Higher-order networks : an introduction to simplicial complexes / Ginestra Bianconi.

Format:

Book

Author/Creator:

Bianconi, Ginestra, author.

Series:

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

Language:

English

Subjects (All):

System theory.

Physical Description:

1 online resource (140 pages) : digital, PDF file(s).

Edition:

1st ed.

Place of Publication:

Cambridge : Cambridge University Press, 2021.

Summary:

Higher-order networks describe the many-body interactions of a large variety of complex systems, ranging from the the brain to collaboration networks. Simplicial complexes are generalized network structures which allow us to capture the combinatorial properties, the topology and the geometry of higher-order networks. Having been used extensively in quantum gravity to describe discrete or discretized space-time, simplicial complexes have only recently started becoming the representation of choice for capturing the underlying network topology and geometry of complex systems. This Element provides an in-depth introduction to the very hot topic of network theory, covering a wide range of subjects ranging from emergent hyperbolic geometry and topological data analysis to higher-order dynamics. This Elements aims to demonstrate that simplicial complexes provide a very general mathematical framework to reveal how higher-order dynamics depends on simplicial network topology and geometry.

Contents:

Cover

Title Page

Copyright Page

Higher-Order Networks

Contents

1 The Relevance of Higher-Order Networks in Network Science

1.1 Simplicial Complexes as Generalized Network Structures

1.2 Simplicial Complexes and Hypergraphs

1.3 A Topological Approach to Complex Interacting Systems

1.4 A Geometrical Approach to Higher-Order Networks

1.5 The Advantages of Using Simplicial Complexes and the Outline of the Element Structure

2 Combinatorial and Statistical Properties of Simplicial Complexes

2.1 Mathematical Definitions

2.1.1 Basic Properties of Simplicial Complexes and Hypergraphs

2.1.2 Cell Complexes

2.2 Generalized Degrees of Simplicial Complexes

2.3 Pure Simplicial Complexes and Their Tensorial Representation

2.4 Generalized Degrees of Pure Simplicial Complexes

2.4.1 General Properties

2.4.2 Case of a Simplicial Complex of Dimension d = 1

2.4.3 Case of a Simplicial Complex of Dimension d = 2

2.5 Clique Complexes of Random Uncorrelated Networks

2.6 Maximum Entropy Ensembles of Simplicial Complex

2.6.1 Microcanonical and Canonical Ensembles of Simplicial Complexes

2.6.2 Canonical Ensemble of Simplicial Complexes with Given Generalized Degree Sequence of the Nodes

2.6.3 Configuration Model of Simplicial Complexes

2.6.4 From Simplicial Complex Models to the Clique Complex of Their Skeleton

2.7 From Ensembles of Pure Simplicial Complexes to Ensembles of Hypergraphs

2.8 Temporal Simplicial Complexes

2.8.1 Temporal Social Networks

2.8.2 Entropy of Temporal Simplicial Complexes

2.8.3 Face-to-Face Simplicial Interactions

3 Simplicial Network Topology

3.1 Introduction to Simplicial Network Topology

3.2 A Brief Introduction to Algebraic Topology

3.2.1 Oriented Simplices and the m-Chains

3.2.2 The Boundary Maps.

3.2.3 Betti Numbers and the Euler Characteristic

3.2.4 Incidence Matrices

3.2.5 Higher-Order Laplacians and Hodge Decomposition

3.3 Topological Data Analysis of Simplicial Complexes

3.3.1 Topological Clustering of Node Neighborhoods

3.3.2 Persistent Homology

3.4 Clique Communities and Motif Conductance

3.4.1 Clique Communities and m-Connectedness

3.4.2 Motif Conductance

3.5 Emergent Community Structure

4 Simplicial Network Geometry

4.1 Discrete Manifolds

4.2 Curvature

4.2.1 Regge Curvature

4.2.2 Gromov -Hyperbolicity

4.3 The Hausdorff Dimension of Network Models

4.4 The Spectral Dimension

4.5 Higher-Order Spectral Dimension

5 Emergent Geometry

5.1 The Quest for Emergent Geometry

5.2 Emergent 2-Dimensional Simplicial Network Geometry

5.3 Network Geometry with Flavor (Neutral Model)

5.3.1 The Definition of the NGF Model (Neutral Model)

5.3.2 Combinatorial Properties of NGFs

5.3.3 Emergent Hyperbolic Geometry

5.3.4 Emergent Spectral Dimension of NGFs

5.4 Generalization of NGF to Cell Complexes

5.4.1 Combinatorial Properties of the NGF Cell Complexes

5.4.2 Emergent Community Structure of NGFs

5.4.3 Spectral Dimension of NGFs Remains as in Cell Complexes

5.4.4 Other Generalizations of NGFs

5.5 Network Geometry with Flavor (with Fitness)

5.5.1 Introducing the Energy of the Faces

5.5.2 Probability of an NGF Evolution

5.5.3 Emergent of Quantum Statistics

5.5.4 Topological Phase Transition for for β &gt

β[sub(c)]

6 Higher-Order Dynamics: Synchronization

6.1 Simplicial Synchronization

6.2 Kuramoto Model on Simple Networks

6.3 Kuramoto Models on Simplicial Network Geometry

6.4 Kuramoto Model with Many-Body Interactions

6.5 Higher-Order Kuramoto Model

6.6 Coupling Topological Signals of Different Dimensions.

7 Higher-Order Dynamics: Percolation

7.1 Interplay Between Percolation and Simplicial Network Geometry and Topology

7.2 Link Percolation in Hyperbolic Manifolds And Pseudo-Fractal Networks

7.3 Topological Percolation

8 Higher-Order Dynamics: Contagion Models

8.1 Higher-Order Contagion

8.2 Higher-Order Contagion on the Configuration Model of Hypergraphs

9 Outlook

Appendix A Maximum Entropy Ensembles of Simplicial Complexes

A1 Microcanonical and Canonical Ensembles of Simplicial Complexes

A2 Canonical Ensembles of Simplicial Complexes with a Given Sequence of Generalized Degree of the Nodes

Appendix B The Hodge Decomposition

Appendix C Spectral Dimension of Euclidean Lattices

Appendix D Topological Moves

Appendix E Emergent Preferential Attachment

Appendix F Generalized Degree Distributions of NGFs (Neutral Model)

Appendix G Apollonian and Pseudo-Fractal Simplicial Complexes

References

Acknowledgements.

Notes:

Title from publisher's bibliographic system (viewed on 25 Nov 2021).

ISBN:

9781108805421

1108805426

9781108770996

1108770991

OCLC:

1291317854