## 1 Introduction  
An important issue of concern after monitoring an event or information is the location of the event or the collection of information in a variety of wireless network applications [1; 2; 3], such as the area of the enemy's movement on the battlefield, the location of forest fires, etc., which is the foundation of further measures and decision-making. Due to cost, lack of indoor signals and other reasons, it is difficult to lay GPS localization modules on wireless nodes. Therefore, the wireless network needs a localization algorithm which conforms to its own characteristics, and obtains the exact location of the unknown node through communication between the nodes.  
Depending on the different focus, localization algorithms can be divided into various types [4; 5], such as, on the basis of whether it is necessary to measure the distance information between nodes, localization algorithms can be divided into range-based and range-free; or according to the localization process, localization algorithms can be divided into centralized computing and distributed computing. In the practical application, various demands invoke different algorithms, in the light of their quality, stability, and efficiency. The range-free algorithm is mainly based on the connectivity of the network and the information fed back by the neighboring nodes. The unknown node calculates its own estimated position. The range-free algorithm has advantages of low network construction cost and small communication cost, and distributed computing has advantages of small computational burden and insensitivity to the network scale. If the complex environment and the node size of the monitoring area are taken into account, distributed computing can better adapt to the real needs. Therefore, the range-free algorithm is highly suitable for large-scale wireless network localization [6]. However, as with many key technologies, the range-free multi-hop localization algorithm is still plagued by technical problems in practical application. The most problematic is that multi-hop localization can only be achieved in an isotropic network with high node density and uniform distribution, while in the anisotropy network such as nonuniform distribution of nodes or sparse, the localization effect is very poor [4; 7; 8; 9].  
Anisotropy indicates that the quality of physical properties has different values when measured along axes in different directions. Anisotropy is most easily observed in the wireless network. As shown in Figure 1, we use \(d_{AB}\) (the dashed line) to represent the physical distance between nodes \(A\) and \(B\) in the network, and use \(h_{AB}\) (the straight line with arrows) to indicate the approximate length of the shortest hop-counts between nodes \(A\) and \(B\). When the nodes are isotropic and uniformly distributed in the network, we can see that \(d_{AB}\propto h_{AB}\); however, when there is an \"obstacle\" that affects the signal transmission in the network, \(d_{AB}<<h_{AB}\). Obviously, if no \"obstacle\" is present in the network, the physical distance between the nodes is proportional to the hop-count between them, that is \(d_{AB}\propto h_{AB}\). When there is an obstacle in the network, this proportional relationship is no longer true. This paper focuses on handling anisotropic network problems in large scale multi-hop localization.  
For the problems existing in the multi-hop localization, this paper presents a novel multi-hop localization method, called Multi-hop Localization though Regularized Extreme Learning Machine (ML-RELM), which is more suitable for practical applications because of its relatively lower computational complexity, higher localization accuracy, and stronger adaptability. The ML-RELM method aims to improve the performance of the localization algorithm and to adapt it to different environments by using the topology of the wireless network and its own characteristics.