The organization of constituent network layers to multiplex networks has recently attracted a lot of attention. Here, we show empirical evidence for the existence of relations between the layers of real multiplex networks that go beyond degree correlations. These relations consist of correlations in hidden metric spaces that underlie the observed topology. We discuss the impact and applications of these relations for trans-layer link prediction, community detection, navigation, game theory, and especially for the robustness of multiplex networks against random failures and targeted attacks. We show that these relations lead to fundamentally new behaviors, which emphasizes the importance to consider organizational principles of multiplex networks beyond degree correlations in future research.

1. Structure and dynamics of multiplex networks:
beyond degree correlations
Kaj Kolja Kleineberg | kkleineberg@ethz.ch
@KoljaKleineberg | koljakleineberg.wordpress.com

2. The World Economic Forum
Risks Interconnec�on Map

4. Introduction Multiplex geometry Applications and implications Summary & outlook
Multiplex networks can describe interdependencies
between different networked systems
Several networking layers
4

5. Introduction Multiplex geometry Applications and implications Summary & outlook
Multiplex networks can describe interdependencies
between different networked systems
Several networking layers
Same nodes exist in different
layers
4

6. Introduction Multiplex geometry Applications and implications Summary & outlook
Multiplex networks can describe interdependencies
between different networked systems
Several networking layers
Same nodes exist in different
layers
One-to-one mapping between
nodes in different layers
4

7. Introduction Multiplex geometry Applications and implications Summary & outlook
Multiplex networks can describe interdependencies
between different networked systems
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 one more!
Degree correlations and overlap have been studied extensively:
Nature Physics 8, 40–48 (2011); Phys. Rev. E 92, 032805 (2015); Phys. Rev.
Lett. 111, 058702 (2013); Phys. Rev. E 88, 052811 (2013); ...
4

8. Hidden metric spaces

9. Introduction Multiplex geometry Applications and implications Summary & outlook
Hidden metric spaces underlying real complex networks
provide a fundamental explanation of their observed topologies
Nature Physics 5, 74–80 (2008)
6

10. Introduction Multiplex geometry Applications and implications 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

11. Introduction Multiplex geometry Applications and implications Summary & outlook
Hyperbolic geometry emerges from Newtonian model
and similarity × popularity optimization in growing networks
S1
p(κ) ∝ κ−γ
7

12. Introduction Multiplex geometry Applications and implications Summary & outlook
Hyperbolic geometry emerges from Newtonian model
and similarity × popularity optimization in growing networks
S1
p(κ) ∝ κ−γ
r = 1
1+
[
d(θ,θ′)
µκκ′
]1/T
PRL 100, 078701
8

13. Introduction Multiplex geometry Applications and implications Summary & outlook
Hyperbolic geometry emerges from Newtonian model
and similarity × popularity optimization in growing networks
S1 H2
p(κ) ∝ κ−γ ri = R − 2 ln κi
κmin
r = 1
1+
[
d(θ,θ′)
µκκ′
]1/T
PRL 100, 078701
9

14. Introduction Multiplex geometry Applications and implications Summary & outlook
Hyperbolic geometry emerges from Newtonian model
and similarity × popularity optimization in growing networks
S1 H2
p(κ) ∝ κ−γ
ρ(r) ∝ e
1
2
(γ−1)(r−R)
r = 1
1+
[
d(θ,θ′)
µκκ′
]1/T
PRL 100, 078701
10

15. Introduction Multiplex geometry Applications and implications Summary & outlook
Hyperbolic geometry emerges from Newtonian model
and similarity × popularity optimization in growing networks
S1 H2
p(κ) ∝ κ−γ
ρ(r) ∝ e
1
2
(γ−1)(r−R)
r = 1
1+
[
d(θ,θ′)
µκκ′
]1/T p(xij) = 1
1+e
xij−R
2T
PRL 100, 078701 PRE 82, 036106
11

16. Introduction Multiplex geometry Applications and implications Summary & outlook
Hyperbolic geometry emerges from Newtonian model
and similarity × popularity optimization in growing networks
S1 H2 growing
p(κ) ∝ κ−γ
ρ(r) ∝ e
1
2
(γ−1)(r−R) t = 1, 2, 3 . . .
r = 1
1+
[
d(θ,θ′)
µκκ′
]1/T p(xij) = 1
1+e
xij−R
2T
mins∈[1...t−1] s · ∆θst
PRL 100, 078701 PRE 82, 036106 Nature 489, 537–540
12

