geometric correlations in multiplexes and how they make them more robust

1. Structure and dynamics of multiplex networks: beyond degree correlations & Geometric correlations mitigate the extreme vulnerability of multiplex networks against targeted attacks Kaj Kolja Kleineberg | [email protected] @KoljaKleineberg | koljakleineberg.wordpress.com

2. The World Economic Forum Risks Interconnecon Map

3. Introduction Multiplex geometry Applications Robustness Summary & outlook Multiplex: nodes are simultaneously present in different network layers Several networking layers 4

4. Introduction Multiplex geometry Applications Robustness Summary & outlook Multiplex: nodes are simultaneously present in different network layers Several networking layers Same nodes exist in different layers 4

5. Introduction Multiplex geometry Applications Robustness Summary & outlook Multiplex: nodes are simultaneously present in different network layers Several networking layers Same nodes exist in different layers One-to-one mapping between nodes in different layers 4

6. Introduction Multiplex geometry Applications Robustness Summary & outlook Multiplex: nodes are simultaneously present in different network layers Several networking layers Same nodes exist in different layers One-to-one mapping between nodes in different layers Typical features: Edge overlap & degree-degree correlations & and geometric correlations! Degree correlations and overlap have been studied extensively: Nature Physics 8, 4048 (2011); Phys. Rev. E 92, 032805 (2015); Phys. Rev. Lett. 111, 058702 (2013); Phys. Rev. E 88, 052811 (2013); ... 4

7. Hidden metric spaces

8. Introduction Multiplex geometry Applications Robustness Summary & outlook Hidden metric spaces underlying real complex networks provide a fundamental explanation of their observed topologies Nature Physics 5, 7480 (2008) 6

9. Introduction Multiplex geometry Applications Robustness Summary & outlook Hidden metric spaces underlying real complex networks provide a fundamental explanation of their observed topologies We can infer the coordinates of nodes embedded in hidden metric spaces by inverting models. 6

10. Introduction Multiplex geometry Applications Robustness Summary & outlook Hyperbolic geometry emerges from Newtonian model and similarity popularity optimization in growing networks S1 p() 7

11. Introduction Multiplex geometry Applications Robustness Summary & outlook Hyperbolic geometry emerges from Newtonian model and similarity popularity optimization in growing networks S1 p() r = 1 1+ [ d(,) ]1/T PRL 100, 078701 8

12. Introduction Multiplex geometry Applications Robustness Summary & outlook Hyperbolic geometry emerges from Newtonian model and similarity popularity optimization in growing networks S1 H2 p() ri = R 2 ln i min r = 1 1+ [ d(,) ]1/T PRL 100, 078701 9

13. Introduction Multiplex geometry Applications Robustness Summary & outlook Hyperbolic geometry emerges from Newtonian model and similarity popularity optimization in growing networks S1 H2 p() (r) e 1 2 (1)(rR) r = 1 1+ [ d(,) ]1/T PRL 100, 078701 10

14. Introduction Multiplex geometry Applications Robustness Summary & outlook Hyperbolic geometry emerges from Newtonian model and similarity popularity optimization in growing networks S1 H2 p() (r) e 1 2 (1)(rR) r = 1 1+ [ d(,) ]1/T p(xij) = 1 1+e xijR 2T PRL 100, 078701 PRE 82, 036106 11

15. Introduction Multiplex geometry Applications Robustness Summary & outlook Hyperbolic geometry emerges from Newtonian model and similarity popularity optimization in growing networks S1 H2 growing p() (r) e 1 2 (1)(rR) t = 1, 2, 3 . . . r = 1 1+ [ d(,) ]1/T p(xij) = 1 1+e xijR 2T mins[1...t1] s st PRL 100, 078701 PRE 82, 036106 Nature 489, 537540 12

16. Introduction Multiplex geometry Applications Robustness Summary & outlook Hyperbolic maps of complex networks: Poincar disk Nature Communications 1, 62 (2010) Polar coordinates: ri : Popularity (degree) i : Similarity Distance: xij = cosh1 (cosh ri cosh rj sinh ri sinh rj cos ij) Connection probability: p(xij) = 1 1 + e xijR 2T 13

17. Introduction Multiplex geometry Applications Robustness Summary & outlook Hyperbolic maps of complex networks: Poincar disk Internet IPv6 topology Polar coordinates: ri : Popularity (degree) i : Similarity Distance: xij = cosh1 (cosh ri cosh rj sinh ri sinh rj cos ij) Connection probability: p(xij) = 1 1 + e xijR 2T 13

