  Centrality measures quantify the importance of a node in a network based on
different geometric or diffusive properties, and focus on different scales.
Here, we adopt a geometrical viewpoint to define a multi-scale centrality in
networks. Given a metric distance between the nodes, we measure the centrality
of a node by its tendency to be close to geodesics between nodes in its
neighborhood, via the concept of triangle inequality excess. Depending on the
size of the neighborhood, the resulting Gromov centrality defines the
importance of a node at different scales in the graph, and recovers as limits
well-known concept such as the clustering coefficient and closeness centrality.
We argue that Gromov centrality is affected by the geometric and boundary
constraints of the network, and illustrate how it can help distinguish
different types of nodes in random geometric graphs and empirical
transportation networks.
