Table 1.

Hypergraph construction algorithm pseudo-code

Algorithm: Hypergraph construction

Input: Disturbance value $R$; Coupling coefficients ${\mu }_1$ and ${\mu }_2$; Number of the nearest neighbor nodes $K$; Flow matrix ${\mathbf{M}}_{\mathbf{F}}$; Distance matrix ${\mathbf{M}}_{\mathbf{D}}$; Node set $\left\{\mathrm{1,\; 2},\dots ,n\right\}$.

Output: Hypergraph matrix $\mathcal{H}$.

1 Initialize the hypergraph. Establish the empty hypergraph matrix $\mathcal{H}={\left\{0\right\}}_{n\times n}$, where rows represent hyperedges and columns represent nodes.

2 For node $i$ in node set $\left\{\mathrm{1,\; 2},\dots ,n\right\}$:

3    Initialize node states. Using Eq. (5) in the main text to calculate each node state at time $t=0$. Denote as $\left\{x_1\left(0\right),x_2\left(0\right),\dots ,x_n\left(0\right)\right\}$.

4    Apply the disturbance. Add $R$ to the state value of node $i$, $x_i\left(0\right)+=R$.

5    While $\exists x_i\left(t\right)>1,\forall i\in \left\{\mathrm{1,\; 2},\dots ,n\right\}$:

6     Disturbance propagation. According Eq. (1) in the main text, update each node state set in the next time step.

7     Flow propagation. According Eqs. (6) and (7) in the main text, the flow of all $x_i\left(t+1\right)\notin \left(\mathrm{0,\; 1}\right]$ nodes are allocated to a maximum of $K$ neighbor nodes whose state values are in the interval $\left(\mathrm{0,\; 1}\right]$.

8     Node failure. For any node $i$, if $x_i\left(t+1\right)>1$, we set $x_i\left(t+1\right)=0$, $f_{ij}=f_{ji}=0$, $d_{ij}=d_{ji}=1$, $\forall j\in \left\{\mathrm{1,\; 2},\dots ,n\right\}$, $t+=1$.

9   End

10     Update hypergraph matrix $\mathcal{H}$. If node $i$ causes the failure of node $j$ ($x_j\left(t\right)=0$), then set the value in the $i$th row and $j$th column of $\mathcal{H}$ to 1.

11 End