20. Multiplex geometry

21. Introduction Multiplex geometry Applications Robustness Summary & outlook Metric spaces underlying different layers of real multiplexes could be correlated 15

25. Introduction Multiplex geometry Applications Robustness Summary & outlook Metric spaces underlying different layers of real multiplexes could be correlated Uncorrelated 15

26. Introduction Multiplex geometry Applications Robustness Summary & outlook Metric spaces underlying different layers of real multiplexes could be correlated Uncorrelated Correlated 15

27. Introduction Multiplex geometry Applications Robustness Summary & outlook Metric spaces underlying different layers of real multiplexes could be correlated Uncorrelated Correlated Are there metric correlations in real multiplex networks? 15

28. Introduction Multiplex geometry Applications Robustness Summary & outlook Radial and angular coordinates are correlated between different layers in many real multiplexes Degreecorrelations Random superposition of constituent layers 16

29. Introduction Multiplex geometry Applications Robustness Summary & outlook Radial and angular coordinates are correlated between different layers in many real multiplexes Degreecorrelations Random superposition of constituent layers 16

30. Introduction Multiplex geometry Applications Robustness Summary & outlook Radial and angular coordinates are correlated between different layers in many real multiplexes Degreecorrelations Random superposition of constituent layers What is the impact of the discovered geometric correlations? 16

31. Communities

32. Introduction Multiplex geometry Applications Robustness Summary & outlook Reshuffling of node IDs destroys correlations but preserves the single layer topologies Real system Reshuffled counterpart: Artificial multiplex that corresponds to a random superposition of the constituent layer topologies of the real system. 18

33. Introduction Multiplex geometry Applications Robustness Summary & outlook Reshuffling of node IDs destroys correlations but preserves the single layer topologies Real system Reshuffled Reshuffled counterpart: Artificial multiplex that corresponds to a random superposition of the constituent layer topologies of the real system. 18

34. Introduction Multiplex geometry Applications Robustness Summary & outlook Reshuffling of node IDs destroys correlations but preserves the single layer topologies Real system Reshuffled Reshuffled counterpart: Artificial multiplex that corresponds to a random superposition of the constituent layer topologies of the real system. 18

35. Introduction Multiplex geometry Applications Robustness Summary & outlook Sets of nodes simultaneously similar in both layers are overabundant in real systems Real system 0 2 1 0 2 2 100 200 Reshufed 0 2 1 0 2 2 100 200 19

36. Introduction Multiplex geometry Applications Robustness Summary & outlook Sets of nodes simultaneously similar in both layers are overabundant in real systems Real system 0 2 1 0 2 2 100 200 Reshufed 0 2 1 0 2 2 100 200 Angular correlations are related to multidimensional communities. 19

37. Link prediction

38. Introduction Multiplex geometry Applications Robustness Summary & outlook Distance between pairs of nodes in one layer is an indicator of the connection probability in another layer Hyperbolic distance in IPv4 Connectionprob.inIPv6 P(2|1) 0 5 10 15 20 25 30 35 40 10-4 10-3 10-2 10-1 100 21

39. Introduction Multiplex geometry Applications Robustness Summary & outlook Distance between pairs of nodes in one layer is an indicator of the connection probability in another layer Hyperbolic distance in IPv4 Connectionprob.inIPv6 P(2|1) 0 5 10 15 20 25 30 35 40 10-4 10-3 10-2 10-1 100 Pran(2|1) 21

40. Introduction Multiplex geometry Applications Robustness Summary & outlook Distance between pairs of nodes in one layer is an indicator of the connection probability in another layer Hyperbolic distance in IPv4 Connectionprob.inIPv6 P(2|1) 0 5 10 15 20 25 30 35 40 10-4 10-3 10-2 10-1 100 Pran(2|1) Geometric correlations enable precise trans-layer link prediction. 21

41. Navigation

42. Introduction Multiplex geometry Applications Robustness Summary & outlook Mutual greedy routing allows efficient navigation using several network layers and metric spaces [Credits: Marian Boguna] Forward message to contact closest to target in metric space Delivery fails if message runs into a loop (define success rate P) 23

47. Introduction Multiplex geometry Applications Robustness Summary & outlook Mutual greedy routing allows efficient navigation using several network layers and metric spaces [Credits: Marian Boguna] Forward message to contact closest to target in metric space Delivery fails if message runs into a loop (define success rate P) Messages switch layers if contact has a closer neighbor in another layer 23

