A Topology Optimization Method for Reducing Communication Overhead in the Kalman Consensus Filter

Featured Application: This work can be used in network topology optimization for reducing com ‐ munication overhead in the Kalman consensus filter. Abstract: Distributed estimation and tracking of interested objects over wireless sensor networks (WSNs) is a hot research topic. Since network topology possesses distinctive structural parameters and plays an important role for the performance of distributed estimation, we first formulate the communication overhead reduction problem in distributed estimation algorithms as the network topology optimization in this paper. The effect of structural parameters on the algebraic connectivity of a network is overviewed. Moreover, aiming to reduce the communication overhead in Kalman consensus filter (KCF) ‐ based distributed estimation algorithm, we propose a network topology op ‐ timization method by properly deleting and adding communication links according to nodes’ local structural parameters information, in which the constraint on the communication range of two nodes is incorporated. Simulation results show that the proposed network topology optimization method can effectively improve the convergence rate of KCF algorithm and achieve a good trade ‐ off between the estimate error and communication overhead.


Introduction
Wireless sensor networks (WSNs), which are composed of a large number of stationary and/or mobile sensor nodes in a self-organizing and multi-hop manner, perceive, collect, process, and transmit collaboratively the information of interested objects in the geographical area covered by the network [1]. The performance of WSNs is directly affected by the network topology. For example, the planar network structure where all nodes are equivalent has better robustness, and the hierarchical network structure extended by the planar network structure has better expansibility [2]. Hence, in order to improve the performance, the problem of network topology optimization has become a hot research topic.
Distributed estimation over a network, one of the most fundamental information processing problems in WSNs, aims to estimate and track the state of interested objects in the noisy environment through the cooperation between sensor nodes [3,4]. Kalman filtering has been an effective algorithm for tracking dynamic processes for over four decades. In [5], the author introduced the Kalman consensus filter (KCF) algorithm, which relies on average consensus, and it is pointed out that this algorithm is applicable to sensor networks with variable topology. The KCF algorithm also has a lot of applications in other To the best of our knowledge, there are few works on reducing the communication overhead of the KCF algorithm from the perspective of network topology optimization. Moreover, due to the unstableness of WSNs, it is more desirable to adjust communication links for the performance improvement. In addition, the communication links are adjusted based on local structural parameters of the network, in terms of the degree of network nodes, average path length, and clustering coefficient. Compared with some existing heuristic search algorithms, the proposed method in this paper outperforms them in terms of the computational complexity and communication overhead.
The remainder of the paper is organized as follows. Section 2 introduces the system model and formulates the problem of network topology optimization. In Section 3, a network topology optimization method is proposed. Simulation results are given in Section 4. Finally, we conclude the paper in Section 5.

System Model and Problem Formulation
In this section, the system model is introduced, and the problem to be resolved is formulated.

Network Model
This work considers a WSN with topology Adjacency matrix N denotes the neighborhood relationship of a network. If node i and node j are adjacent, the corresponding entry in N is   is also called the algebraic connectivity of graph.

Kalman Consensus Filter Algorithm
Considering a dynamic process (or target) with a linear time-varying model as are the state and input noise of the process in time k, respectively, and A is the state transition matrix, and B is the control matrix.
The measurement model at node i in time k can be expressed as The error covariance matrices are defined as In the type-III KCF algorithm [22], node i in time k produces its local weighted measurement matrix   i k u and local information matrix The estimate value of node i in time k, From the KCF algorithm mentioned above, one finds that node i should receive messages from its neighboring nodes Hence, the communication overhead in the KCF algorithm is relatively heavy.

Problem Formulation
To formulate the problem of communication overhead reduction in the KCF algorithm, we first make the following assumption.
Therefore, the problem of communication overhead reduction in the KCF algorithm can be denoted as where n is the number of iterations. When the number of sensor nodes and the state vector of interested object are given, only the number of iterations will affect the solution of (P.1). Obviously, n is related to the convergence rate of KCF algorithm.
As described in [5,22], the estimate error dynamics of the KCF algorithm is a globally asymptotically stable system with a Lyapunov function which is related to the algebraic connectivity of network topology According to (2), we can optimize the network topology to improve the convergence rate of distributed estimation. However, since sensor nodes are power-limited, sensor nodes are not preferable to be communicated to nodes far away. In [23], the power relation between an idealized transmitting node and a receiving node behaves quadratically to the communication distance. Therefore, it is essential to constrain the communication range between sensor nodes.
Hence, the problem formulated in (P.1) can be relaxed and converted into where   , E i j is the Euclidean distance between nodes i and j, and i D is the communication range of node i. Therefore, the algebraic connectivity of network topology is maximized without changing the number of nodes and the number of communication links in the network, in which the constraint on the communication range of two neighboring nodes is incorporated. The optimized network topology is expressed as

Network Topology Optimization Method
In this section, a network topology optimization method is presented to reduce the communication overhead in the KCF algorithm.

