# ρ-Geodesic Betweenness Centrality

#### Definition

The $\rho$-geodesic betweenness centrality is a variation of the traditional betweenness that uses both shortest and quasi shortest paths. The idea is to increase the importance of nodes that do not necessarily fall on shortest paths, but can still be considered critical to the network operation, reducing the reorganization and costs of the network upon failure of a central node. This centrality defines as follow:

$$B_\rho(v_k) = {\underset {\delta_{i,k}+\delta_{k,j} - \delta_{i,j}^* \le \rho}{\underset{i\in |v| j\in|v|}{\sum \sum}}} {n_{i,j}^*(v_k) + n_{i,j} (v_k)\over n_{i,j}^* + n_{i,j}} \times {\delta_{i,j}^*\over \delta_{i,k} + \delta_{k,j}}$$

This method considers a single metric the number of shortest and quasi-shortest paths between all pairs of nodes, as well as the cost of each path.These costs are introduced as a ratio between the cost of the shortest path, $\delta_{i,j}^*$, and the cost of the quasi-shortest path through $v_k$, given by $\delta_{i,j}=\delta_{i,k}+ \delta_{k,j}$. Hence, the $\rho$ geodesic betweenness weights the paths proportionally to their costs, assigning higher importance to nodes on shorter paths. The maximum cost of the quasi-shortest path depends on the spreadness factor $\rho$.

$$B_\rho(v_k) = {\underset {\delta_{i,k}+\delta_{k,j} - \delta_{i,j}^* \le \rho}{\underset{i\in |v| j\in|v|}{\sum \sum}}} {n_{i,j}^*(v_k) + n_{i,j} (v_k)\over n_{i,j}^* + n_{i,j}} \times {\delta_{i,j}^*\over \delta_{i,k} + \delta_{k,j}}$$

This method considers a single metric the number of shortest and quasi-shortest paths between all pairs of nodes, as well as the cost of each path.These costs are introduced as a ratio between the cost of the shortest path, $\delta_{i,j}^*$, and the cost of the quasi-shortest path through $v_k$, given by $\delta_{i,j}=\delta_{i,k}+ \delta_{k,j}$. Hence, the $\rho$ geodesic betweenness weights the paths proportionally to their costs, assigning higher importance to nodes on shorter paths. The maximum cost of the quasi-shortest path depends on the spreadness factor $\rho$.