48. Introduction Multiplex geometry Applications Robustness Summary & outlook Geometric correlations determine the improvement of mutual greedy routing by increasing the number of layers Migaon factor: Number of failed message deliveries compared to single layer case reduced by a constant factor (independent of temperature parameter) Details: Nat. Phys. 12, 10761081 (2016) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.80 0.82 0.84 0.86 0.88 0.90 P 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.980 0.985 0.990 0.995 P Angular correlaons Radialcorrelaons Angular correlaons Radialcorrelaons T = 0.8 T = 0.1 24

49. Pattern formation

50. Introduction Multiplex geometry Applications Robustness Summary & outlook Geometric correlations can lead to the formation of coherent patterns among different layers GN ON +T+S C D Layer 1: Evolutionary games Stag Hunt, Prisoners Dilemma & imitation dynamics Layer 2: Social influence Voter model & bias towards cooperation Coupling: at each timestep, with probability (1 ) perform respective dynamics in each layer nodes copy their state from one layer to the other 26

51. Introduction Multiplex geometry Applications Robustness Summary & outlook Self-organization into clusters of cooperators only occurs if angular correlations are present 27

52. Interdependent systems Robustness

53. Introduction Multiplex geometry Applications Robustness Summary & outlook Robustness of multiplexes against targeted attacks: percolation properties as a proxy Order parameter: Mutually connected component (MCC) is largest fraction of nodes connected by a path in every layer using only nodes in the component Targeted attacks: - Remove nodes in order of their Ki = max(k (1) i , k (2) i ) (k (j) i degree in layer j = 1, 2) - Reevaluate Kis after each removal Control parameter: Fraction p of nodes that is present in the system 29

54. Introduction Multiplex geometry Applications Robustness Summary & outlook Robustness of multiplexes against targeted attacks: percolation properties as a proxy Order parameter: Mutually connected component (MCC) is largest fraction of nodes connected by a path in every layer using only nodes in the component Targeted attacks: - Remove nodes in order of their Ki = max(k (1) i , k (2) i ) (k (j) i degree in layer j = 1, 2) - Reevaluate Kis after each removal Control parameter: Fraction p of nodes that is present in the system Are real systems more robust than a random superposition of their constituent layer topologies? 29

55. Introduction Multiplex geometry Applications Robustness Summary & outlook Racall: Reshuffling of node IDs destroys correlations but preserves the single layer topologies Real system Reshuffled Reshuffled counterpart: Artificial multiplex that corresponds to a random superposition of the individual layer topologies of the real system. 30

56. Introduction Multiplex geometry Applications Robustness Summary & outlook Real systems are more robust than their reshuffled counterparts Original Reshued 0.80 0.85 0.90 0.95 1.00 0.00 0.25 0.50 0.75 1.00 p MCC Arxiv Original Reshued 0.5 0.6 0.7 0.8 0.9 1.0 0.00 0.25 0.50 0.75 1.00 p MCC CElegans Original Reshued 0.80 0.85 0.90 0.95 1.00 0.00 0.25 0.50 0.75 1.00 p MCC Drosophila Original Reshued 0.85 0.90 0.95 1.00 0.00 0.25 0.50 0.75 1.00 p MCC Sacc Pomb 32

57. Introduction Multiplex geometry Applications Robustness Summary & outlook Real systems are more robust than their reshuffled counterparts Original Reshued 0.80 0.85 0.90 0.95 1.00 0.00 0.25 0.50 0.75 1.00 p MCC Arxiv Original Reshued 0.5 0.6 0.7 0.8 0.9 1.0 0.00 0.25 0.50 0.75 1.00 p MCC CElegans Original Reshued 0.80 0.85 0.90 0.95 1.00 0.00 0.25 0.50 0.75 1.00 p MCC Drosophila Original Reshued 0.85 0.90 0.95 1.00 0.00 0.25 0.50 0.75 1.00 p MCC Sacc Pomb Why are real systems more robust than their reshuffled counterparts? 32

58. Introduction Multiplex geometry Applications Robustness Summary & outlook Geometric (similarity) correlations mitigate failures cascades and can lead to a smooth transition a) b) c) d) e) f) g) h) i) 33

59. Introduction Multiplex geometry Applications Robustness Summary & outlook Geometric (similarity) correlations mitigate failures cascades and can lead to a smooth transition a) b) c) d) e) f) g) h) i) Does the strength of similarity correlations predict the robustness of real systems? 33

