Geometry in networks

The concept of hidden geometry has proven to be a powerful tool in analyzing and understanding networks. The idea is that nodes are situated in some metric space and connections are preferred between nodes that are close in this space. A classical example are random geometric graphs. Currently more advanced models exist, which use different geometries, for example hyperbolic geometry, often in combination with other hidden features of nodes. A key strength of using geometry is that it naturally gives rise to strong clustering, a feature present in many complex networks, which most classical models such as the Configuration Model or the Erdos-Renyi graph are unable to reproduce.

Part of my research focuses on a specific class of models which use hyperbolic geometry. I am particularly interested in understanding the behavior of structural properties associated with geometry, such as clustering or graph curvature.

This model is actually part of a larger class of graph models with geometry called spatial inhomogeneous random graphs. Together with Ph.D. student Neeladri Maitra, we are studying several structural features of these models, using mathematical concepts such as local weak convergence.

Current projects:

• Structural analysis of random graphs with geometry using local weak limits

• Scaling of the local clustering function in spatial inhomogeneous random graphs

Future projects:

• Ollivier curvature in random graph using hyperbolic geometry

Relevant papers:

• Local limits of spatial inhomogeneous random graphs

• Clustering in a hyperbolic model of complex networks

• Ollivier-Ricci curvature convergence in random geometric graphs

• Ollivier curvature of random geometric graphs converges to Ricci curvature of their Riemannian manifolds