Geometric correlations in multiplexes and how they make them more robust
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- 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 | kkleineberg@ethz.ch @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
- 18. 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
- 19. 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
- 22. Introduction Multiplex geometry Applications Robustness
Summary & outlook Metric spaces underlying different layers of
real multiplexes could be correlated 15
- 23. Introduction Multiplex geometry Applications Robustness
Summary & outlook Metric spaces underlying different layers of
real multiplexes could be correlated 15
- 24. 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
- 43. 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
- 44. 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
- 45. 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
- 46. 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
- 66. 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
- 67. 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: kkleineberg@ethz.ch
@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: kkleineberg@ethz.ch
@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: kkleineberg@ethz.ch
@KoljaKleineberg Slides koljakleineberg.wordpress.com Data &
Model