## 4 Simulation Model of Metro Congestion Propagation Process
### Metro Network with Graph Cellular Automata  
CA is a kind of mathematic model that capable of simulating discrete dynamic systems where interactions of the system's basic elements can lead to global changes in time and space [26]. The main structure of these CA models can be defined as:  
\[CA=(G,Q,V,f) \tag{6}\]  
Figure 4: Structure of augmented metro network.  
where the cell space is represented by graph \(G=(N,L)\). \(N(G)\) is the set of stations \(\{n_{11},\)\(n_{12},\)\(\ldots,\)\(n_{ij},\)\(\ldots,\)\(n_{nn}\}\) in the metro graph \(G\). \(L(G)\) is the set of links between different stations, \(l_{ij}\) is the link between station \(i\) and station \(j\). \(N(G)\) and \(L(G)\) are the basic elements in the graph. \(Q\) is the states of cells, which is a finite set containing the states of the cells and each cell's state is related to the state of its \(u\) neighbors. \(V\) is the neighbor \(\{v_{1},\)\(v_{2},\)\(\ldots,\)\(v_{u}\}\) of each cell in the graph where the adjacency matrix can be used to show the connections between each cell. Function \(f\) is the cell state transformation function used to describe the changes in cells' states from time step \(t\) to the next time step \(t+1\). Therefore, the state of station \((i,j)\) in time \(t+1\) is represented by \(Q_{ij}^{t+1}\), which can be described as:  
\[Q_{ij}^{t+1}=f\big{(}Q_{ij}^{t},Q_{ab}^{t}\big{)} \tag{7}\]  
The states of link \(l\) in time \(t+1\), \(Q_{l,\;ij}^{t+1}\), are determined by the states of the stations they are connected with (i.e., station \(i\) and station \(j\)). If both of the station \(i\) and station \(j\) are congested, link \(l\) is congested as well. If only one of these two stations is congested, then, the state of link \(l\) is either exposed or infected. It is worth noting that one situation is very rare, that is link \(l\) is congested while both of the end stations of link \(l\) are not congested. In the simulation environment built by the graph-CA, this situation is assumed impossible to appear under metro emergencies.  
With the definitions above, the augmented metro network in Figure 4 can be transformed into cell spaces shown in Figure 5.  
As a useful generalization and implementation of traditional CA models on graphs, graph-CA provides a method to solve the tough problems of the relationship between structure and process, form and function [33]. Graph-CA's derivation from both graph theory and CA formalism makes it suitable for the simultaneous use of well-developed ways of describing the model structure and process dynamics. The graph structure enables graph-CA with more flexibility in describing the relationship in a complex system, and the dynamically changing neighborhoods make it capable of reflecting the realistic characteristics in the simulations. In traditional CA, the state of a given cell is defined by the states of cells surround it, that is to say, that the neighborhood relationship is only defined by the positions of the cells. On the contrary, in the augmented network, the neighborhood is defined by both cell positions and their graph relationships (links, arcs, or weights) [38].  
Figure 5: The augmented metro network in graph-CA.