satisfactory manner. Finally, we compare the results with three different cases: |I| = 10, 20 and 30 type-I sensors, respectively, regarding the average of the root-mean-square error over all type-I sensors |I| T 1 1 RMSE = xi (k) − ζ (k)2 , T − [T /2] + 1 |I| i=1

k=[T /2]

where T is the total number of iterations and [T ] is the rounding integer. Table 1 presents the comparison results of RMSE in these three cases, where for each case, we randomly place |I| type-I sensors separated by at least 2.5 m. The parameters stay the same for all three cases as adopted previously.

RMSE

|I| = 10

|I| = 20

|I| = 30

0.3525

0.2391

0.3222

TABLE 1 RMSE of the abient signal ζ (t) for the proposed estimators (2) and (3) under the sensor deployment strategies Algorithm 1 and Algorithm 2.

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The differences of RMSE are small which indicate that the estimation problem can be solved in all the three cases. And the RMSE is the smallest when the number of type-I sensors is 20. This is because only the type-I sensors make observations about the ambient signal ζ . If the number of type-I sensors is small, few information could be sensed by the sensors. While if the number of type-I sensors is large, more noise might be injected to the data about the signal ζ . Therefore, in both cases, lower RMSE is certainly not expected. This further indicates that the network topology itself matters more than the number of type-I sensors for the distributed estimation in HWSNs. Extensive simulations and the optimal number of type-I sensors, especially the theoretical analysis, for the distributed estimation based on the estimation algorithms (2) and (3) is one of our future works.

6 CONCLUSIONS The sensor deployment problem has been addressed for distributed estimation of unknown signals in HWSNs. Two classes of sensors with different processing abilities are taken into consideration: high-quality type-I sensors and low-end type-II sensors. With the proposed estimator model accounting for the computational heterogeneity, detailed properties of the estimator are analyzed. In order to ensure the performance of the distributed estimation, we present two sensor deployment algorithms to guarantee the connectivity and estimation performance. Simulation results have validated the theoretical analysis results.

ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers and editors for their valuable comments. The work was partially supported by National Basic Research Program of China under the grant no. 2010CB731803, NSF of China under 60804030, 60934003 and 60974123, 61174127 and NSF of Hebei Province under F2011203226, Science and Technology Commission of Shanghai Municipality (STCSM), China under 09PJ1406100, 10XD1402100, 10dz1500402, “Chenguang” Program under 09CG06 and Shanghai Jiao Tong University Innovation Fund for Postgraduates.

