An Improved MDS-MAP Localization Algorithm Based on Weighted Clustering and Heuristic Merging for Anisotropic Wireless Networks with Energy Holes

: The MDS-MAP (multidimensional scaling-MAP) localization algorithm utilize almost merely connectivity information, and therefore it is easy to implement in practice of wireless sensor networks (WSNs). Anisotropic networks with energy hole, however, has blind communication spots that cause loss of information in the merging phase of MDS-MAP. To enhance the positioning accuracy, the authors propose an MDS-MAP (CH) algorithm which can improve the clustering and merging strategy. In order to balance the effect of energy consumption and the network topology stabilization, we present a weighted clustering scheme, which considers the residual energy, the degree of connectivity nodes and node density. As the original MAD-MAP method poses a limitation of merging condition, the authors relax the merging requirement and present a heuristic estimation method for lost connectivity over energy holes. Simulation results show that the improved MDS-MAP (CH) localization algorithm has achieved higher localization accuracy, better-balanced energy consumption and stronger network robustness.


Introduction
WSNs (Wireless Sensor Networks) are ad-hoc networks composed of sensor nodes with limited computational and communicational capabilities. Sensor nodes are usually randomly scattered into the filed. Thus it is necessary to report data with geometrical information. Nowadays, localization of sensor nodes in WSN has become a fundamental and essential service which benefits many industrial and civilian cyber-physical system applications, such as environmental monitoring, industrial control and battle fields [Adissi, Lima, Gomes et al. (2017); Sundararajan, Redfern, Schneider et al. (2005); Perumal, Utharaj and Christo (2014)]. Although localization is crucial in determining the source of triggering events, over complex algorithms are not supported by the light weight process unit in the network. Therefore, extensive research has been focusing on designing localization algorithms that are energy and cost efficient. Mounting GPS on each sensor in the network is an easy solution but too expensive to be feasible in practice. GPS receivers have degraded performance in indoor places and dense The rest of this paper is organized as follows: In Section 2, background and related works are summarized. In Section 3, the motivation and details of the proposed MDS-MAP (CH) are presented. Section 4 presents the experimental results and performance evaluation. Finally, we state the conclusion of this paper and thoughts of future work in Section 5.

Background and related work
The classical MDS-MAP based localization algorithm was developed in the past decades and the idea is to take an input of between-nodes distance matrix and output a coordinate matrix by minimizing a loss function. The popular MDS-MAP was proposed in Hightower [Hightower (2001)] and the localization method was based on multidimensional scaling (MDS). In this centralized approaches, the final network graph requires a transformation into absolute coordinates over the anchors. Shang et al. [Shang and Ruml (2004)] proposed the distributed MDS-MAP (P) which aims at improving the performance by avoiding the inaccurate shortest path. Shi et al. presented a heuristic MDS algorithm in Shi et al. [Shi, Meng, Zhang et al. (2017)], which explores virtual nodes to construct the shortest paths between nodes and this can be a distinct advantage in anisotropic networks. There are two drawbacks of the classical MDS-MAP algorithm. Firstly, poor positioning results are shown for anisotropic networks. Secondly, the merging strategy restriction is too strict. In Shang et al. [Shang and Ruml (2004)], MDS-MAP (P) uses paths within 2 to 3 hops to estimate the distances between nodes, which can cause significant errors when merging nodes are out of the communication range. Jia et al. [Cheng, Qian and Wu (2008)] proposed an improved merging strategy for relaxing the previous restrictions to more general conditions. A relative map segmentation method based on rigid subset is proposed in Yu et al. [Yu, Zhou and Zhang (2017)] to overcome the flip ambiguity problem in merging, which adopts the graphic rigidity theory for unique merging solution. Another segmentation strategy is proposed by Kim et al. [Kim, Woo and Kim (2007)], which only consider local maps composed of nodes within the reliable communication range. Therefore, the accuracy is enhanced, but the merging condition is neglected. Clustering has been considered an effective method to distribute computation overhead and degrade the complexity. Weighted clustering further helps with decision making [Edwards, Stillwell and Seaver (1981)]. A number of cluster-based MDS algorithms are available in recent literature [Biljana, Danco and Andrea (2008)], [Minhan, Minho and Hyunseung (2012)] and they have proven this technique outperforms MDS in terms of accuracy especially in irregular networks. Saeed et al. [Saeed and Nam (2016)] proposed an algorithm for a cognitive network that uses the fuzzy C-means clustering technique to reduce shortest path error and the computational complexity is distributed into clusters. Chen et al. [Cheng, Qian and Wu (2008)] proposed an energy based weight-clustering which improved clustering formation by taking energy efficiency into consideration.

