# Predicting and Modeling Wildfire Propagation Areas with BAT and Maximum-State PageRank
## 2 Overview of Wildfire Propagation, Scale-Free Network, Barabasi-Albert Model, and BAT
### Scale-Free and Barabasi-Albert Model  
In scale-free networks [42; 43; 44; 45], the node degree follows a power-law distribution [44], such that the network size is independent of the scale. Hence, the node degree is the core of a scale-free network. Two different nodes are observed in scale-free networks: hub nodes and non-hub nodes, wherein the degrees of the nodes are greater than and less than a certain threshold, respectively [44].  
The scale-free networks adopted here are generated from a minimal model called the Barabasi-Albert model proposed by [PERSON] and [PERSON] in 1999 [45]. Inspired by the observation that World Wide Web (WWW) exhibits power-law (or scale-free) degree distributions, the Barabasi-Albert model [45] was proposed as the first scale-free model in developing the network based on the following two steps:  
1. Growth: Nodes are individually added into the network such that the network size, i.e., the number of nodes, increases over time.
2. Preferential attachment: The probability of the newly added node being connected with an old node is dependent on the degree of the old node.  
Growth and preferential attachment are extremely common in existing real-life networks. Growth indicates that the network grows from time to time by adding new nodes, and preferential attachment describes the growth of the network, i.e., the connection between new nodes and old nodes [45]. Furthermore, STEP B2 indicates that the degree of nodes follows the power law and preferential attachment is a probabilistic mechanism [45].  
We assume that the number of nodes is \(n\): 0, 1, 2..., \((n-1)\), Deg(\(i\)) is the degree of node \(i\), and Pr(\(i\)) is the probability that a new node is connected to node \(i\). STEP B2 is based on Pr(\(i\)) to predict the probability of the new node being added to the network in connection with node \(i\). Pr(\(i\)) can be defined as  
\[\Pr(i)=\frac{\text{Deg}(i)}{\sum\limits_{j=0}^{n-1}\text{Deg}(j)} \tag{3}\]  
It can be seen from Equation (3) that in the scale-free network, most of the nodes are non-hub nodes with a few links. Based on Step B2, a few high-degree non-hub nodes can gradually transform into hubs because new nodes are more likely to connect with them [45]. This is called a rich-get-richer phenomenon [42; 43; 44; 45]. In other words, the number of edges among all nodes approximately follows the power law property [44].  
Table 1 presents the adjacency matrix [60] that can be used to determine the degree of each node in Figure 1, which is used as an input to the proposed method.  
\begin{table}
2 & & & & & 1 & 1 & 1 \\
3 & & & & & 1 & 1 & 1 \\
4 & & & & & 1 & 1 & 1 \\
5 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
7 & 1 & 1 & & 1 & 1 & 1 & 1 \\ \hline Deg(i) & 2 & 3 & 2 & 3 & 3 & 6 & 6 & 5 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Adjacency matrix for Figure 1.  
The normalized adjacency matrix presented in Table 2 is calculated by dividing the element values in Table 1 by their degrees. The normalized adjacency matrix is required to calculate the PageRank values discussed in the next section. The original concept of BAT is first proposed and used in Table 2 of [61] without naming it.