# A New BAT and PageRank Algorithm for Propagation Probability in Social Networks
## 2 Preliminaries
### Scale-Free Networks and the Barabasi-Albert Model  
As mentioned in Section 2.1, many scholars have proved that social networks are scale-free networks. In a scale-free network, when a node has \(k\) connections, their degree distribution follows a power distribution [PERSON] (1999), as in Equation (1). \[P(k)\sim k^{-\eta} \tag{1}\]  
The power distribution has a higher incidence when the value of \(k\) is small, and occurrences are very rare when its value grows [PERSON] (1999); [PERSON] (1998). In the power distribution function, even if the value of \(k\) only increases by a small amount, the effect on \(P(k)\) can be affected by the power series; when the power is negative, that is \(k^{-\eta}\) as shown in Figure 3.  
The scale-free web following a power distribution was discovered by [PERSON] and [PERSON] in their collaborative study of the WWW in 1998. The WWW consists of several highly connected pages. Most of these pages have no more than 4 hyperlinks, but very few pages have thousands of links [PERSON] (1999). Social networks in the real world will grow in size by adding new nodes, which start from a small source node. By adding new nodes, the number of nodes increases regularly throughout the network [PERSON] (1999); [PERSON] (1999); [PERSON] (1999); [PERSON] (1998). For example, the WWW grows exponentially with the addition of new pages, and research citations  
Figure 2: Facebook depicts all connected community users. grow with the publication of new papers, which [PERSON] and [PERSON] call the scale-free network [PERSON] and [PERSON] (2015). In networks in the real world, newly added nodes are generally connected to the nodes with more connections in the original network; therefore, this phenomenon is called \"rich-get-richer\" [PERSON] and [PERSON] (1988); [PERSON] (1989); [PERSON] and [PERSON] (2000); [PERSON] (2001). For example, new webpages tend to link to already-popular webpages such as the homepage of the website because such webpages are easy to find and widely known. The generation of new nodes in the network and the preferential connection of nodes with more connections in the original network prompted [PERSON] and [PERSON] to propose the Barabasi-Albert model in 1999 [PERSON] and [PERSON] (2000). The algorithm of the Barabasi-Albert model is based on two modes [PERSON] and [PERSON] (1988); [PERSON] (1989), as follows:  
1. Growth mode: The network in the real world continues to expand and grow, such as by adding new friends on social platforms or creating new pages on the Internet. 2. Preferential connection mode: New nodes tend to connect to nodes with more connections in the original network when they join. When selecting a new node to connect to, it is assumed that there are \(n\) nodes, where \(n\) = 0, 1, 2,..., (\(n-1\)); the probability of the new node connecting to node \(i\), namely \(P_{i}\), depends on the degree of node \(i\), namely \(k_{i}\), as in Equation (2). \[P_{i}=\frac{k_{i}}{\sum_{j=0}^{n-1}k_{j}} \tag{2}\]  
To understand how scale-free networks propagate, [PERSON] and [PERSON] presented a simple process for generating scale-free networks in Figure 4 [PERSON] and [PERSON] (2000). Starting from three connected nodes (\(t\) = 1), the original nodes in the layer are represented by solid lines, and the newly added nodes are represented by hollow circles. Figure 4: Scale-free network generation process. Figure 3: Power distribution plot. Due to the growth mode and priority connection mode, the newly added nodes prefer to connect to the original nodes with more connections, leading to the natural output of several highly connected nodes, which means that the probability of selecting a new node is proportional to the number of node connections. A typical feature of scale-free networks is that the node degree obeys a power distribution. Most nodes in the network are only connected to a few nodes, and very few nodes are connected to most nodes; such key nodes are called hubs.