The course focuses on the latent-geometric interpretation of the structure and function of real networks. The hidden space network models with hyperbolic geometry reproduce and explain a variety of important properties of many real networks. In these models, nodes are assigned popularity and similarity coordinates so that more popular and similar nodes are more likely to be connected. The combination of the popularity and similarity coordinates turns out to be equivalent to the hyperbolic distance in the latent space. The models can be reverse-engineered using statistical inference and machine learning techniques to produce cartographic maps of real networks. These maps can then be used in applications such as efficient decentralized navigation, the detection of communities of similar nodes, and a geometric renormalization that unravels the self-similar multiscale structure of networks. The latter provides a theoretically principled procedure for network rescaling and a practical method to produce scaled-down network replicas.

The course consists of the following three parts that: 1) introduce all the necessary background information behind the basic concepts and models in network geometry; 2) teach how to employ tools for hyperbolic network embedding to obtain geometric maps of real networks; 3) analyze these maps in terms of navigability, community structure, multiscale organization, and other important applications.

Instructors

Dima Krioukov

Northeastern University

M. Ángeles Serrano

University of Barcelona

Student preparation

Students should be familiar with basic network terminology. A basic level of mathematical proficiency is expected. All mathematical derivations are presented in an accessible manner. Programming experience in data analysis is very helpful.

The course consists of theoretical parts and computational hand-on activities. Students should have a personal computer (laptop) with Python or a similar software installed. The computational activities include the mapping of real network data using tools available online, such as https://github.com/networkgeometry/mercator.

About the instructors

Dima Krioukov

Dmitri Krioukov graduated from Saint Petersburg State University with Diploma in Physics in 1993. In 1998, he received his PhD in Physics from Old Dominion University, and moved to the networking industry as a network architect with Dimension Enterprizes. Upon their acquisition by Nortel Networks in 2000, he accepted a research scientist position at Nortel. In 2004, he moved back to academia as a Senior Research Scientist at the Cooperative Association for Internet Data Analysis (CAIDA) at the University of California, San Diego (UCSD).

Since 2014, he is an Associate Professor at the Departments of Physics, Mathematics, and Electrical & Computer Engineering at Northeastern University, and a core member of the Network Science Institute, where he is the Director of the DK-Lab. DK-Lab research deals mostly with theory and fundamental aspects of complex networks. Research topics of particular interest to the lab are latent network geometry, maximum-entropy ensembles of random graphs and simplicial complexes, random geometric graphs, causal sets, navigation in networks, and fundamental aspects of network dynamics.

M. Ángeles Serrano

M. Ángeles Serrano is an ICREA Research Professor at the Department of Condensed Matter Physics of the University of Barcelona and External Faculty at the Complexity Science Hub Vienna CSH. She obtained her PhD in Physics at the University of Barcelona, a master in mathematics for finance from the CRM-Autonomous University of Barcelona, and completed her postdoctoral research at Indiana University (USA), the École Polytechnique Fédérale de Lausanne (Switzerland) and IFISC Institute (Spain). She is interested in unraveling the universal principles and laws underlying the structure, function, and evolution of complex networks. Her research covers theoretical developments and applications to a variety of real systems, from international trade to the Internet and the brain. M. Ángeles obtained the Outstanding Referee Award of the American Physical Society. She is a founding member of Complexitat, the Catalan network for the study of complex systems, and a promoter member of UBICS, the Universitat de Barcelona Institute of Complex Systems.