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[3] S. Barbarossa and G. Scutari. (2007). Decentralized maximum-likelihood estimation for sensor networks composed of nonlinearly coupled dynamical systems. IEEE Trans. Signal Process., 55(7):3456–3470. [4] F. Bian, D. Kempe, and R. Govindan. (April 2006). Utility-based sensor selection. In Proc. 5th Int’l Conf. Inf. Process. Sensor Networks (IPSN), pages 11–18, Nashville, USA. [5] J. L. Bredin, E. D. Demaine, M. T. Hajiaghayi, and D. Rus. (2010). Deploying sensor networks with guaranteed fault tolerance. IEEE/ACM Trans. Netw., 18(1):216–228. [6] X. Han, X. Cao, E. L. Lloyd, and C. C. Shen. (2010). Fault-tolerant relay node placement in heterogeneous wireless sensor networks. IEEE Trans. Mobile Comput., 9(5):643–656. [7] W. B. Heinzelman, A. P. Chandrakasan, and H. Balakrishnan. (2002). An applicationspecific protocol architecture for wireless microsensor networks. IEEE Trans. Wireless Commun., 1(4):660–670. [8] R. A. Horn and C. R. Johnson. (1985). Matrix analysis. Cambridge University Press, Cambridge. [9] D. B. Jourdan and N. Roy. (2008). Optimal sensor placement for agent localization. ACM Trans. Sensor Networks, 4(3):13:1–13:40. [10] S. M. Kay. (1993). Fundamentals of Statistical Processing: Estimation Theory. Prentice Hall, Upper Saddle River, NJ. [11] C. W. Ko, J. Lee, and M. Queyranne. (1995). An exact algorithm for maximum entropy sampling. Operations Res., 43(4):684–691. [12] A. Krause, A. Singh, and C. Guestrin. (2008). Near-optimal sensor placements in Gaussian processes: Theory, effcient algorithms and empirical studies. J. Mach. Learn. Res., 9:235– 284. [13] X. Li, H. Frey, N. Santoro, and I. Stojmenovic, (2010). Strictly localized sensor selfdeployment for optimal focused coverage. IEEE Trans. Mobile Comput., 10(11):1520– 1533. [14] X. Li, A. Nayak, D. Simplot-Ryl, and I. Stojmenovic. (2010). Sensor placement in sensor and actuator networks. In A. Nayak and I. Stojmenovic, editors, Wireless Sensor and Actuator Networks: Algorithms and Protocols for Scalable Coordination and Data Communication, chapter 10, pages 263–294. John Wiley & Sons, Inc, Hoboken, New Jersey. [15] X. R. Li and V. P. Jilkov. (July-August 2001). A survey of maneuvering target tracking-Part III: Measurement models. In Proc. SPIE Conf. Signal Data Process. Small Targets, pages 423–446, San Diego, USA. [16] K. M. Lynch, I. B. Schwartz, P. Yang, and R. A. Freeman. (2008). Decentralized environmental modeling by mobile sensor networks. IEEE Trans. Robot., 24(3):710–724. [17] S. Misra, S. D. Hong, G. Xue, and J. Tang. (April 2008). Constrained relay node placement in wireless sensor networks to meet connectivity and survivability requirements. In Proc. IEEE Conf. Computer Communications (INFOCOM), pages 879–887, Phoenix, USA. [18] R. Olfati-Saber, J. A. Fax, and R. M. Murray. (2007). Consensus and cooperation in networked multi-agent systems. Proc. IEEE, 95(1):215–233. [19] R. Olfati-Saber and J. S. Shamma. (December 2005). Consensus filters for sensor networks and distributed sensor fusion. In Proc. 44th IEEE Conf. Decision Control, Eur. Control Conf., pages 6698–6703, Seville, Spain. [20] I. D. Schizas, G. B. Giannakis, S. I. Roumeliotis, and A. Ribeiro. (2008). Consensus in Ad Hoc WSNs with noisy links-Part II: Distributed estimation and smoothing of random signals. IEEE Trans. Signal Process., 56(4):1650–1666. [21] I. D. Schizas, A. Ribeiro, and G. B. Giannakis. (2008). Consensus in Ad Hoc WSNs with noisy links-Part I: Distributed estimation of deterministic signals. IEEE Trans. Signal Process., 56(1):342–356.

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APPENDIX 1 Proof of Proposition 1 Proof. For CC Gk , assume that the set of nodes are numbered as Vk = {k1 , . . . , km }. From (3), we have m

δki kj γki kj (xki − xkj ) = 0,

(16)

j=1

where M(A) = [δij ] is the indicator matrix of A[8]. Define the weighted Laplacian matrix Lk = [˜lki kj ] of CC Gk as follows

j = i, k k γk k , ˜lk k = −δ mi j i j i j δ γ , j = i. j =i ki kj ki kj

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(17)


25

There always exists a permutation matrix P such that Vk ∩I˜ s

P Lk P = T

Vk ∩I˜ s Vk \I˜ s

Vk \I˜ s

Lk2 , Lk4

Lk1 Lk3

which can be done via renumbering the nodes. Hence, (16) becomes Xˆ Lk3 Lk4 = 0, X

(18)

