# Directed Network Disassembly Method Based on Non-Backtracking Matrix
## 1 Introduction
In 2019, [PERSON] proposed a general network disassembly (hereafter referred to as the GND) method for undirected weight networks [14]. In addition, some relatively new disassembly methods and analyses of disassembly [15, 16] are provided, such as the disassembly algorithm based on the message passing model (2020) [17] and the sensitive disassembly method of neighbor connection (2020) [18]. Most of the existing disassembly algorithms are carried out in undirected networks, while there are few disassembly algorithms for directed networks. However, in many real-world networks (such as WWW networks, acquaintance relationships, network email networks, text association networks and article citation networks, etc. ), the edges between nodes are unidirectionally connected, and there is no mutual relationship in undirected networks [19]. The existing disassembly methods are sometimes not suitable for the disassembly of these directed networks. Compared with disassembly in the undirected network, disassembly in the directed network needs to consider the direction of the edges between the nodes in the network. For example, in a public opinion network, when an incoming node (Innode) connected by a directed edge \(e\) (also known as an arc in graph theory) is a communicator, the outgoing node of this edge (Outnode) is not likely to be propagated by the node. When applying the undirected network-based disassembly method to disassemble the network, this one-way relationship between nodes is sometimes ignored, resulting in the removal of this edge \(e\), and causing unnecessary disassembly to affect the disassembly effect. The influence of the internal mechanism of this network on the disassembly is ignored in the traditional disassembly method. To solve this problem, the non-backtracking matrix representing the edge adjacency relationship is applied as the operator of spectral division, whereby it retains the directionality of the node relationship, and combines the disassembly of the directed network with the spectral division method. An improved spectral disassembly method for directed networks (hereafter referred to as the DIR method) is proposed; the edges in the directed network are directly applied as the disassembly unit, and the strongest connected subgraph of the directed network is used as the disassembly subject. The spectral characteristics of the edge adjacency matrix (non-backtracking matrix) are applied to the bipartition of the edge modules in the maximum strongly connected subset. The overlapping node set of the node sets connecting the two edge modules is then found as the minimum disassembly set to disassemble the network until the disassembly scale reaches the specified disassembly scale of the network (the maximum number of nodes in the strongly connected subgraph). The excellent characteristics of the non-backtracking matrix can be made full use of by using the DIR method, and the DIR method can greatly protect the topology of the network during the disassembly process. Furthermore, it is verified by experiments that the DIR method is suitable for the disassembly of directed networks. Finally, the influence of different disassembly methods on the network structure is analyzed by analyzing the changes of network indexes such as the clustering coefficient, assortativity coefficient and modularity function in the disassembly process, and it is verified that the application of edge module partition to disassemble the network can greatly retain the structural information in the networks.