shortest paths from node $i$ to node $j$, while $\sigma_{ij}(e)$ is the number of such shortest paths which contain the link $e$. The quantity $g(e)$ essentially measures the number of times a link is used in the shortest paths connecting any pair of nodes in the network, and is thus a measure of the contribution of a link in the organisation of flows in the network. The BC of a node is defined in a similar way $$g_v(i)=\sum_{s,t\in V}\frac{\sigma_{st}(i)}{\sigma_{st}}$$ where $\sigma_{st}(i)$ denotes here the number of shortest path from node $s$ to $t$ going through the node $i$.

In order to evaluate the impact of a new link on the overall distribution of the betweenness centrality we use the betweenness centrality impact defined in [@Strano:2012]. In the graph at time $t$, we first compute the average betweenness centrality of all the links of $G_t$ as: $$\overline{g}(G_t) = \frac{1}{(N(t) - 1) (N(t) - 2)} \sum_{e\in E_t} g(e)$$ where $g(e)$ is the betweenness centrality of the edge $e$ in the graph $G_t$. Then, for each link $e^* \in \Delta E_t$, i.e. for each newly added link in the time window $]t-1,t]$ we consider the new graph obtained by removing the link $e^*$ 