(19)

where X˜ and X are the stacked columnwise vectors of xi , i ∈ Vk ∩ I˜ s and xj , j ∈ Vk \I˜ s , respectively. Since Gk is connected and ki ∈ I˜ s are pendant nodes from step 3), we can see that induced graph Gk [Vk \I˜ s ] is still connected which is equivalent to say that Lk4 is irreducible [8]. In addition, Lk4 is diagonally dominant, i. e. ˜lk k , ki , kj ∈ Vk \I˜ s , |˜lki ki | ≥ i j j =i

and at least one inequality above holds strictly by virtue of (17) and step 3), since graph G is connected. Then it follows from Taussky’s theorem [8] that Lk4 is invertible. Thus we derive from (19) that −1 ˜ X = −Lk4 Lk3 X. −1 −1 Now, it remains to show that −Lk4 Lk3 is stochastic. Since Lk4 = (I − −1 Ak4 ) , where Ak4 is the adjacent matrix associated with L , Neumann series k4 j −1 −1 = ∞ expansion yields that Lk4 j=0 Ak4 . We thus known that Lk4 is nonnegative because Ak4 is a nonnegative matrix. Note that −Lk3 is also nonnegative. This −1 Lk3 is nonnegative as well. Furthermore, it follows from the assures that −Lk4 −1 Lk3 1 = 1. definition of Lk that Lk 1 = 0 which reveals −Lk4 Based on the discussions above, for each i ∈ Vk \I˜ s , xi can be expressed as xi = γîj xj , (20) j∈Vk ∩I˜ s

where 0 ≤ γîj ≤ 1 and j∈Vk ∩I˜ s γîj = 1. Remember that for all nodes ij , j = 1, 2, . . . , |Ni− | that belong to I˜ s , by referring to Setp 3), we have xij = xi . Therefore, we can rewrite (20) as xi = γ˜ij xj , j∈Vk ∩I˜

where 0 ≤ γ˜ij ≤ 1 and

j∈Vk ∩I˜

γ˜ij = 1. This completes the proof.

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APPENDIX 2 Proof of Proposition 2 ˘ has K Proof. For the homogeneous Markov chain MC, noticing that G˜s [I] CCs, then the transition probability matrix can be written as 

P1 0   P =  ...  0 q1

0 P2 .. .

... ... .. .

0 0 .. .

0 q2

. . . PK . . . qK

 0 0  ..  , .  0 1

where Pk is the transition probability matrix from subset {xi , i ∈ Vk } to itself, ∀k = 1, 2, . . . , K. It follows from (16) and (18) that Pk , ∀k = 1, 2, . . . , K can be expressed as follows Pk =

0 −Lk3

0 , Ak4

where −Lk3 and Ak4 are the transition probability matrices from subsets SkI II k and SkII to S k , respectively. Moreover, qk = [q0i ] is a column vector satisfying k q0i = 1 − j∈I˜ s ∪I c pji , ∀k = 1, 2, . . . , K. By proceeding the mathematical induction, we can obtain that     P =  

P1n 0 .. .

0 P2n .. .

0 n−1

0 n−1

n

j j=0 q1 P1

j j=0 q2 P2

... ... .. .

0 0 .. .

... Pn n−1 K j ... j=0 qK PK

 0 0  ..  , .  0 1

where accordingly Pkn =

0 −An−1 k4 Lk3

0 , ∀k = 1, 2, . . . , K. Ank4

Consequently, the n-step transition probability matrix from SkI to SkII is the sub-matrix −An−1 1, 2, . . . , K. Then the ever visiting probability k4 Lk3 , ∀k = j matrix from SkI to SkII is − ∞ j=0 Ak4 Lk3 , ∀k = 1, 2, . . . , K. From the proof −1 Lk3 . Note that xij = xi , of Proposition 1, this can further be reduced to −Lk4 − ∀j = 1, . . . , |Ni |. Therefore, we can finish the proof following the same arguments as in the proof of Proposition 1.

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