## 5 Study on the Evolution Network Model of the Landslide Disaster Chain
### Vulnerability Analysis of Connected Edges of the Landslide Disaster Chain Network Model  
(1) Edge betweenness \(B\)  
Edge betweenness represents the count of shortest paths passing through an edge within a disaster chain network. It signifies the number of impacts from connected edges in the disaster chain network. A higher value indicates a more substantial influence of the edge on the structural resilience of the network [57; 58]. \[\text{B}=\sum_{i,j\in N}U_{i-j}(\text{m})\]  
\(U_{i-j}(m)\)--The number of paths between node \(V_{i}\) and node \(V_{j}\) passing through the connecting edge m.  
(2) Connectivity \(H\)  
Connectivity represents the proportion of nodes that a given node can connect to out of the total number of nodes. In the evolutionary network of the landslide disaster chain, removing one of the connecting edges reduces connectivity, causing certain hazardous events in the original disaster chain network to no longer occur, thereby decreasing systemic risk. \[\text{H}=\frac{N_{i}}{N}\]  
\(N_{i}\)--The number of nodes that can be connected from node \(V_{i}\);  
\(N\)--The total number of nodes in the disaster chain network. (3) Average path length \(L\)  
The average path length represents the distance from the current disaster event to other disaster events, offering insights into the speed of disaster evolution within the disaster chain network. When a connected edge is removed, if the network's average path length becomes longer, it indicates the greater significance of that edge. \[L=\frac{2}{N(N-1)}\sum_{i,j\in N}D_{i-j}\]  
\(D_{i-j}\)--The number of connecting edges in the shortest path between nodes \(V_{i}\) and \(V_{j}\);  
\(N\)--Total number of nodes in the disaster chain network. (4) Vulnerability of connecting edges \(W\)  
The vulnerability of a connected edge pertains to the consequences of removing it from a disaster chain network and evaluating its influence on the overall network damage. This indicator serves as a comprehensive measure that effectively illustrates the network's reliance on each connecting edge. \[\text{W}_{i,j\in\text{N}}\,=\frac{B_{i-j}\cdot L_{i-j}}{H_{i}-j}\]  
\(B_{i-j}\)--The edge betweenness number of hazards connecting edge \(i-j\);  
\(L_{i-j}\) and \(H_{i-j}\), respectively, represent the average path length and connectivity of the network after removing the connecting edge \(i-j\). Through MATLAB programming, calculations were performed for the edge intermediary number, connectivity, average path length, and vulnerability of the connecting edges within the landslide disaster chain evolution network model. In the landslide hazard chain model, the boundary interface number reflects the influence of the connecting edge of two disasters in the whole network. Connectivity can be used to study the spread and flow of disaster information in the network so as to predict the spread speed and scope and provide a basis for disaster control and management. The average path length reflects the average distance between any two disaster events and is obtained by calculating the shortest path length of all node pairs in the network and taking the average value. Subsequently, the vulnerability was normalized, resulting in a value denoted as \(W_{r}\), ranging from 0 to 1. The computed results for these various indicators are presented in Table 3.  
According to the above analysis of the vulnerability of an edge in the network, we know that the greater the vulnerability of a certain side, the more vulnerable the edge. As can be seen from Formula (8), the vulnerability of an edge in the network is related to the interface number, connectivity, and average path length of the edge. From the above table, it can be seen that the value and vulnerability of each individual index (the number of edge interfaces, the connectivity degree, and the average path length) do not show a single corresponding co-increase or co-decrease relationship.