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The Structure of Complex NetworksTheory and Applications$
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Ernesto Estrada

Print publication date: 2011

Print ISBN-13: 9780199591756

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780199591756.001.0001

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Metric and topological structure of networks

Metric and topological structure of networks

(p.47) 3 Metric and topological structure of networks
The Structure of Complex Networks

Ernesto Estrada

Oxford University Press

This chapter explains the following concepts: shortest path distance, average path length, diameter, and the small-world properties of networks in networks; resistance distance in networks; and the mathematical relationship between this concept and those of the pseudo-inverse of the Laplacian matrix, random walks, and electric resistance. Then, a generalisation of both metric concepts is carried out through the presentation of the forest distance. Next, the chapter analyses topological properties of networks starting with the concept of network planarity. The embeddings of nonplanar networks into surfaces is then considered, and the concepts of genus and skewness of the network presented. This is followed by some illustrative examples of the problem of planarity in some real-world scenarios. The chapter ends with a short discussion of the topic of embedding complex networks into hyperbolic spaces.

Keywords:   graph metrics, shortest path distance, resistance distance, forest distance, planarity, embeddings, hyperbolic spaces, network thickness

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