structure and dynamics of multiplex networks: beyond degree correlations
TRANSCRIPT
Structure and dynamics of multiplex networks:beyond degree correlations
Kaj Kolja Kleineberg | [email protected] @KoljaKleineberg | koljakleineberg.wordpress.com
The World Economic ForumRisks Interconnec�on Map
Introduction Multiplex geometry Applications and implications Summary & outlook
Multiplex networks can describe interdependenciesbetween different networked systems
Several networking layers
Same nodes exist in differentlayers
One-to-one mapping betweennodes 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
Introduction Multiplex geometry Applications and implications Summary & outlook
Multiplex networks can describe interdependenciesbetween different networked systems
Several networking layers
Same nodes exist in differentlayers
One-to-one mapping betweennodes 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
Introduction Multiplex geometry Applications and implications Summary & outlook
Multiplex networks can describe interdependenciesbetween different networked systems
Several networking layers
Same nodes exist in differentlayers
One-to-one mapping betweennodes 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
Introduction Multiplex geometry Applications and implications Summary & outlook
Multiplex networks can describe interdependenciesbetween different networked systems
Several networking layers
Same nodes exist in differentlayers
One-to-one mapping betweennodes 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
Hidden metric spaces
Introduction Multiplex geometry Applications and implications Summary & outlook
Hidden metric spaces underlying real complex networksprovide a fundamental explanation of their observed topologies
Nature Physics 5, 74–80 (2008)
6
Introduction Multiplex geometry Applications and implications Summary & outlook
Hidden metric spaces underlying real complex networksprovide a fundamental explanation of their observed topologies
We can infer the coordinates of nodes embedded inhidden metric spaces by inverting models.
6
Introduction Multiplex geometry Applications and implications Summary & outlook
Hyperbolic geometry emerges from Newtonian modeland similarity× popularity optimization in growing networks
S1
p(κ) ∝ κ−γ
7
Introduction Multiplex geometry Applications and implications Summary & outlook
Hyperbolic geometry emerges from Newtonian modeland similarity× popularity optimization in growing networks
S1
p(κ) ∝ κ−γ
r = 1
1+[d(θ,θ′)µκκ′
]1/TPRL 100, 078701
8
Introduction Multiplex geometry Applications and implications Summary & outlook
Hyperbolic geometry emerges from Newtonian modeland 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/TPRL 100, 078701
9
Introduction Multiplex geometry Applications and implications Summary & outlook
Hyperbolic geometry emerges from Newtonian modeland similarity× popularity optimization in growing networks
S1 H2
p(κ) ∝ κ−γ ρ(r) ∝ e12(γ−1)(r−R)
r = 1
1+[d(θ,θ′)µκκ′
]1/TPRL 100, 078701
10
Introduction Multiplex geometry Applications and implications Summary & outlook
Hyperbolic geometry emerges from Newtonian modeland similarity× popularity optimization in growing networks
S1 H2
p(κ) ∝ κ−γ ρ(r) ∝ e12(γ−1)(r−R)
r = 1
1+[d(θ,θ′)µκκ′
]1/T p(xij) =1
1+exij−R
2T
PRL 100, 078701 PRE 82, 036106
11
Introduction Multiplex geometry Applications and implications Summary & outlook
Hyperbolic geometry emerges from Newtonian modeland similarity× popularity optimization in growing networks
S1 H2 growing
p(κ) ∝ κ−γ ρ(r) ∝ e12(γ−1)(r−R) t = 1, 2, 3 . . .
r = 1
1+[d(θ,θ′)µκκ′
]1/T p(xij) =1
1+exij−R
2T
mins∈[1...t−1] s ·∆θst
PRL 100, 078701 PRE 82, 036106 Nature 489, 537–540
12
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 + exij−R
2T
13
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 + exij−R
2T
13
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 + exij−R
2T
13
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 + exij−R
2T
13
Multiplex geometry
Introduction Multiplex geometry Applications and implications Summary & outlook
Metric spaces underlying different layersof real multiplexes could be correlated
Uncorrelated Correlated
Are theremetric correlations in real multiplexes,and what is the impact?
