The document discusses how geometric correlations between layers in multiplex networks can mitigate their vulnerability to targeted attacks. It finds that while degree correlations provide some robustness to random failures, they do not prevent catastrophic cascades under targeted attacks. However, geometric or similarity correlations, which place similar nodes close together in an underlying metric space representing each layer, can significantly increase robustness to targeted attacks. This effect is demonstrated through a model incorporating such correlations, as well as analyses of real-world multiplex networks that exhibit stronger geometric correlations.

1. Geometric correlations mitigate the extreme vulnerability
of multiplex networks against targeted attacks
Kaj Kolja Kleineberg | kkleineberg@ethz.ch
@KoljaKleineberg | koljakleineberg.wordpress.com

2. Percolation reveals robustness of complex networks:
Scale-free networks are robust yet fragile
Network Science, Barabasi

3. Percolation in multiplex networks:
Discontinuous hybrid transition
Order parameter: Mutually connected component (MCC) is
largest fraction of nodes connected by a path in every layer using
only nodes in the component

4. Percolation in multiplex networks:
Discontinuous hybrid transition
Order parameter: Mutually connected component (MCC) is
largest fraction of nodes connected by a path in every layer using
only nodes in the component
M. Angeles Serrano et al. New J. Phys. 17 053033 (2015)

5. Percolation in multiplex networks:
Discontinuous hybrid transition
Order parameter: Mutually connected component (MCC) is
largest fraction of nodes connected by a path in every layer using
only nodes in the component
M. Angeles Serrano et al. New J. Phys. 17 053033 (2015)
Degree correlations mitigate catastrophic failure
cascades in mutual percolation.

6. 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 Ki’s after each removal
Control parameter: Fraction p of nodes that is present in the
system

7. 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 Ki’s after each removal
Control parameter: Fraction p of nodes that is present in the
system
How robust/fragile against targeted attacks are real
multiplexes?

8. 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.

10. Real systems are more robust
than their reshuffled counterparts
Original
Reshuffled
0.80 0.85 0.90 0.95 1.00
0.00
0.25
0.50
0.75
1.00
p
MCC
Arxiv
Original
Reshuffled
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
Reshuffled
0.80 0.85 0.90 0.95 1.00
0.00
0.25
0.50
0.75
1.00
p
MCC
Drosophila
Original
Reshuffled
0.85 0.90 0.95 1.00
0.00
0.25
0.50
0.75
1.00
p
MCC
Sacc Pomb

11. Real systems are more robust
than their reshuffled counterparts
Original
Reshuffled
0.80 0.85 0.90 0.95 1.00
0.00
0.25
0.50
0.75
1.00
p
MCC
Arxiv
Original
Reshuffled
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
Reshuffled
0.80 0.85 0.90 0.95 1.00
0.00
0.25
0.50
0.75
1.00
p
MCC
Drosophila
Original
Reshuffled
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?

12. Hypothesis: Geometric correlations are
responsible for the robustness of real
multiplexes against targeted attacks.

13. Hidden metric spaces underlying real complex networks
provide a fundamental explanation of their observed topologies
Nature Physics 5, 74–80 (2008)

14. 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.

15. Scale-free clustered networks
can be embedded into hyperbolic space
“Hyperbolic geometry of complex networks” [PRE 82, 036106]
Distribute:
ρ(r) ∝ e
1
2
(γ−1)r
Connect:
p(xij) =
1
1 + e
xij−R
2T
xij = cosh−1
(cosh ri cosh rj
− sinh ri sinh rj cos ∆θij)

16. Scale-free clustered networks
can be embedded into hyperbolic space
“Hyperbolic geometry of complex networks” [PRE 82, 036106]
Distribute:
ρ(r) ∝ e
1
2
(γ−1)r
Connect:
p(xij) =
1
1 + e
xij−R
2T
xij = cosh−1
(cosh ri cosh rj
− sinh ri sinh rj cos ∆θij)

17. Scale-free clustered networks
can be embedded into hyperbolic space
“Hyperbolic geometry of complex networks” [PRE 82, 036106]
Distribute:
ρ(r) ∝ e
1
2
(γ−1)r
Connect:
p(xij) =
1
1 + e
xij−R
2T
xij = cosh−1
(cosh ri cosh rj
− sinh ri sinh rj cos ∆θij)
Real networks can be embedded into hyperbolic
space by inverting the model.