60. Introduction Multiplex geometry Applications Robustness Summary & outlook Strength of geometric correlations predicts robustness of real multiplexes against targeted attacks Arx12Arx42 Arx41 Arx28 Phys12 Arx52 Arx15 Arx26 Internet Arx34 CE23 Phys13 Phys23 Sac13 Sac35 Sac23 Sac12 Dro12 CE13 Sac14 Sac24 Brain Rattus CE12 Sac34 AirTrain 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.2 0.4 0.6 0.8 1.0 NMI Datasets AirTrain Sac34 CE12 Raus Brain Sac24 Sac14 CE13 Dro12 Sac12 Sac23 Sac35 Sac13 Phys23 Phys13 CE23 Arx34 Internet Arx26 Arx15 Arx52 Phys12 Arx28 Arx41 Arx42 Arx12 Relative mitigation of vulnerability: = N Nrs N + Nrs NMI: Normalized mutual information, measures the strength of similarity (angular) correlations 34

61. Introduction Multiplex geometry Applications Robustness Summary & outlook Targeted attacks lead to catastrophic cascades even with degree correlations 35

62. Introduction Multiplex geometry Applications Robustness Summary & outlook Geometric correlations mitigate this extreme vulnerability and can lead to continuous transition 36

63. Introduction Multiplex geometry Applications Robustness Summary & outlook Edge overlap is not responsible for the mitigation effect id an rs un 103 104 105 106 100 101 102 103 104 N N N0.822 N0.829 -47.6+0.696 log[x]2.304 N-0.011 id an rs un 103 104 105 106 100 101 102 103 104 N Max2ndcomp id an rs un 103 104 105 106 10-1 100 N Relavecascadesize Largest cascade id an rs un 103 104 105 106 10-2 10-1 N Relavecascadesize 2nd largest cascade 37

64. Take home

65. Introduction Multiplex geometry Applications Robustness Summary & outlook Constituent network layers of real multiplexes exhibit significant hidden geometric correlationsFrameworkResultBasis Implications Network geometry Networks embedded in hyperbolic space Useful maps of complex systems Structure governed by joint hidden geometry Perfect navigation, increase robustness, ... Importance to consider geometric correlations Geometric correlations between layers Nat. Phys. 12, 10761081 Connection probability depends on distance Multiplexes not random combinations of layers Multiplex geometry Geometric correlations induce new behavior PRE 82, 036106 PRL 118, 218301 39

68. Marian Bogu M. Angeles Serrano Fragkiskos Papadopoulos Lubos Buzna Roberta Amato

69. References: Hidden geometric correlations in real multiplex networks Nat. Phys. 12, 10761081 (2016) K-K. Kleineberg, M. Bogu, M. A. Serrano, F. Papadopoulos Geometric correlations mitigate the extreme vulnerability of multiplex networks against targeted attacks PRL 118, 218301 (2017) K-K. Kleineberg, L. Buzna, F. Papadopoulos, M. Bogu, M. A. Serrano Interplay between social influence and competitive strategical games in multiplex networks Scientific Reports 7, 7087 (2017) R. Amato, A. Daz-Guilera, K-K. Kleineberg Kaj Kolja Kleineberg: [email protected] @KoljaKleineberg koljakleineberg.wordpress.com

70. References: Hidden geometric correlations in real multiplex networks Nat. Phys. 12, 10761081 (2016) K-K. Kleineberg, M. Bogu, M. A. Serrano, F. Papadopoulos Geometric correlations mitigate the extreme vulnerability of multiplex networks against targeted attacks PRL 118, 218301 (2017) K-K. Kleineberg, L. Buzna, F. Papadopoulos, M. Bogu, M. A. Serrano Interplay between social influence and competitive strategical games in multiplex networks Scientific Reports 7, 7087 (2017) R. Amato, A. Daz-Guilera, K-K. Kleineberg Kaj Kolja Kleineberg: [email protected] @KoljaKleineberg Slides koljakleineberg.wordpress.com

71. References: Hidden geometric correlations in real multiplex networks Nat. Phys. 12, 10761081 (2016) K-K. Kleineberg, M. Bogu, M. A. Serrano, F. Papadopoulos Geometric correlations mitigate the extreme vulnerability of multiplex networks against targeted attacks PRL 118, 218301 (2017) K-K. Kleineberg, L. Buzna, F. Papadopoulos, M. Bogu, M. A. Serrano Interplay between social influence and competitive strategical games in multiplex networks Scientific Reports 7, 7087 (2017) R. Amato, A. Daz-Guilera, K-K. Kleineberg Kaj Kolja Kleineberg: [email protected] @KoljaKleineberg Slides koljakleineberg.wordpress.com Data & Model

geometric correlations in multiplexes and how they make them more robust

Science