17. Introduction Multiplex geometry Applications and implications Summary & outlook
Hyperbolic maps of complex networks:
Poincaré disk
Nature Communications 1, 62 (2010)
Polar coordinates:
ri : Popularity (degree)
θi : Similarity
Distance:
xij = cosh−1
(cosh ri cosh rj
− sinh ri sinh rj cos ∆θij)
Connection probability:
p(xij) =
1
1 + e
xij−R
2T
13

18. Introduction Multiplex geometry Applications and implications Summary & outlook
Hyperbolic maps of complex networks:
Poincaré disk
Internet IPv6 topology
Polar coordinates:
ri : Popularity (degree)
θi : Similarity
Distance:
xij = cosh−1
(cosh ri cosh rj
− sinh ri sinh rj cos ∆θij)
Connection probability:
p(xij) =
1
1 + e
xij−R
2T
13

19. Introduction Multiplex geometry Applications and implications Summary & outlook
Hyperbolic maps of complex networks:
Poincaré disk
Internet IPv6 topology
Polar coordinates:
ri : Popularity (degree)
θi : Similarity
Distance:
xij = cosh−1
(cosh ri cosh rj
− sinh ri sinh rj cos ∆θij)
Connection probability:
p(xij) =
1
1 + e
xij−R
2T
13

20. Introduction Multiplex geometry Applications and implications Summary & outlook
Hyperbolic maps of complex networks:
Poincaré disk
Internet IPv6 topology
Polar coordinates:
ri : Popularity (degree)
θi : Similarity
Distance:
xij = cosh−1
(cosh ri cosh rj
− sinh ri sinh rj cos ∆θij)
Connection probability:
p(xij) =
1
1 + e
xij−R
2T
13

21. Multiplex geometry

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

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

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

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

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

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

28. Introduction Multiplex geometry Applications and implications Summary & outlook
Metric spaces underlying different layers
of real multiplexes could be correlated
Uncorrelated Correlated
Are there metric correlations in real multiplexes,
and what is the impact?
15

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

31. Introduction Multiplex geometry Applications and implications 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

32. Communities

33. Introduction Multiplex geometry Applications and implications Summary & outlook
Sets of nodes simultaneously similar in both layers
are overabundant in real systems
Real system
0
π
2 π
θ1
0
π
2 π
θ2
100
200
Reshuffled
0
π
2 π
θ1
0
π
2 π
θ2
100
200
18

34. Introduction Multiplex geometry Applications and implications Summary & outlook
Sets of nodes simultaneously similar in both layers
are overabundant in real systems
Real system
0
π
2 π
θ1
0
π
2 π
θ2
100
200
Reshuffled
0
π
2 π
θ1
0
π
2 π
θ2
100
200
Angular correlations are related to
multidimensional communities.
18

35. Link prediction

36. Introduction Multiplex geometry Applications and implications 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
20

37. Introduction Multiplex geometry Applications and implications 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)
20

38. Introduction Multiplex geometry Applications and implications 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.
20

39. Navigation

40. Introduction Multiplex geometry Applications and implications 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)
22

41. Introduction Multiplex geometry Applications and implications 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)
22

42. Introduction Multiplex geometry Applications and implications 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)
22

43. Introduction Multiplex geometry Applications and implications 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)
22

44. Introduction Multiplex geometry Applications and implications 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)
22

45. Introduction Multiplex geometry Applications and implications 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
22

46. Introduction Multiplex geometry Applications and implications Summary & outlook
Geometric correlations determine the improvement of
mutual greedy routing by increasing the number of layers
Mi�ga�on factor: Number
of failed message deliveries
compared to single layer
case reduced by a constant
factor
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 correla�ons
Radialcorrela�ons
Angular correla�ons
Radialcorrela�ons
T = 0.8 T = 0.1
23

47. Interdependent systems
Robustness

48. Introduction Multiplex geometry Applications and implications Summary & outlook
Mutual percolation is a proxy of the vulnerability
of the system against random failures
Mutually connected component (MCC) is largest fraction of nodes
connected by a path in every layer using only nodes in the
component
25

49. Introduction Multiplex geometry Applications and implications Summary & outlook
Mutual percolation is a proxy of the vulnerability
of the system against random failures
Mutually connected component (MCC) is largest fraction of nodes
connected by a path in every layer using only nodes in the
component
Radial or angular correlations mitigate catastrophic
failure cascades in mutual percolation.
25

50. Introduction Multiplex geometry Applications and implications Summary & outlook
In real systems failures may not always be random,
but the result of targeted attacks
Targeted attacks:
- Rank nodes according to Ki = max(k
(1)
i , k
(2)
i ) (k
(j)
i degree in
layer j = 1, 2)
- Remove nodes with higher Ki first (undo ties at random)
- Reevaluate Ki’s after each removal
a)
bcd
c) d)b)
26

