Geometric correlations mitigate the extreme vulnerability of multiplex networks against targeted attacks
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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.
![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)](https://image.slidesharecdn.com/netsci18percolation-180613132456/85/Geometric-correlations-mitigate-the-extreme-vulnerability-of-multiplex-networks-against-targeted-attacks-15-320.jpg)
![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)](https://image.slidesharecdn.com/netsci18percolation-180613132456/85/Geometric-correlations-mitigate-the-extreme-vulnerability-of-multiplex-networks-against-targeted-attacks-16-320.jpg)
![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.](https://image.slidesharecdn.com/netsci18percolation-180613132456/85/Geometric-correlations-mitigate-the-extreme-vulnerability-of-multiplex-networks-against-targeted-attacks-17-320.jpg)
![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)]](https://image.slidesharecdn.com/netsci18percolation-180613132456/85/Geometric-correlations-mitigate-the-extreme-vulnerability-of-multiplex-networks-against-targeted-attacks-34-320.jpg)
![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](https://image.slidesharecdn.com/netsci18percolation-180613132456/85/Geometric-correlations-mitigate-the-extreme-vulnerability-of-multiplex-networks-against-targeted-attacks-40-320.jpg)
Geometric correlations mitigate the extreme vulnerability of multiplex networks against targeted attacks
- 1. Geometric correlations mitigate the extreme vulnerability of multiplex networks against targeted attacks Kaj Kolja Kleineberg | [email protected] @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: • [email protected] • @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: • [email protected] • @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: • [email protected] • @KoljaKleineberg ← Slides • koljakleineberg.wordpress.com ← Data & Model