18. 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

19. 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

20. 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

21. 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

22. Metric spaces underlying different layers
of real multiplexes could be correlated

23. Metric spaces underlying different layers
of real multiplexes could be correlated

24. Metric spaces underlying different layers
of real multiplexes could be correlated

25. Metric spaces underlying different layers
of real multiplexes could be correlated

26. Metric spaces underlying different layers
of real multiplexes could be correlated
Uncorrelated

27. Metric spaces underlying different layers
of real multiplexes could be correlated
Uncorrelated Correlated

28. Metric spaces underlying different layers
of real multiplexes could be correlated
Uncorrelated Correlated
Are there metric correlations in real multiplex
networks?

29. Radial and angular coordinates are correlated
between different layers in many real multiplexes
Degreecorrelations
Random superposition
of constituent layers

30. Radial and angular coordinates are correlated
between different layers in many real multiplexes
Degreecorrelations
Random superposition
of constituent layers

31. Radial and angular coordinates are correlated
between different layers in many real multiplexes
Degreecorrelations
Random superposition
of constituent layers
What is the impact of geometric correlations for the
robustness of multiplexes against targeted attacks?

32. Model with geometric (similarity) correlations
behaves similar to real multiplexes
(with similarity correla�ons)
(without similarity correla�ons)
Model

33. Largest cascade size decreases with system size
only if similarity correlations are present

34. Without similarity correlations the removal of a single node
triggers a large cascade
∆N: Number of nodes whose removal reduces size M of MCC
from 0.4M to less than M0.4.
[Science 323, 5920, pp. 1453-1455 (2009)]

35. Distribution of component sizes behaves very different
depending on the existence of similarity correlations
Without similarity correla�ons With similarity correla�ons

36. Scaling of the size of the second largest component
for the case with similarity correlations

37. 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
Ω
Datasets
AirTrain
Sac34
CE12
Ra�us
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:
Ω =
∆N − ∆Nrs
∆N + ∆Nrs
NMI: Normalized mutual
information, measures the
strength of similarity (angular)
correlations

38. Targeted attacks lead to catastrophic cascades
even with degree correlations

39. Geometric correlations mitigate this extreme vulnerability
and can lead to continuous transition

40. Edge overlap is not responsible
for the mitigation effect
id
an
rs
un
103
104
105
106
100
101
102
103
104
N
ΔN
∝ 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
Rela�vecascadesize
Largest cascade
id
an
rs
un
103
104
105
106
10-2
10-1
N
Rela�vecascadesize
2nd largest cascade

41. Geometric correlations can explain the robustness
of real multiplexes against targeted attacks
Summary:
- Multiplexes are vulnerable against random failures and
targeted attacks (discontinuous transition)
- Degree correlations mitigate vulnerability against random
failures (percolation), lead to continuous transition
- Degree correlations fail to mitigate vulnerability against
targeted attacks
- Geometric (similarity) correlations mitigate vulnerability
against targeted attacks, may lead to continuous transition

42. Reference:
»Geometric correlations mitigate the extreme vulnerability of multiplex
networks against targeted attacks«
PRL 118, 218301 (2017)
K-K. Kleineberg, L. Buzna, F. Papadopoulos, M. Boguñá, M. A. Serrano
Kaj Kolja Kleineberg:
• kkleineberg@ethz.ch
• @KoljaKleineberg
• koljakleineberg.wordpress.com

43. Reference:
»Geometric correlations mitigate the extreme vulnerability of multiplex
networks against targeted attacks«
PRL 118, 218301 (2017)
K-K. Kleineberg, L. Buzna, F. Papadopoulos, M. Boguñá, M. A. Serrano
Kaj Kolja Kleineberg:
• kkleineberg@ethz.ch
• @KoljaKleineberg ← Slides
• koljakleineberg.wordpress.com

44. Reference:
»Geometric correlations mitigate the extreme vulnerability of multiplex
networks against targeted attacks«
PRL 118, 218301 (2017)
K-K. Kleineberg, L. Buzna, F. Papadopoulos, M. Boguñá, M. A. Serrano
Kaj Kolja Kleineberg:
• kkleineberg@ethz.ch
• @KoljaKleineberg ← Slides
• koljakleineberg.wordpress.com ← Data & Model