Modularity and dynamics on complex networks / Renaud Lambiotte, Michael T. Schaub.

Format:

Book

Author/Creator:

Lambiotte, Renaud, author.

Schaub, Michael T., author.

Series:

Cambridge elements. Elements in structure and dynamics of complex networks 2516-5763

Language:

English

Subjects (All):

System analysis.

Modular groups.

Physical Description:

1 online resource (92 pages) : digital, PDF file(s).

Edition:

1st ed.

Place of Publication:

Cambridge : Cambridge University Press, 2021.

Summary:

Complex networks are typically not homogeneous, as they tend to display an array of structures at different scales. A feature that has attracted a lot of research is their modular organisation, i.e., networks may often be considered as being composed of certain building blocks, or modules. In this Element, the authors discuss a number of ways in which this idea of modularity can be conceptualised, focusing specifically on the interplay between modular network structure and dynamics taking place on a network. They discuss, in particular, how modular structure and symmetries may impact on network dynamics and, vice versa, how observations of such dynamics may be used to infer the modular structure. They also revisit several other notions of modularity that have been proposed for complex networks and show how these can be related to and interpreted from the point of view of dynamical processes on networks.

Contents:

Cover

Title Page

Copyright Page

Modularity and Dynamics on Complex Networks

Contents

1 Introduction

Network Dynamics and Network Structure

Outline of This Element

Notation

2 Background Material

2.1 Graph Theory

2.2 Random Graph Models

2.3 Network Dynamical Systems and Linear Dynamics

2.3.1 Linear Dynamics on Networks

2.3.2 Linear Network Dynamical Systems versus General Linear Systems

2.3.3 Spectral Decomposition

2.4 Laplacians, Diffusion and Consensus

2.4.1 The Combinatorial Graph Laplacian

2.4.2 The Normalised and the Random-Walk Graph Laplacian

2.5 Distances, Similarities, and Kernels

2.5.1 Distance Measures on Graphs

2.5.2 Similarities and Kernels

2.6 Further Discussion and References

3 Modularity, Community Detection, and Clustering in Networks

3.1 Communities as Clusters: Modularity and Assortative Communities

3.2 Communities Defined via Sparse Cuts: Graph Partitioning and Spectral Methods

3.3 Communities Defined by Node Equivalences: Disassortative Communities and Block Structures

3.4 Further Discussion and References

4 Timescale Separation and Dynamics on Modular Networks

4.1 Timescale Separation for General Dynamics

4.2 Timescale Separation for Linear Network Dynamics

4.3 Assortative Modular Network Structure and Timescale Separation

4.3.1 Matrix Perturbation Theory for Assortative Modular Networks

4.3.2 Stochastic Assortative Modular Structure and Separation of Timescales

4.4 Non-assortative Network Structures and Directed Networks

4.4.1 Beyond Assortative Network Sructures

4.4.2 Timescale Separations on Directed Networks

4.5 Further Discussion and References

5 Symmetries and Dynamics on Modular Networks

5.1 Equivalence Classes and Symmetries of Nodes

Symmetries and Node Equivalence under Linear Dynamics.

5.2 Equitable and Externally Equitable Partitions

5.2.1 Dynamical Implications of EEPs

5.3 Stochastic Symmetries and Equivalences

5.4 Differences and Relationships between EEPs and Timescale Separation

5.5 Further Discussion and References

6 Dynamical Methods for Assortative Communities

6.1 Basics of Markov Stability

6.2 Time as (Nonlinear) Resolution Parameter

6.3 Flow-Based versus Structure-Based Methods in Directed Networks

6.4 Different Dynamics Lead to Different Quality Functions

6.5 Further Discussions and References

7 Dynamical Methods for Disassortative Communitiesand General Block Structures

7.1 Kernels and Embeddings for Assortative Communities

7.2 Dynamical Embeddings for General Linear Dynamics

7.3 Dimensionality Reduction and Detection of General Block Structures via Dynamical Similarities

7.4 Node Equivalence Classes from Linear Dynamics

7.5 Further Discussions and References

8 Perspectives

References

Acknowledgements.

Notes:

Title from publisher's bibliographic system (viewed on 22 Dec 2021).

ISBN:

9781108808651

1108808654

9781108774116

1108774113

OCLC:

1493013249