Energy Transmission and Synchronization in Complex Networks (eBook)

Nicolas Rubido received his Ph.D. in 2014 from the University of Aberdeen, which he owes to the 2011 the Scottish Universities Physics Alliance studentship prize. He received his B.Sc. (2008) and M.Sc. (2010) degrees from the Universidad de la República, Uruguay, where he is now an Adjunct Professor. There, he was awarded a research initiation scholarship (2009) to study turbulence phenomena and a post-graduate scholarship (2010) to research into synchronization phenomena in gregarious fireflies, which resulted in various publications. In general, his research focuses on understanding the structure and behaviours of different complex systems from a mathematical and data-analysis perspective.

Supervisor's Foreword 7
Abstract 9
Acknowledgments 10
Contents 12
Notations 14
Observations on Notations 14
Notations for Methods 14
Notation for Transmission of Energy 15
Notation for Synchronisation 16
1 Introduction 17
1.1 Background 17
1.1.1 Complexity: A Shift from Reductionism 17
1.1.2 Complex Systems: Structure and Behaviour 20
1.2 Motivation 22
1.2.1 Transmission of Energy 22
1.2.2 Synchronisation 23
1.3 Summary of Results 24
References 27
2 Complex Networks 29
2.1 Networks 29
2.1.1 Notions and Definitions 30
2.1.2 Matrix Representations 33
2.1.3 Network Characterisation Methods 36
2.1.4 Archetypal Networks 38
2.2 The Laplacian Matrix 42
2.2.1 General Properties 42
2.2.2 Eigenvalues and Eigenvectors 44
2.2.3 Spectral Theorems 47
2.2.4 Spectral Properties of Laplacian Matrices 49
2.3 Resistance Distance 51
2.3.1 Laplacian Matrix Pseudo-inverse 52
2.3.2 Circulant Networks Spectral Characteristics 54
2.3.3 Equivalent Resistance, Its Bounds, and the Kirchhoff Index 56
References 59
3 Transmission of Energy 60
3.1 Flow Networks 61
3.1.1 AC/DC Flow Networks 61
3.1.2 DC Conservative Flow-Network Solutions 64
3.1.3 Configurational Edge-Capacity 67
3.1.4 Steady-State Cascade of Failures 69
3.1.5 Configurational Edge Capacity Margins 73
3.1.6 Community Detection via Flow Networks 76
3.2 Power Networks 78
3.2.1 Swing Equations: Power-Grid Model 78
3.2.2 Variational Equations 83
3.2.3 Stability Criteria for Networks of Generators 85
3.2.4 Equilibrium Solutions for Networks of Generators 89
3.2.5 Stability of Equilibrium Solutions in Circulant Networks 92
3.2.6 Energy Manifold for Networks of Generators 94
References 96
4 Synchronisation 99
4.1 Phase Oscillators 100
4.1.1 Model and Frameworks 101
4.1.2 Collective Behaviour: Conditions for Existence 103
4.1.3 Phase Master Stability Function 108
4.1.4 Fixed-Points Explicit Solutions 111
References 115
5 General Conclusions 117
Appendix 119
Curriculum Vitae 122