Time-Varying Hyperbolic Geometric Graphs–TV-HGGs
Project summary: A decade ago we discovered a powerful and unique geometric framework explaining the ubiquitous common structure of complex networks and linking this structure to the optimality of their common functions. In this framework, network nodes are mapped to points in hyperbolic spaces, which lie beneath the observable topologies.
The analysis of complex networks is then simplified significantly, as their discrete complex structure can be studied in purely geometric terms.
This framework, known as hyperbolic geometric graphs (HGGs), has attracted a great deal of interest in mathematics, physics, computer science and biology.
However, despite a decade of research, our knowledge and understanding of network geometry is essentially still limited to static HGGs and methods that can only infer the geometry of network snapshots.
But real networks are complex dynamical systems, evolving over time with the addition and deletion of nodes and links, and there currently exists no principled theory that can model and predict their dynamics — a grand-challenge open problem in modern network theory. The main aim of this project is to address this challenge by mapping the general problem of predicting network dynamics to the specific problem of predicting the motion of nodes in their hidden hyperbolic spaces.
To this end, we aim to:
(i) capture the motion of nodes in the hyperbolic spaces of real networks using Langevin dynamics;
(ii) incorporate the discovered motion equations into sound generative models of time-varying hyperbolic geometric graphs (TV-HGGs), which will satisfy certain desired statistical properties, and which will explain both the structural and dynamical properties of real networks; and
(iii) develop novel statistical inference methods based on these models and Fokker-Planck equations, which will be able to accurately forecast future connections and disconnections in real networks over different time scales.
Funded by: Cyprus Research & Innovation Foundation.