Shortest Path to Network Geometry: A Practical Guide to Basic Models and Applications

Real networks have non-trivial connectivity patterns, making them difficult to map. However, many networks have a hidden geometry that enables meaningful maps in the two-dimensional hyperbolic plane. This field of network geometry has applications in decentralized routing protocols, geometric community detection, and the self-similar multiscale unfolding of networks.

\n Format: Paperback / softback
\n Length: 75 pages
\n Publication date: 06 January 2022
\n Publisher: Cambridge University Press
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Real networks, comprising hundreds to millions of interacting elements, permeate various contexts, ranging from technology to biology to society. These intricate networks exhibit non-trivial connectivity patterns, such as the small-world phenomenon, where nodes are separated by a limited number of intermediate links. Consequently, networks lack a clear metric structure, making them challenging to map accurately. However, many networks possess a hidden geometry that enables meaningful mapping in the two-dimensional hyperbolic plane. The discovery of this hidden geometry and its understanding have become fundamental questions in network science, giving rise to the field of network geometry.

This Element provides a comprehensive review of fundamental models and methods for geometrically describing real networks. It focuses on applications of real network maps, including decentralized routing protocols, geometric community detection, and the self-similar multiscale unfolding of networks through geometric renormalization. By exploring these topics, this Element aims to shed light on the rich geometric properties of real networks and their potential applications in various fields.

The first section introduces the concept of real networks and highlights their complexity and diversity. It discusses the various elements that contribute to the structure and dynamics of networks, such as nodes, links, and communities. The small-world phenomenon, a key characteristic of real networks, is explored in detail, along with its implications for network behavior and mapping.

The second section delves into fundamental models and methods for geometrically describing real networks. It discusses network embedding techniques, such as node embedding and link embedding, which aim to represent networks as low-dimensional vectors or matrices. These embeddings provide a compact and interpretable representation of networks, enabling efficient analysis and visualization.

The third section explores the application of real network maps in various fields. It discusses decentralized routing protocols, which optimize network communication by leveraging the geometric structure of the network. Geometric community detection, a technique used to identify cohesive groups of nodes, is also discussed, along with its significance in understanding network dynamics and behavior.

The fourth section focuses on the self-similar multiscale unfolding of networks by geometric renormalization. This technique involves mapping networks onto a lower-dimensional space, where they exhibit self-similar properties and can be analyzed using geometric methods. The application of this method in understanding network structure, dynamics, and evolution is explored.

In conclusion, real networks possess a hidden geometry that offers significant opportunities for understanding and analyzing their structure and dynamics. The discovery of this hidden geometry and the development of effective methods for mapping and analyzing networks have opened up new avenues for research and innovation in various fields. By leveraging the geometric properties of real networks, we can gain insights into complex systems, optimize communication protocols, and develop more efficient and effective algorithms for a wide range of applications.

\n Weight: 96g\n
Dimension: 152 x 228 x 7 (mm)\n
ISBN-13: 9781108791083\n
Edition number: New ed\n