The Structural Parameters of Network Topology
It is indicated in [24] that the algebraic connectivity of network topology   2   is affected by structural parameters of network topology. Therefore, it is reasonable to take the structural parameters as the basis of a network topology optimization method.
The structural parameters of network topology, such as node's degree (d), the average path length (l), and clustering coefficient (c), play a key role in the recent development of complex network theory.
The average value of the shortest path between any two sensor nodes in a network topology, l, determines the effective "size" of the network.
The clustering coefficient of node i, i c , is defined as the ratio between the actual number of links, i E , and the maximum number of links, , between neighboring nodes, and

Network Topology Optimization Method
Combining with Propositions 1-3, we can properly delete and add communication links according to some rules to solve problem (P.2).
The rules for deleting communication links are: For any node i, i{1, 2, …, I}, Step 1: To determine the initial deleting communication link at node i: , | max 1 where i l  denotes the variation of l at node i before and after deleting link (i, j). , | min 1 where i c  denotes the variation of c at node i before and after deleting link (i, j).
(1-4) The first link belonging to set ,3 i  is taken as the initial deleting communication link at node i, and Step 2: To determine the final deleting communication link at node i: The rules for adding communication links are: For any node p, p{1, 2, …, I}, Step 1′: To determine the initial adding communication link at node p: (1′-1) The links set where p l  denotes the variation of l at node p before and after adding link (p, s).

(1′-3) The links set
where p c  denotes the variation of c at node p before and after adding link (p, s).
(1′-4) The first link belonging to Step According to the rules for deleting and communication links described above, we propose a network topology optimization method summarized in Method 1 as follows, in which the communication range of each node is assumed to be the same.

Performance Analysis
In Method 1, the network topology is optimized according to nodes' local structural parameters information. Hence, compared with other network topology optimization methods [21,29] based on global topology information, the communication overhead of the proposed method is smaller. Meanwhile, due to the structural parameters information, the computational complexity of the proposed method is lower than other heuristic search methods [21,29,30]. The concrete analysis is as follows.
In Method 1, the maximum number of optimization adjustments is the communication links K. In the b-th adjustment, deleting links or adding links are determined by the rules described above. Taking deleting links as an example, where node i receives initial deleting links information from its neighboring nodes, i.e., In the b-th adjustment, taking deleting links as an example, where node i should calculate its initial deleting links information, and combine with the initial deleting links information of neighboring nodes to determine the communication link to be deleted. Hence, the computational complexity for determining the communication link to be de- The communication overhead and computational complexity of the proposed method in this paper and the WWKJ method proposed in [29] are given in Table 1. The WWKJ method optimizes the network topology based on the global network information, and performs an iterative search for feasible solutions, where m T is the maximum number of iterations for each simulation m.

Communication Overhead Computational Complexity
The proposed method The WWKJ method [29]  

Performance Evaluation and Discussion
In this section, the performance of the proposed network topology optimization method is evaluated.
This section considers a WSN    Figure 2 shows the corresponding adjustments of the original network  , where black lines and blue lines represent the deleted links and the added links, respectively. The normalized topology structural parameters as the corresponding adjustments of  are illustrated in Figure 3. From Figure 3, we observe that the degree distribution of nodes (d) becomes more uniform, and the average path length decreases (l), while clustering coefficient (c) decreases. Figure 4 shows the optimized topology   , where Figures 1-3, the proposed network topology optimization method is effective in that the algebraic connectivity of network topology is improved. Furthermore, the performance improved by d and l is larger than the performance decreased by c because of the locality of clustering coefficient (c).
The performance of three cases, (1) the traditional KCF algorithm [5] under original topology  , (2) the traditional KCF algorithm [5] under optimized topology   , (3) the RC-KCF algorithm [14] under original topology  , is evaluated, and P denotes the probability that each node receives messages from its neighboring nodes. Figure 5 shows the total communication overhead at time k. From Figure 5, we observe that the number of iterations for three cases is 2425, 1556, and 3010, respectively, which means that the proposed method in this paper improves the convergence rate of the KCF algorithm effectively. Figures 6 and 7 show the mean square estimate error at time k and the mean square estimate disagreement at time k, respectively. From Figures 6 and 7 Figures 5-7, we conclude that the proposed network topology optimization method achieves a good trade-off between the estimation error and communication overhead.   We also evaluated the performance of the proposed network topology optimization method using 20 WSNs with I = 15, K = 40, and D = 60 m, and the results are given in Table  2. In these simulations, the number of iterations of the KCF algorithm under the optimized network topologies is reduced by 10%-40% compared with that of the original network topologies. In the meantime, the proposed network topology optimization method achieves a trade-off between the estimate error and the communication overhead compared to other methods by probabilistic sending.   2779  2818  2194  4020  1669  1467  2709  1884  1557  3112  1637  1360  2773  2119  1914  3539  2418  1601  3303  2647  1861  3671  2336  2099  3986  1823  1596  2906  1857  1607  3211  1826  1601  3044  1750  1183  2492  1974  1301  3129  1819  1373  2655  1620  1246  2497  1915  1279  2828  2156  1799  3238  2345  1583  3240  1527 1375 2636 Figure 8 shows the impact of communication range over networks on the algebraic connectivity 2  , the value given in the figure is averaged on 20 simulations. From Figure   8, we observe that the value of 2  is increased as the value of D increases. The reason is that sensor nodes can receive deleting or adding links information from more different nodes within a larger communication range, which is beneficial to the network topology optimization. As D = 110 m, sensor nodes receive enough information, and then the algebraic connectivity tends to be stable.

Conclusions
In this paper, we proposed a network topology optimization method. Firstly, we define the communication overhead of the KCF algorithm, which can be transformed into a  network topology optimization problem. Secondly, the network topology optimization method is introduced, where the network topology is optimized by properly deleting and adding communication links, as well as considering the constraint on the communication range of two nodes. Simulation results show that the proposed optimization method can improve the algebraic connectivity of network topology and increase the convergence rate of the KCF algorithm under optimized network topologies. Moreover, the proposed network topology optimization method achieves a good trade-off between the estimate error and the communication overhead.
In the future, the network topology optimization method can be further combined with more structural parameters. Meanwhile, the network topology optimization method can also be considered to improve the stability of the topology.