Summary of MDS-MAP
In this subsection, we briefly summarize the 3 phases of the MDS-MAP based localization algorithms as follows: Phase 1: A distance matirx for the MDS is constructed within each cluster based on the shortest paths between nodes.
The distance matrix, which is also called dissimilarity matrix D, can be estimated with algorithms, e.g., Dijkstra shortest path. Assume that there are nodes lying on a 2-D plane. Then distance matrix 2 ( ) is constructed with squa value of distance measurements, where denotes the coordinate estimation matrix. 2 ( ) is expressed as: where d ij represents the Euclidean distance between node and node , = (1,1, ⋯ ,1) , Apply a decantation process with matrix = ( − −1 ): Perform singular value decomposition on matrix B: Phase 2: The MDS utilizes the distance matrix to generate relative maps. The relative coordinates can be represented by Phase 3: Relative maps are merged into the absolute map with sufficient anchors. Various merging strategies can be used to stitch patches of local maps to form a global relative map. Finally, build the absolute map by performing a transformation process on the relative map: where is the scaling factor and 0 is the translation coefficient, and (•) is the rotation and translation matrix. , is the relative coordinates of node before merging. After all clusters are merged, the relative coordinates of the nodes can be adjusted into the global map by anchor nodes, where each node gets its absolute coordinates at 0, .

Merging strategy
In phase 1 of MDS-MAP framework, the network is divided into segments or clusters. Those local patches should be stitched together in merging phase 3, using information on the borders or overlapping areas. Merging can only be done under certain conditions. Yu et al. [Yu, Zhou and Zhang (2017)] utilizes the rigid subset characteristic to avoid the ambiguity in local map construction and merging, which implicitly requires a sufficient number of common nodes and connectivity between local graphs. Merging restriction analysis has been a neglected problem in MDS-MAP (P) related algorithms. In Kim et al. [Kim, Woo and Kim (2007)], the paper proposed a merging method that constructs the global map with local maps at a common node. This algorithm requires three common nodes between every two segmented maps during merging. Most cluster-based algorithms [Biljana, Danco and Andrea (2008); Minhan, Minho and Hyunseung (2012)] also require adequate information within communication range on the boundary. Assuming that there are nodes deployed in a 2-D plane and for each node ∈ S, where node set is = {1,2, ⋯ , }. The distance measurement between node and is denoted as . A common node merging condition can be summarized as follows: Assume that two cluster nodes sets are 1 and 2 and the set of common nodes is = 1 ∩ 2 , and the set of non-common nodes of 1 and 2 are represented as 1 = ( 1 ∪ 2 )\ 2 and • Condition 1: If | | ≥ + 1 , two maps can be uniquely merged. The rigid subset [Yu, Zhou, Zhang (2017)] strategy falls into this category.
• Condition 2: If | | < + 1, additional connectivity information has to be found to form a local rigid boundary. In a 2-D plane there requires a number of (m + 1 − | |) connectivity.
Merging conditions can be illustrated by Fig. 1(a): In the 2-D space, there have to be at least 3 common nodes between two clusters to meet the merging condition [Jia, Li, Wang et al. (2016)]. Fig. 1(b) shows that if the number of non-collinear common nodes is less than 3, there has to be some connectivity information for the two clusters to merge into a unique global formation. Moreover, Fig. 1(c) shows that if there is no common nodes, there has to be at least three known connectivity (For example: 1 1 , 1 2 , 2 3 ). In an anisotropic network, energy holes often block communication over the border, which causes loss of common nodes and connectivity information. In order to solve this problem, a heuristic method of distance estimation is proposed in Section 3.3 to compensate the merging rules.

