# A Topology Optimization Method for Reducing Communication Overhead in the Kalman Consensus Filter
## 1 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 fields such as autonomous vehicles [6], indoor positioning systems (IPS) [7],wearable devices [8], visual tracking [9], underwater environment sensing [10], and trace applications [11; 12]. Therefore, we focus on the KCF algorithm in this work, where each sensor node only communicates with its neighboring nodes in the network to generate an estimate, and the data exchanging constitutes the main reason of energy consumption. However, due to the limited energy of sensor nodes in WSNs, it is necessary to reduce communication overhead in distributed estimation algorithms. In the past few years, there have been several methods [13; 14; 15; 16; 17] that reduced the communication overhead of distributed estimation algorithms, which can be classified, typically, into two categories. First, the sensor nodes probabilistically receive estimation information from neighbors. For example, in [13], sensor nodes choose to exchange a subset of intermediate estimation information with neighboring nodes. In [14], authors proposed a distributed estimation algorithm that relies on limited local information for average consensus where sensor nodes exchange messages with their neighboring nodes less frequently as nodes' estimations approach to a stable value. [PERSON] et al. provided a rule to build the optimal set of information neighbors for every node in [15]. Second, dimension reduction is also a method used to reduce communication overhead in distributed estimation algorithms. In [16], a distributed estimation algorithm based on compressed sensing is proposed, where sensor nodes only exchange their compressed estimation to reduce the communication overhead over the network. Obviously, all of these methods reduce the communication overhead at the cost of the precision of estimation. It is pointed out in [5] that the structure of network topology plays an important role in distributed estimation algorithms, where the algebraic connectivity of network topology affects the upper bound of convergence rate. However, network topology optimization is a complex combinatorial optimization problem [18]. In recent years, many heuristic network topology optimization algorithms have been proposed. In [19], one finds that the convergence rate in random networks can be improved by redirecting a small portion of communication links. In [20], a method is presented to design a network topology for distributed estimation over a tree graph, where the convergence rate and security is the trade-off. In [21], a network topology optimization method based on the combination of a greedy algorithm and Tabu search is proposed, where the constraint on the distance of two nodes is incorporated.