## 2 Materials and Methods
### Methodology
#### 2.2.3 Robustness Evaluation Scheme  
In this paper, we adopt the average vertex connectivity, maximum connectivity component, and network efficiency as evaluation indicators, drawing on complex network theory and plankton community structure. However, these traditional single indicators present limitations in terms of robustness analysis and measurement. To address this issue, we introduce a novel multi-indicator weighted robustness evaluation scheme. (1) Average vertex connectivity: vertex connectivity measures the ability of network vertices to remain connected after deleting some number of vertices [30] based on connectivity [31]. In marine plankton communities, a plankton type's steady-state biomass is closely related to the nutrients it can uptake. If a plankton type feeds only on one food type, any sharp population decline risks severe harm, as a reduction in food availability threatens this food source. Feeding on varied foods reduces this risk, as loss of one food source can be replaced with others. From an energy transfer perspective, fewer bioenergy sources are easily lost through disturbance and destruction. Community stability is often related to interaction diversity, and vertex connectivity measures a similar network property. In network \(G\), the vertex connectivity \(rm_{ij}\) between vertices \(i\) and \(j\) denotes the minimum vertex number needed to disconnect vertices \(i\) and \(j\). Then, the average vertex connectivity \(AC\) is  
\[AC=\frac{\sum_{i,j\in V}rm_{ij}}{{n\choose 2}},\quad i\
eq j. \tag{11}\]  
Generally, higher \(AC\) indicates better network robustness. To eliminate dimensional influence on the robustness analysis, the value can be normalized as follows:  
\[AC^{{}^{\prime}}=\frac{AC-\min(AC)}{\max(AC)-\min(AC)}. \tag{12}\]  
(2) Maximum connected component: the maximum connected component refers to the largest subnetwork formed by vertices and edges. In robustness analysis, it reflects a network's damage degree [32]. When the removed vertex or edge ratio reaches a critical value, the maximum connected component changes [33]. Plankton interactions play a critical role in shaping community characteristics and functions. However, disrupting plankton interaction clusters and hindering material transfer between them significantly changes characteristic functions, leading to collapse of the original plankton community. The maximum connected component change effectively reflects the network structure's tolerance to attack, providing a good evaluation of the network cluster structure. In network \(G\), the initial vertex number is \(N\). After attack, the largest connected subnetwork's vertex number is \(N^{\prime}\). Then, the largest connected component's relative value is  
\[MP=\frac{N^{\prime}}{N} \tag{13}\]  
Here, \(MP\) reflects the network structure's tolerance degree to attacks. To eliminate dimensional influence on the robustness analysis, the values can be normalized as follows:  
\[MP^{\prime}=\frac{MP-\min(MP)}{\max(MP)-\min(MP)}. \tag{14}\]  
(3) Network efficiency: network efficiency refers to the efficiency of information exchange throughout the network [34; 35]. Energy transfer between plankton often involves losses. The narrower the energy transfer path between any two plankton, the less conducive it is to gaining energy and the more prone it is to being disconnected under disturbances and destruction. In contrast, a wide path is clearly more conducive to maintaining the community in its steady state. Given the shortest path weight \(d_{st}\) in network \(G\), the network efficiency \(NE\) is  
\[NE=\frac{\sum_{s,t\in V}d_{st}}{\binom{n}{2}},\quad s\
eq t. \tag{15}\]  
Generally, a higher \(NE\) indicates better network robustness. To eliminate dimensional influence, the value can be normalized as follows:  
\[NE^{\prime}=\frac{NE-\min(NE)}{\max(NE)-\min(NE)} \tag{16}\]  
(4) Robustness coefficient: three indicators (average vertex connectivity, maximum connected component, and network efficiency) have been introduced above.