# Proactive Caching at the Edge Leveraging Influential User Detection in Cellular D2D Networks
## 4 Proposed Methodology
### System Model  
Assume that there are \(n\) number of users who generate number of requests \(R=\{r_{1},r_{2},...r_{n}\}\), number of files \(F=\{f_{1},f_{2},...f_{n}\}\) with lengths \(L=\{l_{1},l_{2},...l_{n}\}\), and bit rate \(B_{r}=\{b_{1},b_{2},...b_{n}\}\). Note that bit rate of the file represents the transferring rate of the file per unit time. The requests are generated by the users in different time frames \(T=\{t_{1},t_{2},...t_{n}\}\). A request \(r_{1}\in R\) generated by user \(n\) is said to be satisfied iff D2D link capacity is higher than the bit rate of the file. The user satisfaction can be formulated as follows:  
\[B_{r}\leq\frac{l_{i}}{t_{i}^{\prime}-t_{i}}, \tag{1}\]  
where \(l_{i}\) is the size(length) of the file, \(t_{i}^{\prime}\) and \(t_{i}\) is the end and starting time of delivery of file \(i\). \(B_{r}\) represents the bit rate of file \(f\). Based on Equation (1), the user satisfaction rate \(S_{r}\) can be defined as:  
\[S_{r}=\frac{1}{R}\sum_{r\in R}x_{r}, \tag{2}\]  
where  
\[x_{r}= \begin{cases}1,&\text{if }\frac{L}{t_{z}^{\prime}-t_{z}}\geq B_{r},\\ 0,&\text{otherwise}.\end{cases} \]  
In order to reduce backhaul traffic load, our main goal is to maximize \(S_{r}\). The users are considered to be connected with other users of their community in a D2D mode of communication. Thus, the request \(r\) of the user \(i\) for a file \(f\) is directed to its nearest user to search the requested file and will be served if the file exists in the nearby cache. Thus, a good proactive caching strategy can satisfy more requests at the user end, which results in reducing the backhaul traffic load. The architecture considered in this work assumes the D2D communication enabled small cellular network. Every community in the network finds a central node that can serve a maximum number of D2D users with a minimum distance as shown in Figure 1. There are a total of \(n\) number of users that are connected with each other via D2D communication with a link capacity \(C_{b}\). The users are also connected with the small cell base station with a link capacity \(C_{w}\). Assume that the \(\sum C_{w}\leq\sum C_{b}\), then the average backhaul traffic load due to each file of length \(l_{i}\) can be formulated as follows [13]:  
\[B_{l}(r)=\frac{1}{R}\sum_{r\in R}\frac{1}{l_{i}}\sum_{t=tr}^{t=t^{\prime}} \lambda_{r}(t), \tag{3}\]  
where \(\lambda_{r}(t)\) is the backhaul data rate during the content delivery for request \(r\). We consider the network as a graph \(G=(V,E)\) where \(V\) represents set of users \(V=\{v_{1},v_{2},v_{3},...v_{n}\}\) and \(E\) is set of social links (\(E\subseteq\binom{r}{2}\)) between the users. For generating the social network for our experimental setup, we used a well known model called preferential attachment proposed by [PERSON] [17] to form the small cellular network. This model basically represents degree distribution of the random generated network i.e., how users connect to each other. The core idea behind the preferential attachment model is that new users prefer to connect to well-connected users over less-connected users proportional to the probability of their degree. The generated network using this model follows power law distribution with exponent \(\alpha=3\). This process is also known as rich-get-richer or cumulative advantage or Matthew effect. The process starts with some initial subgraph. Each new user comes with some initial \(m\) links. The probability \(\Pi(u)\) of connecting user \(u\) with other users \(j\) in the network is:  
\[\Pi(u)=m\frac{k_{u}}{\sum_{j}k_{j}}, \tag{4}\]  
where \(\Pi(u)\) represents the probability of connecting user \(u\) with other users \(j\) in the network. \(m\) represents the number of links to which the new user joins the network and \(k\) represents the degree of each user in the network. This process results in a network with a power-law distribution with exponent = 3.