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Introduction to network geometry

by M. Àngels Serrano,

 ICREA Research Professor, Department of Condensed Matter Physics & Institute of Complex Systems (UBICS), Universitat de Barcelona,
Martí i Franquès, 1 08028 Barcelona, Spain



Complex networks of interactions permeate reality and have important implications. Examples are all around us---the Internet, food webs, international trade, online and offline social networks... ---, and inside us –biochemical interactions in our cells, the brain connectome... Surprisingly, these networks talk a common language, regardless of their origin, and are imprinted with universal features. In particular, neworks are small worlds and nodes are separated by a small minimal number of intermediate links. As a consequence, networks present an apparent lack of metric structure and are difficult to map. Yet, many networks have a hidden geometry that enables to embed them in low-dimensional hyperbolic space. The discovery of such hidden geometry and the understanding of its role have become fundamental questions in network science giving place to the field of network geometry. We will review fundamental concepts, models, and methods for the geometric description of real networks with a focus on applications of real network maps, including decentralized routing protocols, geometric community detection, and the self-similar multiscale unfolding of networks by geometric renormalization.