  Classic measures of graph centrality capture distinct aspects of node
importance, from the local (e.g., degree) to the global (e.g., closeness). Here
we exploit the connection between diffusion and geometry to introduce a
multiscale centrality measure. A node is defined to be central if it breaks the
metricity of the diffusion as a consequence of the effective boundaries and
inhomogeneities in the graph. Our measure is naturally multiscale, as it is
computed relative to graph neighbourhoods within the varying time horizon of
the diffusion. We find that the centrality of nodes can differ widely at
different scales. In particular, our measure correlates with degree (i.e.,
hubs) at small scales and with closeness (i.e., bridges) at large scales, and
also reveals the existence of multi-centric structures in complex networks. By
examining centrality across scales, our measure thus provides an evaluation of
node importance relative to local and global processes on the network.