15
Introduction Multiplex geometry Applications and implications Summary & outlook
Metric spaces underlying different layersof real multiplexes could be correlated
Uncorrelated Correlated
Are theremetric correlations in real multiplexes,and what is the impact?
15
Introduction Multiplex geometry Applications and implications Summary & outlook
Metric spaces underlying different layersof real multiplexes could be correlated
Uncorrelated Correlated
Are theremetric correlations in real multiplexes,and what is the impact?
15
Introduction Multiplex geometry Applications and implications Summary & outlook
Metric spaces underlying different layersof real multiplexes could be correlated
Uncorrelated Correlated
Are theremetric correlations in real multiplexes,and what is the impact?
15
Introduction Multiplex geometry Applications and implications Summary & outlook
Metric spaces underlying different layersof real multiplexes could be correlated
Uncorrelated
Correlated
Are theremetric correlations in real multiplexes,and what is the impact?
15
Introduction Multiplex geometry Applications and implications Summary & outlook
Metric spaces underlying different layersof real multiplexes could be correlated
Uncorrelated Correlated
Are theremetric correlations in real multiplexes,and what is the impact?
15
Introduction Multiplex geometry Applications and implications Summary & outlook
Metric spaces underlying different layersof real multiplexes could be correlated
Uncorrelated Correlated
Are theremetric correlations in real multiplexes,and what is the impact?
15
Introduction Multiplex geometry Applications and implications Summary & outlook
Radial and angular coordinates are correlatedbetween different layers in many real multiplexes
Deg
ree
corr
elat
ions
Random superpositionof constituent layers
What is the impact of the discovered geometriccorrelations?
16
Introduction Multiplex geometry Applications and implications Summary & outlook
Radial and angular coordinates are correlatedbetween different layers in many real multiplexes
Deg
ree
corr
elat
ions
Random superpositionof constituent layers
What is the impact of the discovered geometriccorrelations?
16
Introduction Multiplex geometry Applications and implications Summary & outlook
Radial and angular coordinates are correlatedbetween different layers in many real multiplexes
Deg
ree
corr
elat
ions
Random superpositionof constituent layers
What is the impact of the discovered geometriccorrelations?
16
Communities
Introduction Multiplex geometry Applications and implications Summary & outlook
Sets of nodes simultaneously similar in both layersare overabundant in real systems
Real system
0
π
2π
θ1
0
π
2π
θ2
100
200
Reshuffled
0
π
2π
θ1
0
π
2π
θ2
100
200
Angular correlations are related tomultidimensional communities.
18
Introduction Multiplex geometry Applications and implications Summary & outlook
Sets of nodes simultaneously similar in both layersare overabundant in real systems
Real system
0
π
2π
θ1
0
π
2π
θ2
100
200
Reshuffled
0
π
2π
θ1
0
π
2π
θ2
100
200
Angular correlations are related tomultidimensional communities.
18
Link prediction
Introduction Multiplex geometry Applications and implications Summary & outlook
Distance between pairs of nodes in one layer isan indicator of the connection probability in another layer
Hyperbolic distance in IPv4
Co
nn
ecti
on
pro
b. i
n IP
v6
P(2|1)
0 5 10 15 20 25 30 35 40
10-4
10-3
10-2
10-1
100
Geometric correlations enable precise trans-layerlink prediction.
20
Introduction Multiplex geometry Applications and implications Summary & outlook
Distance between pairs of nodes in one layer isan indicator of the connection probability in another layer
Hyperbolic distance in IPv4
Co
nn
ecti
on
pro
b. i
n IP
v6
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-layerlink prediction.
20
Introduction Multiplex geometry Applications and implications Summary & outlook
Distance between pairs of nodes in one layer isan indicator of the connection probability in another layer
Hyperbolic distance in IPv4
Co
nn
ecti
on
pro
b. i
n IP
v6
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-layerlink prediction.
