# Application-based Centrality

#### Definition

This concept for the first time coined to measure application-based importance of Facebook users. It is based on this concept that "There are more important relationships based on users' behavior and the done activities than those of friendship in online social networks". Accordingly, almost in all the works in the field of analyzing users’ influence, influence is defined as a non-flexible and rigid idea whereas a person may be important by one application, and unimportant by another one

This method considers different activity networks regarding each activity has its own importance in terms of the

considered application. Modified PageRank based centrality measure applied for measuring the importance of nodes for different applications.

$$C_{MLPR}(i)={1\over \sum_{\alpha=1}^LW_\alpha}{\underset {\alpha=1} {\overset{L}{\sum}}}\left ( W_{\alpha}. \left( d_{\alpha}.{\underset{j}{\sum}} A_{ji}^\alpha {C_{MLPR}(j)\over \sum _k A_{jk}^\alpha} +{(1 -d_\alpha)\over n_\alpha}\right) \right)$$

where $W_\alpha$ is the weight of the layer or activity a which is different according to the application. $A_{jk}^\alpha$ is the weight of the link $j\rightarrow k$ in layer $\alpha$. And $L$ is the number of layers of the activity network, which is 4 in facebook Application-based Centrality measurement. In addition, $n_\alpha$ and $d_\alpha$ are the number of nodes and the damping factor of layer $\alpha$ respectively. The different damping factor for different layers can be due to the nature of the considered layer, and also behavior of users based on the considered activities.

This method considers different activity networks regarding each activity has its own importance in terms of the

considered application. Modified PageRank based centrality measure applied for measuring the importance of nodes for different applications.

$$C_{MLPR}(i)={1\over \sum_{\alpha=1}^LW_\alpha}{\underset {\alpha=1} {\overset{L}{\sum}}}\left ( W_{\alpha}. \left( d_{\alpha}.{\underset{j}{\sum}} A_{ji}^\alpha {C_{MLPR}(j)\over \sum _k A_{jk}^\alpha} +{(1 -d_\alpha)\over n_\alpha}\right) \right)$$

where $W_\alpha$ is the weight of the layer or activity a which is different according to the application. $A_{jk}^\alpha$ is the weight of the link $j\rightarrow k$ in layer $\alpha$. And $L$ is the number of layers of the activity network, which is 4 in facebook Application-based Centrality measurement. In addition, $n_\alpha$ and $d_\alpha$ are the number of nodes and the damping factor of layer $\alpha$ respectively. The different damping factor for different layers can be due to the nature of the considered layer, and also behavior of users based on the considered activities.