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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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Adjacency relations in networks

Adjacency relations in networks

Chapter:
(p.24) (p.25) 2 Adjacency relations in networks
Source:
The Structure of Complex Networks
Author(s):

Ernesto Estrada

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199591756.003.0002

This chapter explains the concepts of adjacency between nodes in a network; the adjacency operator of a network and its matrix representation; and the meaning of degree, degree distributions, and degree–degree correlations. These topics are critically analysed and a reinterpretation of the assortativity coefficient is presented to give a structural interpretation to this statistical parameter. The concept of adjacency relationships is then extended beyond nodes to include links and other types of network structures, which is followed by the formal definition of the Laplacian operator of a network. Then, the chapter concentrates on the spectral properties of the adjacency and (unnormalised and normalised) Laplacian operators of networks, which includes results from algebraic and extremal graph theory as well as the study of spectral density functions.

Keywords:   adjacency matrix, laplacian matrix, degree distributions, network assortativity, spectral graph theory

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