20
Navigation
Introduction Multiplex geometry Applications and implications Summary & outlook
Mutual greedy routing allows efficient navigationusing several network layers and metric spaces
[Credits: Marian Boguna]
Forward messageto contact closestto target in metric
space
Delivery failsif message runs into
a loop (definesuccess rate P )
Messages switchlayers if contact hasa closer neighbor in
another layer
22
Introduction Multiplex geometry Applications and implications Summary & outlook
Mutual greedy routing allows efficient navigationusing several network layers and metric spaces
[Credits: Marian Boguna]
Forward messageto contact closestto target in metric
space
Delivery failsif message runs into
a loop (definesuccess rate P )
Messages switchlayers if contact hasa closer neighbor in
another layer
22
Introduction Multiplex geometry Applications and implications Summary & outlook
Mutual greedy routing allows efficient navigationusing several network layers and metric spaces
[Credits: Marian Boguna]
Forward messageto contact closestto target in metric
space
Delivery failsif message runs into
a loop (definesuccess rate P )
Messages switchlayers if contact hasa closer neighbor in
another layer
22
Introduction Multiplex geometry Applications and implications Summary & outlook
Mutual greedy routing allows efficient navigationusing several network layers and metric spaces
[Credits: Marian Boguna]
Forward messageto contact closestto target in metric
space
Delivery failsif message runs into
a loop (definesuccess rate P )
Messages switchlayers if contact hasa closer neighbor in
another layer
22
Introduction Multiplex geometry Applications and implications Summary & outlook
Mutual greedy routing allows efficient navigationusing several network layers and metric spaces
[Credits: Marian Boguna]
Forward messageto contact closestto target in metric
space
Delivery failsif message runs into
a loop (definesuccess rate P )
Messages switchlayers if contact hasa closer neighbor in
another layer
22
Introduction Multiplex geometry Applications and implications Summary & outlook
Mutual greedy routing allows efficient navigationusing several network layers and metric spaces
[Credits: Marian Boguna]
Forward messageto contact closestto target in metric
space
Delivery failsif message runs into
a loop (definesuccess rate P )
Messages switchlayers if contact hasa closer neighbor in
another layer22
Introduction Multiplex geometry Applications and implications Summary & outlook
Geometric correlations determine the improvement ofmutual greedy routing by increasing the number of layers
Mi�ga�on factor: Number of failed message deliveriescompared to single layer case reduced by a constant factor
0.0 0.2 0.4 0.6 0.8 1.00.0
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 1.00.0
0.2
0.4
0.6
0.8
1.0
0.980
0.985
0.990
0.995
P
Angular correla�ons
Rad
ial c
orre
la�
ons
Angular correla�ons
Rad
ial c
orre
la�
ons
T = 0.8 T = 0.1
23
Interdependent systems
Robustness
Introduction Multiplex geometry Applications and implications Summary & outlook
Mutual percolation is a proxy of the vulnerabilityof the system against random failures
Mutually connected component (MCC) is largest fraction of nodesconnected by a path in every layer using only nodes in thecomponent
Radial or angular correlationsmitigate catastrophicfailure cascades in mutual percolation.
25
Introduction Multiplex geometry Applications and implications Summary & outlook
Mutual percolation is a proxy of the vulnerabilityof the system against random failures
Mutually connected component (MCC) is largest fraction of nodesconnected by a path in every layer using only nodes in thecomponent
Radial or angular correlationsmitigate catastrophicfailure cascades in mutual percolation.
25
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 toKi = max(k(1)i , k
(2)i ) (k(j)i degree in
layer j = 1, 2)
- Remove nodes with higherKi first (undo ties at random)
- ReevaluateKi’s after each removal
a)
bcd
c) d)b)
26
Introduction Multiplex geometry Applications and implications Summary & outlook
Strength of geometric correlations predicts robustnessof real multiplexes against targeted attacks
Model Geometric corr. & robustness
Angular correla�ons (NMI)Robu
stne
ss re
al v
s re
shuffl
ed (
)
arXiv:1702.02246
Only geometric correlationsmitigate extremevulnerability against targeted attacks.
