Table of Contents
Preface
1 Basic concepts of single networks. . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
1.2 Degree distribution—How networks are structured? . . . . . . . . . . .2
1.2.1 Types of networks . . . . . . . . . . . . . . . . . . . . . . . . .2
1.2.2 Degree and degree distributions . . . . . . . . . . . . . . . . . .3
1.2.3 Generating functions . . . . . . . . . . . . . . . . . . . . . . . .5
1.3 Percolation transition—How a network collapses? . . . . . . . . . . . .6
1.3.1 Failure and attack . . . . . . . . . . . . . . . . . . . . . . . . .7
1.3.2 Giant component . . . . . . . . . . . . . . . . . . . . . . . . . .9
1.3.3 Critical threshold . . . . . . . . . . . . . . . . . . . . . . . . . .12
1.4 Further network properties . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Spatial networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 From single network to network of networks 25
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 How networks network? . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.1 When a single network is not enough: examples of interacting networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.2 Zoology of network of networks . . . . . . . . . . . . . . . . . . 27
2.3 Key phenomena in network of networks . . . . . . . . . . . . . . . . . . 29
2.3.1 Cascading failures in interdependent networks . . . . . . . . . . 29
2.3.2 Increased resilience of interconnected networks . . . . . . . . . . 31
2.3.3 Dynamics on network of networks . . . . . . . . . . . . . . . . . 32
3 A pair of interdependent networks 41
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Different types of coupling . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.1 Feedback condition . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.2 No-Feedback condition . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.3 Multiple support . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3 Structural percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.1 Fully interdependent networks . . . . . . . . . . . . . . . . . . . 45
3.3.2 Partially interdependent networks . . . . . . . . . . . . . . . . . 49
3.3.3 Networks with multiple support . . . . . . . . . . . . . . . . . . 53
3.3.4 Interconnected networks . . . . . . . . . . . . . . . . . . . . . . 58
3.3.5 Antagonistic networks . . . . . . . . . . . . . . . . . . . . . . . 62
3.4 Targeted attack on partially interdependent networks . . . . . . . . . . 66
4 Robustness of networks of interdependent networks 77
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2 Structures of networks of networks (NON) . . . . . . . . . . . . . . . . 79
4.2.1 Loopless network of networks . . . . . . . . . . . . . . . . . . . 79
4.2.2 Loop-like network of networks . . . . . . . . . . . . . . . . . . . 79
4.2.3 Regular network of random networks . . . . . . . . . . . . . . . 79
4.2.4 Replica nodes in network of networks . . . . . . . . . . . . . . . 79
4.3 Cascading Failures in a Network of Networks . . . . . . . . . . . . . . . 79
4.3.1 Dynamic process . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.3.2 Identical steady state results for all tree-like structures . . . . . 85
4.3.3 General Stationary State . . . . . . . . . . . . . . . . . . . . . . 86
4.4 Percolation of network of networks . . . . . . . . . . . . . . . . . . . . 88
4.4.1 Percolation of tree-like network of networks . . . . . . . . . . . 88
4.4.2 Percolation of a RR NON composed of random networks . . . . 93
4.5 Comparing Feedback and No-Feedback Conditions . . . . . . . . . . . . 99
4.6 Vulnerability of network of networks for large number of networks . . . 101
5 Spatially embedded interdependent networks 131
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.2 The extreme vulnerability of semi-spatial interdependent networks . . . 133
5.2.1 Percolation transition in semi-spatial interdependent networks . 134
5.2.2 In semi-spatial interdependent networks qc = 0 . . . . . . . . . . 135
5.2.3 Critical exponents of the percolation transition . . . . . . . . . 136
5.3 Semi-spatial model of network of networks . . . . . . . . . . . . . . . . 137
5.3.1 Tree-like network of interdependent networks . . . . . . . . . . 137
5.3.2 Star-like network of interdependent networks . . . . . . . . . . 139
5.3.3 Random regular network formed of interdependent spatially embedded networks . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.4 Fully-spatial interdependent networks: propagation of cascading failures 140
5.5 Effect of dependency link length, the r-model . . . . . . . . . . . . . . 141
5.5.1 Theoretical model of spatial failure propagation . . . . . . . . . 142
5.5.2 Cascades propagation due to random failures . . . . . . . . . . 143
5.6 Effect of connectivity link length, the ζ-model . . . . . . . . . . . . . . 144
5.7 Localized attacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.7.1 Localized attacks on spatial networks with short-range dependency links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.7.2 Localized attacks on spatial multiplexes . . . . . . . . . . . . . 149
5.7.3 Spatial multiplexes with community structure . . . . . . . . . . 150
6 Further Features in Networks of Networks 174
6.1 Synchronization and Dynamics on Networks of Networks . . . . . . . . 174
6.2 Different network structures in networks of networks . . . . . . . . . . 176
6.2.1 Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
6.2.2 Node correlations . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.2.3 Directed networks . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.2.4 Community Structure . . . . . . . . . . . . . . . . . . . . . . . 177
6.3 Overlap and Intersimilarity in Networks of Networks . . . . . . . . . . 178
6.4 Different percolation processes in networks of networks . . . . . . . . . 178
6.4.1 Bond Percolation . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.4.2 Reinforced nodes . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.4.3 Combining k-core and interdependent percolation . . . . . . . . 179
6.5 Multimodal transportation . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.6 Games on networks of networks . . . . . . . . . . . . . . . . . . . . . . 180
6.7 Controllability in a network of networks . . . . . . . . . . . . . . . . . 180