hidden geometric correlations in real multiplex networks
TRANSCRIPT
Correlationsin real
MULTIPLEXES
HIDDEN GEOMETRIC
Kaj Kolja KLEINEBERGUniversitat de Barcelona
You have no message
routingthe problem of
map relies on underlying
metric space
real systems interact:
multiplexes
between maps of layers?Relation
Hidden metric spaces underlying real complex networksprovide a fundamental explanation of their observed topologies
Nature Physics 5, 74–80 (2008)
Hidden metric spaces underlying real complex networksprovide a fundamental explanation of their observed topologies
How can we infer the coordinates of nodes embeddedin hidden metric spaces?
Hyperbolic geometry emerges from Newtonian modeland similarity× popularity optimization in growing network
S1
H2 growing
1
2
3
p(κ) ∝ κ−γ
ri = R− 2 ln κi
κmin t = 1, 2, 3 . . .
r = 1
1+[d(θ,θ′)µκκ′
]1/T
p(xij) =1
1+exij−R
2T
mins [s×∆θst]
PRL 100, 078701
PRE 82, 036106 Nature 489, 537–540
Hyperbolic geometry emerges from Newtonian modeland similarity× popularity optimization in growing network
S1 H2
growing
1
2
3
p(κ) ∝ κ−γ ri = R− 2 ln κi
κmin
t = 1, 2, 3 . . .
r = 1
1+[d(θ,θ′)µκκ′
]1/T p(xij) =1
1+exij−R
2T
mins [s×∆θst]
PRL 100, 078701 PRE 82, 036106
Nature 489, 537–540
Hyperbolic geometry emerges from Newtonian modeland similarity× popularity optimization in growing network
S1 H2 growing
1
2
3
p(κ) ∝ κ−γ ri = R− 2 ln κi
κmin t = 1, 2, 3 . . .
r = 1
1+[d(θ,θ′)µκκ′
]1/T p(xij) =1
1+exij−R
2T
mins [s×∆θst]
PRL 100, 078701 PRE 82, 036106 Nature 489, 537–540
Constituent network layers of real multiplex systemsare embedded into separate hyperbolic spaces
Invert model→ embed real networks: PRE 92, 022807 (2015)
Internet Air and train Drosophila
C. Elegans Human brain arXiv
Rattus Physicians SacchPomb
Constituent network layers of real multiplex systemsare embedded into separate hyperbolic spaces
Invert model→ embed real networks: PRE 92, 022807 (2015)
Internet Air and train Drosophila
C. Elegans Human brain arXiv
Rattus Physicians SacchPomb
Constituent network layers of real multiplex systemsare embedded into separate hyperbolic spaces
Internet IPv4 network Internet IPv6 network
Constituent network layers of real multiplex systemsare embedded into separate hyperbolic spaces
Internet IPv4 network Internet IPv6 network
Are coordinates of same nodes in different layerscorrelated?
Radial and angular coordinates are correlatedbetween different layers in many real multiplexes
0
π
2π
θ1
0
π
2π
θ2
50100150200
Multiplexes are not random superpositions of layersbut instead are dictated by geometric correlations.
Radial and angular coordinates are correlatedbetween different layers in many real multiplexes
0
π
2π
θ1
0
π
2π
θ2
50100150200
Multiplexes are not random superpositions of layersbut instead are dictated by geometric correlations.
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.
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.