51. Introduction Multiplex geometry Applications and implications Summary & outlook
Strength of geometric correlations predicts robustness
of real multiplexes against targeted attacks
Model Geometric corr. & robustness
Angular correla�ons (NMI)
Robustnessrealvsreshuffled()
arXiv:1702.02246
27

52. Introduction Multiplex geometry Applications and implications Summary & outlook
Strength of geometric correlations predicts robustness
of real multiplexes against targeted attacks
Model Geometric corr. & robustness
Angular correla�ons (NMI)
Robustnessrealvsreshuffled()
arXiv:1702.02246
Only geometric correlations mitigate extreme
vulnerability against targeted attacks.
27

53. Pattern formation

54. Introduction Multiplex geometry Applications and implications Summary & outlook
Geometric correlations can lead to the formation
of coherent patterns among different layers
γ
β
GN
ON
+T+S
C D
Layer 1: Evolutionary games
Stag Hunt, Prisoner’s Dilemma
& imitation dynamics
Layer 2: Social influence
Voter model & bias towards
cooperation
Coupling: at each timestep, with probability
(1 − γ) perform respective dynamics in each layer
γ nodes copy their state from one layer to the other
29

55. Introduction Multiplex geometry Applications and implications Summary & outlook
Geometric correlations give rise to metastable state
of high polarization between groups of different strategies
1
2
1
2
3 3
Game layer Opinion layer
1 1
22
3 3
Game Opinion
30

56. Take home

57. Introduction Multiplex geometry Applications and implications 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, 1076–1081
Connection probability
depends on distance
Multiplexes not random
combinations of layers
Multiplex
geometry
Geometric correlations
induce new behavior
PRE 82, 036106 arXiv:1702.02246
32

58. Introduction Multiplex geometry Applications and implications 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, 1076–1081
Connection probability
depends on distance
Multiplexes not random
combinations of layers
Multiplex
geometry
Geometric correlations
induce new behavior
PRE 82, 036106 arXiv:1702.02246
32

59. Introduction Multiplex geometry Applications and implications 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, 1076–1081
Connection probability
depends on distance
Multiplexes not random
combinations of layers
Multiplex
geometry
Geometric correlations
induce new behavior
PRE 82, 036106 arXiv:1702.02246
32

60. References:
»Hidden geometric correlations in real multiplex networks«
Nat. Phys. 12, 1076–1081 (2016)
K-K. Kleineberg, M. Boguñá, M. A. Serrano, F. Papadopoulos
»Geometric correlations mitigate the extreme vulnerability of multiplex
networks against targeted attacks«
arXiv:1702.02246 (2017)
K-K. Kleineberg, L. Buzna, F. Papadopoulos, M. Boguñá, M. A. Serrano
»Interplay between social influence and competitive strategical games
in multiplex networks«
arXiv:1702.05952 (2017)
R. Amato, A. Díaz-Guilera, K-K. Kleineberg
Kaj Kolja Kleineberg:
• kkleineberg@ethz.ch
• @KoljaKleineberg
• koljakleineberg.wordpress.com

61. References:
»Hidden geometric correlations in real multiplex networks«
Nat. Phys. 12, 1076–1081 (2016)
K-K. Kleineberg, M. Boguñá, M. A. Serrano, F. Papadopoulos
»Geometric correlations mitigate the extreme vulnerability of multiplex
networks against targeted attacks«
arXiv:1702.02246 (2017)
K-K. Kleineberg, L. Buzna, F. Papadopoulos, M. Boguñá, M. A. Serrano
»Interplay between social influence and competitive strategical games
in multiplex networks«
arXiv:1702.05952 (2017)
R. Amato, A. Díaz-Guilera, K-K. Kleineberg
Kaj Kolja Kleineberg:
• kkleineberg@ethz.ch
• @KoljaKleineberg ← Slides & Model (soon)
• koljakleineberg.wordpress.com

62. References:
»Hidden geometric correlations in real multiplex networks«
Nat. Phys. 12, 1076–1081 (2016)
K-K. Kleineberg, M. Boguñá, M. A. Serrano, F. Papadopoulos
»Geometric correlations mitigate the extreme vulnerability of multiplex
networks against targeted attacks«
arXiv:1702.02246 (2017)
K-K. Kleineberg, L. Buzna, F. Papadopoulos, M. Boguñá, M. A. Serrano
»Interplay between social influence and competitive strategical games
in multiplex networks«
arXiv:1702.05952 (2017)
R. Amato, A. Díaz-Guilera, K-K. Kleineberg
Kaj Kolja Kleineberg:
• kkleineberg@ethz.ch
• @KoljaKleineberg ← Slides & Model (soon)
• koljakleineberg.wordpress.com ← Slides & Model