## 1 Introduction  
As a new type of information acquisition system, wireless sensor networks (WSNs) have superiority in high flexibility, fault tolerance, self-control, and rapid deployment [1]. WSNs have a wide range of applications, including military counter-terrorism [2], precision agriculture [3], and healthcare [4]. A fundamental and hot topic in WSNs is localization. In recent years, many scholars have carried out numerous investigations and studies, mainly because many relevant algorithms have been presented in the literature. These algorithms are categorized into two types [5]: range-free and range-based localization. In all these algorithms, a few nodes, called anchor nodes, are equipped with a global positioning system (GPS) module and are configured to discriminate their position [6]. Range-free localization algorithms have low hardware requirements for nodes. They are more suitable for large-scale sensor networks, for example, multi-sequence positioning (MSP) [7], approximate point-in-triangulation test (APIT) [8], DV-Hop [9], etc. A novel framework for the performance analysis of cooperative network synchronization is proposed [10]. A quantitative description of the asymptotic local optimization performance is achieved by introducing generalized random walks with matrix-valued \"pseudo-probabilities\" in [11].  
The DV-Hop is a localization algorithm designed by [PERSON] et al. [9]. It has received much attention because of its low hardware requirements for nodes and simple implementation. The disadvantage is that hop distances are used directly instead of straight-line distances, which makes the ranging error larger. The average hop distance of anchor nodes was corrected by [PERSON] et al., but the error in the estimation of the unknown node position was not considered [12]. [PERSON] et al. introduced a weight model to improve the accuracy but ignored the intrinsic connection between weights and errors [13]. [PERSON] et al. used geometric features to correct the shortest distance of zigzag paths but did not consider the distribution density of nodes [14]. [PERSON] et al. removed some anchor nodes that were far from unknown nodes but did not make good use of the effective information of anchor nodes [15]. In order to improve the accuracy of location, we propose an elaboratedDV-Hop based on the cyclotomic method and weighted normalization (CMWN-DV-Hop). The error sources of the original DV-Hop localization algorithm are analyzed and evaluated. For implementing the global optimization, the DV-Hop algorithm is optimized and revised by utilizing the recursive algorithm of the cyclotomic method, introducing the weighting factor and the weighted recursive least squares. The simulation compares the localization performance of CMWN-DV-Hop and the other four algorithms under three factors.  
The article is structured as follows: In Section 2, we summarize the previous studies in detail in Section 2. Section 3 illustrates the sources of error in DV-Hop, the cyclotomic method, and the recursive algorithm for least-squares parameter estimation. Section 4 introduces the theory of CMWN-DV-Hop. Section 5 covers simulation, curve analysis, and capacity evaluation. Section 6 presents the conclusions of the paper.