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.
navigation?How do geometric correlations
affect
Greedy routing in single network using hyperbolic spaceallows efficient navigation relying only on local knowledge
Greedy routing in single network using hyperbolic spaceallows efficient navigation relying only on local knowledge
Greedy routing in single network using hyperbolic spaceallows efficient navigation relying only on local knowledge
Greedy routing in single network using hyperbolic spaceallows efficient navigation relying only on local knowledge
Greedy routing in single network using hyperbolic spaceallows efficient navigation relying only on local knowledge
Greedy routing in single network using hyperbolic spaceallows efficient navigation relying only on local knowledge
Mutual greedy routing allows efficient navigationusing several network layers and metric spaces
Mutual greedy routing allows efficient navigationusing several network layers and metric spaces
Mutual greedy routing allows efficient navigationusing several network layers and metric spaces
Mutual greedy routing allows efficient navigationusing several network layers and metric spaces
Mutual greedy routing allows efficient navigationusing several network layers and metric spaces
Mutual greedy routing allows efficient navigationusing several network layers and metric spaces
Mutual greedy routing allows efficient navigationusing several network layers and metric spaces
Mutual greedy routing allows efficient navigationusing several network layers and metric spaces
How do geometriccorrelations affectperformance?
Do more layersimprove
performance?
How closeto the optimum isthe Internet?
Develop a model with tunable geometriccorrelations preserving constituent layer topologies.
Mutual greedy routing allows efficient navigationusing several network layers and metric spaces
How do geometriccorrelations affectperformance?
Do more layersimprove
performance?
How closeto the optimum isthe Internet?
Develop a model with tunable geometriccorrelations preserving constituent layer topologies.
Mutual greedy routing allows efficient navigationusing several network layers and metric spaces
How do geometriccorrelations affectperformance?
Do more layersimprove
performance?
How closeto the optimum isthe Internet?
Develop a model with tunable geometriccorrelations preserving constituent layer topologies.
Mutual greedy routing allows efficient navigationusing several network layers and metric spaces
How do geometriccorrelations affectperformance?
Do more layersimprove
performance?
How closeto the optimum isthe Internet?
Develop a model with tunable geometriccorrelations preserving constituent layer topologies.
Geometric multiplex model allows to tunecorrelations independently from layer topologies
Constituent layertopologies by
hyperbolic model
Radial correlationscontrolled byν ∈ [0, 1]
Angular corr.controlled byg ∈ [0, 1]
Geometric multiplex model allows to tunecorrelations independently from layer topologies
Constituent layertopologies by
hyperbolic model
Radial correlationscontrolled byν ∈ [0, 1]
Angular corr.controlled byg ∈ [0, 1]
Geometric multiplex model allows to tunecorrelations independently from layer topologies
Constituent layertopologies by
hyperbolic model
Radial correlationscontrolled byν ∈ [0, 1]
Angular corr.controlled byg ∈ [0, 1]
Geometric correlations determine the improvement ofmutual greedy routing by increasing the number of layers
0.0 0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0Hyperbolic routing
0.980
0.985
0.990
0.995
P
1 2 3 40
2
4
6
Layers
Mit
igat
ion
fact
or
opt. correlateduncorrelated
Angular correlations
Rad
ial c
orr
elat
ion
s
Reduction of failure rate
Routing is perfected by using many networkssimultaneously if correlations are present.
Geometric correlations determine the improvement ofmutual greedy routing by increasing the number of layers
0.0 0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0Hyperbolic routing
0.980
0.985
0.990
0.995
P
1 2 3 40
2
4
6
Layers
Mit
igat
ion
fact
or
opt. correlateduncorrelated
Angular correlations
Rad
ial c
orr
elat
ion
s
Reduction of failure rate
Routing is perfected by using many networkssimultaneously if correlations are present.
Geometric correlations determine the improvement ofmutual greedy routing by increasing the number of layers
0.0 0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0Hyperbolic routing
0.980
0.985
0.990
0.995
P
1 2 3 40
2
4
6
Layers
Mit
igat
ion
fact
or
opt. correlateduncorrelated
Angular correlations
Rad
ial c
orr
elat
ion
s
Reduction of failure rate
Routing is perfected by using many networkssimultaneously if correlations are present.
Internet?Do correlations help to navigate
the real
Internet multiplex model allows to study the performanceof mutual greedy routing for arbitrary correlations
Real Model
• Node only in IPv4 • Node in IPv4 and IPv6
Existing correlations in the Internet multiplex increaseperformance of mutual greedy routing significantly
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
g
νAngular
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
gν
Hyperbolic
0.63 0.67 0.71 0.75
Success rate
0.85 0.91
Success rate
uncorr. emp. opt. uncorr. emp. opt.
