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Table of Contents
Intro; Contents; 1 Introduction; 1.1 From Simple Networks to Multiplex Networks; 2 Multiplex Networks: Basic Definition and Formalism; 2.1 Graph Representation; 2.2 Matrix Representation; 2.2.1 The Supra-Adjacency Matrix; 2.2.2 The Supra-Laplacian Matrix; 2.2.3 Multiplex Walk Matrices; 2.3 Coarse-Graining Representation of a Multiplex Network; 2.3.1 Mathematical Background; 2.3.1.1 Adjacency and Laplacian Matrices; 2.3.1.2 Regular Quotients; 2.3.2 The Aggregate Network; 2.3.3 The Network of Layers; 2.4 Supra-Walk Matrices and Loopless Aggregate Network; 3 Structural Metrics.
3.1 Structure of Triadic Relations in Multiplex Networks3.1.1 Triads on Multiplex Networks; 3.1.2 Expressing Clustering Coefficients Using Elementary 3-Cycles; 3.1.3 Clustering Coefficients for Aggregated Networks; 3.1.4 Clustering Coefficients in Erdős-Rényi (ER) Networks; 3.2 Transitivity in Empirical Multiplex Networks; 3.3 Subgraph Centrality; 3.3.1 Subgraph Centrality, Communicability, and Estrada Index in Single-Layer Networks; 3.3.2 Supra-Walks and Subgraph Centrality for Multiplex Networks; 4 Spectra; 4.1 The Largest Eigenvalue of the Supra-Adjacency Matrix; 4.1.1 Statistics of Walks.
4.2 Dimensionality Reduction and Spectral Properties4.2.1 Interlacing Eigenvalues; 4.2.2 Equitable Partitions; 4.2.3 Laplacian Eigenvalues; 4.3 Network of Layers and Aggregate Network; 4.4 Layer Subnetworks; 4.5 Discussion and Some Applications; 4.5.1 Adjacency Spectrum; 4.5.2 Laplacian Spectrum; 4.6 The Algebraic Connectivity; 5 Structural Organization and Transitions; 5.1 Eigengap and Structural Transitions; 5.2 The Aggregate-Equivalent Multiplex and the Structural Organization of a Multiplex Network; 5.3 Dynamical Consequences and Discussions.
5.4 Structural Transition Triggered by Layer Degradation5.5 Continuous Layers Degradation; 5.5.1 Exact Value of t* for Identical Weights; 5.5.2 General Mechanism; 5.6 Links Failure and Attacks; 5.7 The Shannon Entropy of the Fiedler Vector; 5.7.1 Transition-Like Behavior for No Node-Aligned Multiplex Networks; 6 Polynomial Eigenvalue Formulation; 6.1 Definition of the Problem; 6.1.1 Quadratic Eigenvalue Problem; 6.1.2 2-Layer Multiplex Networks; 6.2 Spectral Analysis; 6.2.1 Bounds; 6.2.2 Comments on Symmetric Problems: HQEP; 6.2.3 Limits for Sparse Inter-Layer Coupling; 6.3 Applications.
3.1 Structure of Triadic Relations in Multiplex Networks3.1.1 Triads on Multiplex Networks; 3.1.2 Expressing Clustering Coefficients Using Elementary 3-Cycles; 3.1.3 Clustering Coefficients for Aggregated Networks; 3.1.4 Clustering Coefficients in Erdős-Rényi (ER) Networks; 3.2 Transitivity in Empirical Multiplex Networks; 3.3 Subgraph Centrality; 3.3.1 Subgraph Centrality, Communicability, and Estrada Index in Single-Layer Networks; 3.3.2 Supra-Walks and Subgraph Centrality for Multiplex Networks; 4 Spectra; 4.1 The Largest Eigenvalue of the Supra-Adjacency Matrix; 4.1.1 Statistics of Walks.
4.2 Dimensionality Reduction and Spectral Properties4.2.1 Interlacing Eigenvalues; 4.2.2 Equitable Partitions; 4.2.3 Laplacian Eigenvalues; 4.3 Network of Layers and Aggregate Network; 4.4 Layer Subnetworks; 4.5 Discussion and Some Applications; 4.5.1 Adjacency Spectrum; 4.5.2 Laplacian Spectrum; 4.6 The Algebraic Connectivity; 5 Structural Organization and Transitions; 5.1 Eigengap and Structural Transitions; 5.2 The Aggregate-Equivalent Multiplex and the Structural Organization of a Multiplex Network; 5.3 Dynamical Consequences and Discussions.
5.4 Structural Transition Triggered by Layer Degradation5.5 Continuous Layers Degradation; 5.5.1 Exact Value of t* for Identical Weights; 5.5.2 General Mechanism; 5.6 Links Failure and Attacks; 5.7 The Shannon Entropy of the Fiedler Vector; 5.7.1 Transition-Like Behavior for No Node-Aligned Multiplex Networks; 6 Polynomial Eigenvalue Formulation; 6.1 Definition of the Problem; 6.1.1 Quadratic Eigenvalue Problem; 6.1.2 2-Layer Multiplex Networks; 6.2 Spectral Analysis; 6.2.1 Bounds; 6.2.2 Comments on Symmetric Problems: HQEP; 6.2.3 Limits for Sparse Inter-Layer Coupling; 6.3 Applications.