# Predicting and Modeling Wildfire Propagation Areas with BAT and Maximum-State PageRank
## 3 Novel Dynamic BAT and States
### The New BAT  
In general, BAT can identify all possible state vectors (cases) that satisfy certain predefined conditions, e.g., a connected path, required amount/information flows, etc., between two specific node subsets in networks, i.e., one source-node to one target-node, one source-node to many target-nodes, many source-nodes to many target-nodes, and many source-nodes to one target-node [46]. However, to satisfy the real-time wildfire propagation conditions, the number of coordinates, the node representing a coordinate, and the state label for a node are unknown. To meet the convenience of the conventional BAT for the practical problem discussed in this study, a novel dynamic BAT is proposed by implementing dynamic trees with dynamic lengths, flexible node orders, and different states. We assume that the wildfire starts in the node (area) \(i\), and we aim to identify all the possible scenarios where the wildfire spreads to at least N\({}_{\text{area}}\) areas and their corresponding probabilities. The procedure of the proposed BAT in determining the scenarios and probabilities are listed in Algorithm 3. The number of coordinates in the state vectors may be different in the proposed BAT and the nodes connected may be present in the same state vectors as shown in Table 8.  
For example, in Table 8, the initial state vector is \(X=(1/3)\) if node 3 is the first identified area with a wildfire. The initial state vector always includes the first burned node and its state label starts from 1, as shown in line 3. Then, \(X=(1/3,0/5)\) due to \(S_{1}(3)=[5]\) and node 3 does not reach its maximum-state. Furthermore, the other areas that are not the first identified area must begin with state 0, as shown in line 5. If N\({}_{\text{area}}=2\), we obtain a possible state vector because the wildfire in node 3 is propagated to a neighbor area, i.e., node 5, by line 6. If N\({}_{\text{area}}>2\), we should increase the state of node 5 by one because the current state of node 5 is zero, i.e., it does not reach another node. The aforementioned analysis is based on line 3 and is similar to that of the conventional BAT. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(i\) & **0** & **1** & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline
**0** & 0.0816538 & 0.194502 & 0.163308 & 0.357809 & & & & \\
**1** & 0.0816538 & 0.194502 & 0.191247 & 0.385748 & 0.163308 & 0.357809 & 0.354554 & 0.549056 \\
**2** & 0.0956233 & 0.194502 & 0.191247 & 0.385748 & & & & \\
**3** & 0.0816538 & 0.194502 & 0.191247 & 0.385748 & 0.163308 & 0.357809 & 0.354554 & 0.549056 \\
**4** & 0.0816538 & 0.194502 & 0.191247 & 0.385748 & 0.163308 & 0.357809 & 0.354554 & 0.549056 \\
**5** & 0.0366988 & 0.0740667 & 0.10116 & 0.175227 & 0.0733977 & 0.147464 & 0.174558 & 0.248624 \\
**6** & 0.0366988 & 0.10116 & 0.0733977 & 0.174558 & 0.10116 & 0.20232 & 0.174558 & 0.275718 \\
**7** & 0.0370333 & 0.0740667 & 0.10116 & 0.175227 & 0.10116 & 0.175227 & 0.20232 & 0.276387 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: PageRank values of the first eight states of each node.