0.88
Summary: Geometry of multiplex networks can makeinteracting decentralized systems perfectly navigable
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
g
ν
0.80
0.82
0.84
0.86
0.88
0.90
Fram
ewor
k
0
π
2 π
θ1
0
π
2 π
θ2
0.20.4
Res
ult
Bas
is
Navigation
Mutual greedy routing(local knowledge)
Geometriccorrelations
Layers embedded inhyperbolic space
Multidimensionalcommunity structure
Precise trans-layerlink prediction
Correlations increaserouting performance
Many correlated layersperfect routing
Distances between nodes are correlated
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)
Nature Physics, doi: 10.1038/NPHYS3782
Node coordinatesare correlated
Multiplexes not randomcombinations of layers
Multiplexorganization
Summary: Geometry of multiplex networks can makeinteracting decentralized systems perfectly navigable
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
g
ν
0.80
0.82
0.84
0.86
0.88
0.90
Fram
ewor
k
0
π
2 π
θ1
0
π
2 π
θ2
0.20.4
Res
ult
Bas
is
Navigation
Mutual greedy routing(local knowledge)
Geometriccorrelations
Layers embedded inhyperbolic space
Multidimensionalcommunity structure
Precise trans-layerlink prediction
Correlations increaserouting performance
Many correlated layersperfect routing
Distances between nodes are correlated
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)
Nature Physics, doi: 10.1038/NPHYS3782
Node coordinatesare correlated
Multiplexes not randomcombinations of layers
Multiplexorganization
Summary: Geometry of multiplex networks can makeinteracting decentralized systems perfectly navigable
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
g
ν
0.80
0.82
0.84
0.86
0.88
0.90
Fram
ewor
k
0
π
2 π
θ1
0
π
2 π
θ2
0.20.4
Res
ult
Bas
is
Navigation
Mutual greedy routing(local knowledge)
Geometriccorrelations
Layers embedded inhyperbolic space
Multidimensionalcommunity structure
Precise trans-layerlink prediction
Correlations increaserouting performance
Many correlated layersperfect routing
Distances between nodes are correlated
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)
Nature Physics, doi: 10.1038/NPHYS3782
Node coordinatesare correlated
Multiplexes not randomcombinations of layers
Multiplexorganization
Geometric correlations in real multiplexes:Reference and contact information
Thanks to: M. Boguñá, M. A. Serrano, F. Papadopoulos
Reference:
»Hidden geometric correlations in real multiplex networks«Nature Physics, doi: 10.1038/NPHYS3782K.-K. Kleineberg, M. Boguñá, M. A. Serrano, F. Papadopoulos
Kaj Kolja Kleineberg:
• @KoljaKleineberg
• koljakleineberg.wordpress.com
Geometric correlations in real multiplexes:Reference and contact information
Thanks to: M. Boguñá, M. A. Serrano, F. Papadopoulos
Reference:
»Hidden geometric correlations in real multiplex networks«Nature Physics, doi: 10.1038/NPHYS3782K.-K. Kleineberg, M. Boguñá, M. A. Serrano, F. Papadopoulos
Kaj Kolja Kleineberg:
• @KoljaKleineberg← Slides
• koljakleineberg.wordpress.com
IMAGE CREDITS
Compass: Martin FischMessage in bottle: SusanneNilssonOld globe: jayneanddCompass: Creative StallCompass (navigate): CreativeStallInternet router: Thomas Uebe
Train: Naomi Atkinsonfly (drosophila): Daan Kauwenbergworm (celegans): anbileru adalerubain network: parkjisuncoauthor: Matt Wassercommunity: Edward BoatmanLink: Rafaël Massé
Icons: thenounproject
Pictures: flickr