# Improving Traffic Load Distribution Fairness in Mobile Social Networks
## 4 TraLDA Design
### Social-Relations Popularity Calculation  
An individual can gain (social) benefits from relationships with her more central or popular acquaintances in a social network. Depending on the substance of the relations, measures of node centrality can be classified as undirected (symmetric) relations, such as friendship and kinship, or directed (asymmetric) relations, such as choice and influence. Moreover, in directed graphs centrality is known as \"prestige\" [48], where the direction of the interaction is a key attribute for this metric. For instance, individuals who are picked as friends by many others have a special status-prestige in the group. In the literature, there exist metrics of prestige which consider both direct and indirect social influences. For instance, the centrality measures in [37; 49] are based on the assumption that the importance of a node in the network is determined by the importance of its neighbours. Thus, these metrics take into account both a node's connectivity to other nodes and its proximity to other important nodes. We now mention one of the widely used centrality measures, the Katz's prestige measure [37]. This defines the prestige of node \(i\) in the graph \(G\), denoted by \(C_{Katz}(i)\), as the sum of the prestige of all \(i\)'s neighbours divided by their degrees. Node \(i\) therefore gains its prestige from having a neighbour \(j\) with higher prestige. This \(i\)'s prestige is however corrected by the number of neighbours of \(j\), so if \(j\) has more relations, then \(i\) gains less prestige from friendship with \(j\). This adjustment might be thought of correcting for \(i\)'s time spent with or relative access to \(j\). As a result, node \(i\)'s Katz centrality in the graph \(G\) is determined as follows:  
\[C_{Katz}(i)=\sum_{j\
eq i}\,g_{ij}\,\frac{C_{Katz}(j)}{d_{j}} \tag{13}\]  
where \(g_{ij}=1\) if there is a relation between \(i\) and \(j\), or \"0\" otherwise, and \(d_{j}\) is the degree of \(j\) representing the number of \(j\)'s neighbours. Inspired by the Katz's centrality measure, we introduce social-relations popularity, the node's popularity derived from relationships with more popular nodes. This distributes the popularity of more (socially) important nodes to their less important neighbours, and thus takes neighbours' popularity into account when calculating a node's popularity. We employ (13) to compute a node's social-relations popularity in a given time window as follows. To begin, we suppose that social influence occurs in only one direction, with nodes with lower popularity can only receive social benefits from their more popular neighbours; for instance, from (13) we can deduce influence from \(j\) towards \(i\), denoted by \(\overrightarrow{g_{ji}}=1\), exists when \(C_{Katz}(j)>C_{Katz}(i)\) or \"0\" otherwise, and therefore \(\overrightarrow{g_{ji}}\
eq\overrightarrow{g_{ji}}\). Second, we assume that the popularity of a more important node is shared by its less important neighbours and is weighted by the strength of their interactions with the given node. As a result, the higher (social) level node gives more effect on the closer neighbours.