Network model
Irregular network topology can obviously degrade the performance of MDS-MAP algorithms. The authors investigate the energy level and network topology characteristics of anisotropic networks. In this paper irregular specifically refers to networks with energy holes caused by battery drainage. The network model and parameters used in the proposed algorithm can be described as follows: =Total number of source nodes on a 2-D plane, =Communication range of sensor node (m) =Average node degree of senor node (number of connected neighbors) ( )=Edge degree of sensor (how far away the sensor is close to the edge of the network) =Energy cost of signal reception in one transmission =Residual energy level of the node Tab. 1 lists other parameters used for the simulation of the research. In the study, sensor nodes are deployed in a two-dimensional square area. We assume that each node has an adjustable communication range to be more energy efficient. The degree of connectivity is dependent on the deployment of the network, which directly affect the network density. Edge degree can be a decent measurement of how close a node is to the local geometric center (or how far away it is to the edges). Nodes on the edge should not be favored as cluster heads because the sub process unit should lie in the center to balance resources in each local map. Fig. 2 shows an anisotropic network with holes which signal does not pass through and no nodes can be sensed within communication range. A local O-shaped area is therefore formed and the distance estimation using shortest path algorithm has to be done with a detour (Fig. 2(b)). The authors used cluster-based graph segmentation method that considers energy level as a major factor so that network hole usually lies on the boundary between clusters. Therefore the inter-cluster communication issue needs to be addressed in the merging phase of the proposed MDS-MAP algorithm. Other parameters used in simulations are listed in Tab. 1.
The degree of a node can be described as the number of its neighbors within transmission range. To save resources of network bandwidth, the deployment of nodes in each cluster should not exceed a certain number, which means that the average degree of nodes connectivity is bounded. If the node degree is too large, there will be a communication bottle neck in the network, while if the node degree is too small, there will be a waste of network bandwidth resources. The sum of the distance of the node i with its neighbors within the range can be calculated as: Where is the distance between node and node .
The degree of all nodes is measured by degree-difference. The degree-difference of the node is expressed as: Where represents the average degree of nodes. Normalize the degree-difference: In the wireless sensor network, a node with a smaller degree-difference means evener distribution around it and it is more likely to be elected as a cluster head. The battery power is largely consumed by signal transmission and amplification. To formulate the energy cost of signal reception in one transmission, we use the wireless communication consumption model [Arumugam and Ponnuchamy (2015)] expressed as： The energy consumed by nodes in each cluster can be expressed as: We assume that node is elected as a cluster head in the process of data collection, the total energy consumed in the cluster in one unit time is: where, represents the energy consumed to send or receive 1-bit data. denotes the energy consumed by the node to fuse 1-bit data. indicates the amplification factor of the amplifier. indicates the number of neighbor nodes and is the length of the data packet, denotes the number of nodes in the network, and represents the number of cluster heads. The number of nodes in each cluster is represented by / .
Assuming that the initial cluster energy is , the remaining energy in the cluster at time can be expressed as: Higher residual energy level of the node means the node fits cluster head better. denotes the normalized value of the residual energy: Usually, the cluster head should be in the center of the cluster, which can be measured by edge-degree [Anchao and Guifen (2017)]. The edge degree is a measurement of how far away one node is to the center, or how close it is to the edge. The distance between each node with the nearest anchor node is recorded as ( ), and the farthest one is recorded as .
The overall weight parameter, considering degree-difference, residual energy, and edge degree is formulated as: 1 , 2 and 3 are the normalized weights of each index. The clustering procedure is conducted as the following steps: • Wireless sensor network battery power level are initialized. Each sensor node calculates its energy consumption; • Within the transmission range , each node sends message of the residual energy, hop count and network ID number to its neighbors. Meanwhile, the node receives the information and put it in the neighbor table; • With the distance measurements obtained by RSSI values, degree-difference and the remaining energy , and the normalized edge-degree ( ) are calculated; • Adjust the weights for each factor according to the scenario. The node with the highest weight among all adjacent nodes is elected cluster head (the number of nodes in the cluster should be more than 4 and each node is connected to each other). After cluster head is selected, clustering can be done by taking in nodes within two hops distance away from the cluster head. After the clustering is done, the local maps are segmented and localization can be calculated using MDS-MAP by the cluster head.

