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Introduction to Modern DynamicsChaos, Networks, Space, and Time$
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David D. Nolte

Print publication date: 2019

Print ISBN-13: 9780198844624

Published to Oxford Scholarship Online: November 2019

DOI: 10.1093/oso/9780198844624.001.0001

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PRINTED FROM OXFORD SCHOLARSHIP ONLINE (oxford.universitypressscholarship.com). (c) Copyright Oxford University Press, 2021. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in OSO for personal use. date: 01 December 2021

Network Dynamics

Network Dynamics

Chapter:
(p.207) 7 Network Dynamics
Source:
Introduction to Modern Dynamics
Author(s):

David D. Nolte

Publisher:
Oxford University Press
DOI:10.1093/oso/9780198844624.003.0007

A language of nodes and links, degree and moments, and adjacency matrix and distance matrix, among others, is defined and used to capture the wide range of different types and properties of network topologies. Regular graphs and random graphs have fundamentally different connectivities that play a role in dynamic processes such as diffusion and synchronization on a network. Three common random graphs are the Erdös–Rényi (ER) graph, the small-world (SW) graph, and the scale-free (SF) graph. Random graphs give rise to critical phenomena based on static connectivity properties, such as the percolation threshold, but also exhibit dynamical thresholds for the diffusion of states across networks and the synchronization of oscillators. The vaccination threshold for diseases propagating on networks and the global synchronization transition in the Kuramoto model are examples of dynamical processes that can be used to probe network topologies.

Keywords:   Kuramoto transition, small-world network, scale-free network, random networks, graph theory, percolation, adjacency matrix

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