A random geometric graph built on a time-varying Riemannian manifold

• A scale-free network model is proposed, which is a random geometric graph built on a time-varying Riemannian manifold.
• The model gives a geometric realization of connections made preferentially to more popular nodes and to more similar nodes.
• The model is used to physically simulate the increasing and connecting phenomena of a type of cancer cell.

The theory of random geometric graph enables the study of complex networks through geometry. To analyze evolutionary networks, time-varying geometries are needed. Solutions of the generalized hyperbolic geometric flow are such geometries. Here we propose a scale-free network model, which is a random geometric graph on a two dimensional disc. The metric of the disc is a Ricci flat solution of the flow. The model is used to physically simulate the growth and aggregation of a type of cancer cell.