# Extending Brandes algorithm to improve betweenness centrality computation in temporal graphs with discrete and continuous time

Mehdi Naima (LIP6, Paris)Betweenness centrality has been a long subject of study in network science since it was introduced by Freeman in 1977. This centrality measure assesses the importance of nodes in a graph, it has been used for example in social, biological and research collaboration networks. Moreover, betweenness centrality has been used in graph partitioning and community detection in the well-known Girvan-Newan algorithm.

This centrality measure is based on the enumeration of shortest paths passing through a node. A simple approach to compute betweenness centrality for all the nodes of a static graph is to use Floyd-Warshall algorithm that runs in \(O(n^3)\). Brandes in 2001 published an algorithm that runs in \(O(nm + n^2 \log n )\) on weighted graphs, it is still considered one of the best theoretical results on the question.

Betweenness centrality has also been extended to temporal graphs. Temporal graphs have edges that bear labels according to the time of the interactions between the nodes. Betweenness centrality has been extended to the temporal graph settings, and the notion of paths has been extended to temporal paths. We will see that we are able to deploy Brandes algorithm to its full extent and improve the running time of recent results to \(O(nmT + n^2T\log(nT))\). We will also discuss how Brandes algorithm can also be generalized to stream graphs which are dynamic graphs with continuous time and dynamicity on the nodes.