Heuristic merging strategy
This section presents the improved heuristic strategy that helps two clusters merge when the energy hole on the boundary blocks necessary information. The previous merging scheme requires a sufficient number of common nodes in this phase. However, in an anisotropic network, energy holes often block the communication between clusters and neither the number of common nodes or distance measurements are enough for merging (as shown in Fig. 3). On the other hand, using the Dijkstra method to calculate the shortest path causes a detour around the hole and the approximation will result in an aggregation of errors. Alternatively, we propose a heuristic method to explore virtual connectivity and calculate the Euclidean distance to compensate the merging condition.  Fig. 3 shows an example of an energy hole, denoted as EH, on the border of two merging clusters. There are only two common nodes in the overlapping area. As described in Section 2.2, in a 2-D plane if only two or less common nodes are found, the relaxed merging condition requires additional connectivity to be measured for two clusters to merge.
A a heuristic method to solve this problem in the relaxed merging condition is proposed. As shown in Fig. 4, we consider there is an energy hole lying between two clusters 1 and 2 . Sensor node 1 ∈ 1 , s 2 ∈ 2 and 3 is unavailable because of the energy hole. c 1 ∈ 1 ∩ 2 is one common node in the overlapping area of two clusters. 1 = | 1 1 | and 2 = | 2 1 |, therefore so 2 should be lying on the arc of circle C S 2 between 3 and 3 ′ , where C S1 ∩ C S2 = { 3 , 3 ′ }.
Now we explore the possibility of compensating a virtual connectivity 3 . As shown in Fig. 4, we consider an estimation of distance measurement 3 required for merging condition when common nodes are not sufficient. And the equation can be expressed as: As we can see the angle between 2 1 and 3 1 , and the angle between 3 1 and 4 1 , is not blocked and can be measured. Therefore, the equation can be expressed as: Assuming that 2 lies on the middle of arc 3 , 4 , the equation can be expressed as: Therefore, 3 is constructed and can be used as additional connectivity information on the border. The relaxed merging condition is met.

The complete MDS-MAP (CH) algorithm
This subsection presents the improved MDS-MAP (CH) localization algorithm for WSNs with energy holes by utilizing the schemes mentioned in previous subsetion 3.1~3.3 and the work flow of the new algorithm is shown in Fig. 5.

Start
Initialize network parameters, including sensor numbers, anchor numbers, nodes energy level and communication range.
Obtain RSSI measurements and residual energy level, communication degree and edgedegree for each nodes.
Propose weighted clustering algorithm to select cluster head and divide whole network into clusters.
End of each nodes calculation

Experimental results and analysis
In order to evaluate the performance of MDS-MAP (CH) algorithm, we simulate a WSN system with Matlab and compare the performance of the proposed algorithm with the classical MDS-MAP algorithm and the improved MDS-MAP (P). The average estimation error in the Eq. (21) is used as an index for evaluating the localization error of the algorithm.
where ( ) denotes the estimated coordinates of the node and ( ) represents the real coordinates of the node , and is the communication range.

