# Optimal Disintegration Strategy in Multiplex Networks under Layer Node-Based Attack
## 1 Introduction  
Recent decades have witnessed a dramatic advance in network sciences [1]. Network robustness, as an important field of complex networks, aims to improve the resilience of systems and to construct more efficient architecture [2; 3; 4]. However, not all networks are beneficial, some of them are harmful to human life, for instance, the epidemic or rumor spreading networks [5; 6]. For these networks, we need to break its functionality by removing some nodes or links. This is the focus of this paper, namely, network disintegration problem, which is also called network dismantling or network attack [7; 8]. Over the past few decades, there have been a large variety of works focusing on this problem. The most traditional solutions are nodes-based methods, where the nodes are removed in decreasing order of their significance determined by some indices, such as degree centrality [9], betweenness centrality [10], and others [11; 12]. Recent years, many algorithms are presented to identify the minimal nodes set whose removal would collapse the giant connected component. Examples include collective influence algorithm [13], CoreHD algorithm [14], Explosive immunization algorithm [15], Min-sum algorithm [7], GND algorithm [16] and so on. In this article, we consider a more generalized problem, named the disintegration strategy problem, where we consider limited attack resource and aim to maximize the disintegration effect under fixed attack length. The methods mentioned above could also be used to develop disintegration strategies when we remove a specified number of nodes in the given order. Moreover, for searching the optimal disintegration strategy, there have beensome combinatorial optimization-based studies in undirected [17] and directed [18] networks. In these studies, some meta-heuristic algorithms were introduced to identify the optimal disintegration strategy. Considering that networks rarely exist independently, the research of network science was extended to a multilayer framework [19]. According to the difference between the types of nodes in each layer and the links between the nodes in different layers [20], multilayer networks could be divided into multiplex networks, interdependent networks and so on [21]. Especially, the multiplex network could be regarded as a set of nodes connected by different types of links. Examples include the individuals connected through various social relationships [22] and the cities linked by different transportation networks [23]. The study of multiplex networks has emerged as one of the significant topics in network science and recent research proved that the co-existence of different interactions in multiplex networks would strongly affect the properties of the systems [24]. Therefore, the network disintegration problem was also generalized to multiplex networks. [PERSON] et al. [25] firstly extended the existing disintegration strategies to multiplex networks, and pointed out that ignoring the multi-layer structure of the systems will cause the result of overestimating the robustness of them. [PERSON] et al. [26] proposed a heuristic strategy named Effective Multiplex Degree for dismantling interdependent or multiplex networks. The meta-heuristic algorithm has also been introduced to determine the optimal disintegration strategy in general multiplex networks [27]. Nevertheless, an important hypothesis in the above studies is that the nodes in each layer would fail after being attacked. Actually, in some realistic cases, the targets that are attacked or those which fail may be just the layer nodes [28]. For instance, in the multiplex social networks, the users are banned from using one or some, but not all social network sites. Correspondingly, in the multiplex transport networks, different traffic stations in a city, like airports or highways, may not fail at the same time. We name this kind of attack as the layer node-based attack. In this paper, we focus on the disintegration strategy problem in multiplex networks.