# Optimal Disintegration Strategy in Multiplex Networks under Layer Node-Based Attack
## 2 Optimization Model of Disintegration Strategy in Multiplex Networks under Layer Node-Based Attack  
In this section, we extend the optimization model of the disintegration strategy in the multiplex network to different attack modes, multiplex node-based attack and layer node-based attack, of which we give an example in Figure 1. The removal of individuals means the removal of all kinds of edges connecting them and is named multiplex node-based attack [28]. Moreover, the network could also be embedded in the framework of the multiplex network with two layers, which are given in Figure 1c,d. The layer node-based attack is the attack of nodes in one layer, which means the removal of all edges connecting the nodes in this layer, and the edges in other layers are not affected. Figure 1: Multiplex node-based attack (**a**) and layer node-based attack (**c**) in the illustrative examples of the multiplex network. The edges in the top and bottom represent different kinds of edges. The two networks after different attacks are given in (**b,d**). We consider a multiplex network composed of \(N\) individuals (multiplex nodes) and \(M\) layers in which edges are unweighted and undirected. It could be formed as a tuple \(G=(V,E)\), where \(V(G)\) is the set of multiplex nodes and \(E(G)\) is the set of edges connecting these multiplex nodes in different layers. Each multiplex node \(v_{i},(i=1,...,N)\) in \(V\) could be represented by a set of layer nodes \([v_{i}^{1},...,v_{i}^{M}]\). Each layer \(\alpha\) is presented by an adjacency matrix \(A^{\alpha}=\{a_{ij}^{\alpha}\}_{N\times N}\), where \(a_{ij}^{\alpha}=1\) if layer nodes \(v_{i}^{\alpha}\) and \(v_{j}^{\alpha}\) are connected through an edge (i.e., multiplex nodes \(v_{i}\) and \(v_{j}\) are connected through an edge on layer \(\alpha\)). The multiplex network could be completely determined by the vector of the adjacency matrixes of the M layers  
\[A=\{A^{1},\ldots,A^{M}\} \tag{1}\]  
We define the degree of a layer node \(v_{i}^{\alpha}\) (or the layer degree of a multiplex node \(v_{i}\) on layer \(\alpha\)) as \(k_{i}^{\alpha}=\sum_{j}a_{ij}^{\alpha}\), which is the number of adjacent edges of layer node \(v_{i}^{\alpha}\), and the layer degree of the multiplex node \(v_{i}\) in all \(M\) layer is the vector  
\[k_{i}=\{k_{i}^{1},\ldots,k_{i}^{M}\} \tag{2}\]  
Corresponding, we define the multiplex degree of multiplex node \(v_{i}\) as \(k_{i}=\sum_{\alpha}k_{i}^{\alpha}\), which is the total number of adjacent edges of multiplex node \(v_{i}\) in all layers. Moreover, we define the aggregated topological adjacency matrix \(\mathcal{A}=\{a_{ij}\}_{N\times N}\) of a multiplex network, where  
\[a_{ij}=\left\{ \begin{array}{ll}1,\text{ if }\exists\alpha:a_{ij}^{\alpha}=1 \\ 0,\text{ otherwise }\end{array} ,i=1,...,N;j=1,...,N\right. \tag{3}\]  
This is also the adjacency matrix of an unweighted network obtained from the multiplex network, which is named aggregated network. The nodes \(v_{i}\) and \(v_{j}\) in the aggregated network are connected when there are edges between them in at least one layer of the multiplex network. In this article, the disintegration approaches focus on the layer node removal, and we assume that if a layer node is removed, the attached edges of it in this layer are removed. Meanwhile, we assume that all the layers in the multiplex network are coupled in a connective/complementary manner. That is to say, the removal of edges in one layer does not affect the layer nodes and edges in other layers and the function of the edges could be replaced by the edges in other layers. Examples include multiplex transportation networks composed of different modes of transportation and multiplex online social networks composed of various social media. Denote by \(\widehat{V}\subseteq V\) the set of layer nodes that are removed and denote by \(\widehat{G}=(V-\widehat{V},\widehat{E})\) the network after the nodes are removed.