Hyperbolic Representations of Complex Networks

Introduction

Network embeddings or mappings of networks to latent geometric spaces are standard tools in the arsenal of data analysis, and are routinely used in machine learning, visualization, network science, and graph theory. In general, a procedure of network embedding is mapping network nodes to points in a suitable latent metric space, such that latent distances between connected node pairs are smaller than those between disconnected node pairs. Latent-geometric distances are often interpreted as generalized measures of similarities: similar nodes are separated by small distances. 

Applications of network embeddings include prediction of missing links, network reconstruction, as well as the analysis of dynamic processes taking place on the network, e.g., routing or epidemic spreading.

Real networks are often characterized by heterogeneous distributions of number of connections per node (node degrees): while the majority of nodes in a network have only a handful connections to other nodes, there are also hubs, nodes with abnormally large number of connections to other nodes [1]. Networks with heterogeneous degree distributions are often referred to as scale-free.

It has recently been shown that scale-free networks are best represented/embedded in spaces with non-Euclidean geometries. One example, to this end, is embedding of networks into spaces with constant negative curvature, i.e., hyperbolic spaces [2, 3]. 

The biggest challenge/bottleneck of hyperbolic network representations is the lack of scalable embedding methods. Existing hyperbolic embedding methods are either (i) accurate and non-scalable or (ii) scalable but not very accurate.

Project

Rely on latest developments in Statistical Inference and Machine Learning to develop a SCALABLE and ACCURATE hyperbolic embedding algorithm. 

Deliverables

Hyperbolic embedding algorithm.

To apply:

Contact Maksim Kitsak, This email address is being protected from spambots. You need JavaScript enabled to view it.

Requirements

This is a very challenging project at the intersection of Machine Learning, Network Science and non-Euclidean geometry. Successful student will have excellent coding skills, prior experience with Euclidean network embeddings, and some knowledge of Statistical Machine Learning. 

References

[1] A-L. Barabási, Network Science, Cambridge University Press, (2016).
[2] D. Krioukov, F. Papadopoulos, M. Kitsak, A. Vahdat, and M. Boguná, Hyperbolic Geometry of Complex Networks, Physical Review E, 82 3 (2010).
[3] M. Boguná, F. Papadopoulos, D. Krioukov, Sustaining the internet with hyperbolic mapping, Nature communications, 1 1 (2010).