# Scale-dependent measure of network centrality from diffusion dynamics

## TL;DR

This paper introduces a multiscale network centrality measure based on diffusion dynamics, capturing local and global importance of nodes by analyzing how they influence diffusion metricity across different scales.

## Contribution

It proposes a novel multiscale centrality measure derived from diffusion geometry, revealing scale-dependent node importance and network structures.

## Key findings

- Centrality varies significantly across scales.
- Correlates with degree at small scales and closeness at large scales.
- Identifies multi-centric structures in complex networks.

## Abstract

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.