Analysis of performance in uniform network
We first consider the performance of our algorithm in a uniformly deployed situation regarding error and energy consumption. We assume a sensor network randomly deployed on a square area and the anchor nodes are scattered randomly around the edges. Path loss is set to a fixed value. The influence of node density and residual energy is relatively lager than other factors. Therefore the order of weight factor should be > > ( ). According to the weight calculation method of the centroid weights [Edwards, Stillwell and Seaver (1981)], the calculated weighted results are 0.61, 0.278, and 0.112. The node performance parameters are shown in Tab. 1.
Path loss of transmit circuit Amplification factor Energy loss of fusion data Packet length We assume that there are 100 ordinary nodes randomly deployed on a 2-D plane and the average network connectivity is 10. There are four anchor nodes and the ranging error is 10%. Fig. 6(a) shows the relative coordinates of nodes after all clusters are merged where the circles denote the ordinary nodes and the diamonds denotes the anchor nodes. Fig. 6(b) displays the global absolute coordinates which were transformed from the map in Fig. 6(a) by rotation and translation with the aids of anchor nodes. Fig. 6(c) is a comparison between the estimated position (denoted by circles) calculated by the MDS-MAP (CH) algorithm, and the actual position (denoted by an asterisk). The average estimation error is around 30% in this case, which can be reduced with the increase of connectivity and the number of anchor nodes and simulation results are illustrated later. The performance regarding energy consumption in large-scale networks of the proposed method is compared with the classical MDS-MAP algorithm. As shown in Fig. 7, after over 100 trails for both algorithms, the new method is significantly superior to the MDS-MAP algorithm in terms of energy consumption and residual energy, and the energy level can be maintained at a high level within the first two hundred runs. Simulation result shows that when the density goes higher, the energy decays faster. However, the proposed method maintains a stable energy decay speed as the residual energy level goes lower.

Effect of anchor percentage
We assume that the percentage of anchor nodes are 4%, 6%, 10% and 20% respectively in Figs. 8(a)-8(d), and the average connectivity are adjusted by changing the transmission ranges. The degrees of connectivity are adjusted from 7 to 30 and the growth step is 3. Comparison of the classical MDS-MAP algorithm, the improved MDS-MAP (P) algorithm and the proposed MDS-MAP (CH) algorithm are conducted regarding to average estimation error after 50 simulations. When the degree of connectivity is less than 8, the proposed MDS-MAP (CH) algorithm significantly outperforms others. With the increase of connectivity degree, the average estimation error of all the algorithms decreases. As shown in Fig. 8. The performance of the other two algorithms can catch up by the increasing the degree of connectivity, but at the cost of introducing redundancy and consuming more energy. With the increase of the number of anchor nodes, as shown in Fig. 8(a) to Fig. 8(d), the localization accuracy of the algorithm goes higher. The average estimation error of the MDS-MAP (CH) algorithm proposed in this paper under each degree of connectivity is smaller than the other two algorithms and the localization results are better.

Effect of different network patterns
In this subsection, the effect of two typical anisotropic network patterns where our MDS-MAP (CH) algorithm is applied are studied. When calculating the virtual connectivity over the energy holes, two patterns are analyzed here: O-shape in Figs. 9(a)-9(b) and Hshape in Figs. 9(c)-9(d). Hop density is assumed to be 10/100 m 2 , the anchor percentage is 20%. We can see the performance in terms of error in Fig. 9, where the estimated position (denoted by circles) calculated by localization algorithm and the actual position (denoted by an asterisk), that compared with MDS-MAP (P) the MDS-MAP (CH) localization result is more accurate. Evaluating the performance with Eq. (23) and the error can be improved by 30-40% compared to MDS-MAP (P).

Conclusions and future work
In this paper, we proposed a MDS-MAP (CH) localization algorithm that aims at balancing energy consumption, reducing communication costs and lowering merging demand for an anisotropic network with energy holes. Because the cluster heads determine the performance of the algorithm, the authors proposed a weight-based clustering scheme. In this clustering scheme, with graph topology feature and energy load regarded as weighting factors, which can be adjusted flexibly according to realistic systems. The heuristic method is used to estimate missing distance information and compensate common nodes set to thus improve the merging rule. Experimental results show that the MDS-MAP (CH) algorithm performs better than the existing algorithms in higher accuracy and lower energy overhead, and suits well in practical large-scale network scenarios. In the future, a lot of our effort will be focusing on further increasing the accuracy of the heuristic merging method and a more dynamic weighted clustering strategy when considering energy level and topology vary at different time.