# HBC - Hyperbolic Betweenness Centrality

#### Definition

BC is defined as a normalized sum of fractions of hop-measured shortest paths between each pair of vertices that pass through the node examined. HBC is a modified version of BC that obtained by replacing shortest paths with greedy paths

$$HBC(v)={\underset{s,t\in V}{\sum}} {\sigma_t^v (s)\over \sigma_t (s)}$$

were $\sigma_t (s)$ is the number of greedy paths with source node $s$ and destination $t$, $\sigma_t^v (s)$ the number of greedy paths with source node $s$ and destination node $t$ that pass via $v\ne s,t$. Note that the greedy paths connecting a source-destination pair need not be all of the same length as the hop-measured shortest paths in the definition of BC.

$$HBC(v)={\underset{s,t\in V}{\sum}} {\sigma_t^v (s)\over \sigma_t (s)}$$

were $\sigma_t (s)$ is the number of greedy paths with source node $s$ and destination $t$, $\sigma_t^v (s)$ the number of greedy paths with source node $s$ and destination node $t$ that pass via $v\ne s,t$. Note that the greedy paths connecting a source-destination pair need not be all of the same length as the hop-measured shortest paths in the definition of BC.

#### References

- Stai E., Sotiropoulos K., Karyotis V., Papavassiliou S., 2017. Hyperbolic Embedding for Efficient Computation of Path Centralities and Adaptive Routing in Large-Scale Complex Commodity Networks. IEEE Transactions on Network Science and Engineering, 4(3), pp.140-153. DOI: 10.1109/TNSE.2017.2690258