27
Introduction Multiplex geometry Applications and implications Summary & outlook
Strength of geometric correlations predicts robustnessof real multiplexes against targeted attacks
Model Geometric corr. & robustness
Angular correla�ons (NMI)Robu
stne
ss re
al v
s re
shuffl
ed (
)
arXiv:1702.02246
Only geometric correlationsmitigate extremevulnerability against targeted attacks.
27
Pattern formation
Introduction Multiplex geometry Applications and implications Summary & outlook
Geometric correlations can lead to the formationof coherent patterns among different layers
γ
β
GN
ON
+T+S
C D
Layer 1: Evolutionary gamesStag Hunt, Prisoner’s Dilemma& imitation dynamics
Layer 2: Social influenceVoter model & bias towardscooperation
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
Introduction Multiplex geometry Applications and implications Summary & outlook
Geometric correlations give rise to metastable stateof 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
Take home
Introduction Multiplex geometry Applications and implications Summary & outlook
Constituent network layers of real multiplexesexhibit significant hidden geometric correlations
Fram
ewor
kR
esul
tB
asis
ImplicationsNetworkgeometry
Networks embedded in hyperbolic space
Useful maps ofcomplex systems
Structure governed byjoint hidden geometry
Perfect navigation,increase robustness, ...
Importance to considergeometric correlations
Geometric correlationsbetween layers
Nat. Phys. 12, 1076–1081
Connection probabilitydepends on distance
Multiplexes not randomcombinations of layers
Multiplexgeometry
Geometric correlationsinduce new behavior
PRE 82, 036106 arXiv:1702.0224632
Introduction Multiplex geometry Applications and implications Summary & outlook
Constituent network layers of real multiplexesexhibit significant hidden geometric correlations
Fram
ewor
kR
esul
tB
asis
ImplicationsNetworkgeometry
Networks embedded in hyperbolic space
Useful maps ofcomplex systems
Structure governed byjoint hidden geometry
Perfect navigation,increase robustness, ...
Importance to considergeometric correlations
Geometric correlationsbetween layers
Nat. Phys. 12, 1076–1081
Connection probabilitydepends on distance
Multiplexes not randomcombinations of layers
Multiplexgeometry
Geometric correlationsinduce new behavior
PRE 82, 036106 arXiv:1702.0224632
Introduction Multiplex geometry Applications and implications Summary & outlook
Constituent network layers of real multiplexesexhibit significant hidden geometric correlations
Fram
ewor
kR
esul
tB
asis
ImplicationsNetworkgeometry
Networks embedded in hyperbolic space
Useful maps ofcomplex systems
Structure governed byjoint hidden geometry
Perfect navigation,increase robustness, ...
Importance to considergeometric correlations
Geometric correlationsbetween layers
Nat. Phys. 12, 1076–1081
Connection probabilitydepends on distance
Multiplexes not randomcombinations of layers
Multiplexgeometry
Geometric correlationsinduce new behavior
PRE 82, 036106 arXiv:1702.0224632
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 multiplexnetworks 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 gamesin multiplex networks«arXiv:1702.05952 (2017)R. Amato, A. Díaz-Guilera, K-K. Kleineberg
Kaj Kolja Kleineberg:
• @KoljaKleineberg
• koljakleineberg.wordpress.com
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 multiplexnetworks 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 gamesin multiplex networks«arXiv:1702.05952 (2017)R. Amato, A. Díaz-Guilera, K-K. Kleineberg
Kaj Kolja Kleineberg:
• @KoljaKleineberg← Slides & Model (soon)
• koljakleineberg.wordpress.com
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 multiplexnetworks 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 gamesin multiplex networks«arXiv:1702.05952 (2017)R. Amato, A. Díaz-Guilera, K-K. Kleineberg
Kaj Kolja Kleineberg:
• @KoljaKleineberg← Slides & Model (soon)
• koljakleineberg.wordpress.com← Slides & Model