## 2 Materials and Methods
### Methodology
#### 2.2.1 Complex Network Construction  
This paper explores the application of complex network theory to model marine plankton community structures [19]. The resulting networks, referred to as marine plankton interaction networks, provide a comprehensive representation of the complex interactions between different plankton species within the marine ecosystem. The network consists of vertices and edges, where the vertices represent plankton community organisms and the edges represent interactions between them. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Plankton Community ID** & **Number of the Genus** & **Average Biomass (mg/m3)** \\ \hline \(a\) & 62 & 29,872.03 \\ \(b\) & 62 & 12,055.91 \\ \(c\) & 63 & 76,150.55 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Statistical data on each plankton community. The marine plankton interaction network can be represented as an undirected weighted network \(G=(V,E)\), where \(V=v_{1},v_{2},v_{3},\ldots,v_{n}\) constitutes the set of vertices, \(n\) denotes the number of vertices, \(E=e_{1},e_{2},e_{3},\ldots,e_{m}\) constitutes the set of edges, and \(m\) represents the number of edges for any edge \(e=<u,v>\in E\), where \(u,v\in V\) and \(u\
eq v\)  
The Spearman rank correlation coefficient (Spearman coefficient) is a statistical measure used to calculate the correlation degree between two variables. It is commonly used in marine biology to determine the correlation between different plankton types. Suppose that two variables \(X\) and \(Y\) represent two plankton abundance datasets, each with \(h\) observations; \(X_{i}\) and \(Y_{i}\) denote the ith values of \(X\) and \(Y\) (\(1\leq i\leq H\)), respectively. We sort both \(X\) and \(Y\) in ascending or descending order, yielding the two variables \(x\) and \(y\). The element \(x_{i}\) is the rank of \(X_{i}\) in \(X\), while \(y_{i}\) is the rank of \(Y_{i}\) in \(Y\). The Spearman rank correlation coefficient between \(X\) and \(Y\) can be obtained by the following formula:  
\[r_{s}=\rho_{x,y}=\frac{\sum_{i=1}^{h}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{ \sum_{i}^{h}(x_{i}-\bar{x})^{2}\sum_{i}^{h}(y_{i}-\bar{y})^{2}}} \tag{1}\]  
where \(\bar{x}=\frac{1}{h}\sum_{i=1}^{h}x_{i}\) is the mean value of \(x\) and \(\bar{y}=\frac{1}{h}\sum_{i=1}^{h}y_{i}\) is the mean value of \(y\). In the marine plankton interaction network defined in this paper, each plankton type is represented as a network vertex. Edge weights are determined by the Spearman rank correlation coefficient between two vertices \(X\) and \(Y\). No edges exist between \(X\) and \(Y\) if the \(p\)-value is \(<\)0.05. The weights \(w_{XY}\) of edges \(<X,Y>\) between vertices \(X\) and \(Y\) are defined as follows:  
\[w_{XY}=\left|\rho_{x,y}\right| \tag{2}\]  
In addition, this paper proposes that the shortest path between vertices \(i\) and \(j\) has path weights \(d_{ij}\), defined as follows:  
\[d_{ij}=\frac{w_{v_{1}v_{1}}+w_{v_{1}v_{2}}+\cdots+w_{v_{t-1}v_{t}}+w_{v_{1}v_ {j}}}{t+1} \tag{3}\]  
Figure 1: The approach to robustness analysis described in this paper consists of three essential components. First, the construction of a complex network relies on derived edge relationships. Second, attack strategies include two types of attacks, namely, random and deliberate. Finally, a robustness assessment scheme is derived from multi-indicator weighting. where \(v_{1},v_{2},\ldots,v_{t}\) are the vertices on the shortest path between \(i\) and \(j\). Complex networks often possess intricate structures and numerous relationships, making their detailed structural modeling a challenge. Network structural characteristics are often used to concisely describe networks effectively. In this section, three common network metrics, namely, the average degree, average path length, and network diameter, are utilized to describe the overall topology of the marine plankton interaction network.