The Hidden Geometry of Multiplex Networks @ Next Generation Network Analytics

The Hidden Geometry of Multiplex
Networks
January 5, 2018
Kaj Kolja Kleineberg
kkleineberg@ethz.ch • @KoljaKleineberg • www.koljakleineberg.wordpress.com
ETH Zurich

Multilayer networks: some reviews
Kivelä et al. Multilayer networks. J. Complex Netw. 2, 203–271
(2014)
Boccaletti el al. The structure and dynamics of multilayer
networks, Physics Reports 544, 1, pp. 1-122 (2014)

The World Economic Forum
Risks Interconnec�on Map

Multiplex: nodes are simultaneously present
in different network layers
Several networking layers

Same nodes exist in different
layers

layers
One-to-one mapping between
nodes in different layers

layers
One-to-one mapping between
nodes in different layers
Typical features: Edge overlap
& degree-degree correlations
& and geometric correlations!
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); ...

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

Hidden metric spaces underlying real complex networks
provide a fundamental explanation of their observed topologies
One can infer the coordinates of nodes embedded in
metric spaces by inverting models [PRE 92, 022807].

Hyperbolic geometry emerges from Newtonian model
and similarity × popularity optimization in growing networks
S1
p(κ) ∝ κ−γ

S1
p(κ) ∝ κ−γ
r = 1
1+
[
d(θ,θ′)
µκκ′
]1/T
PRL 100, 078701

S1 H2
p(κ) ∝ κ−γ ri = R − 2 ln κi
κmin
r = 1
1+
[
d(θ,θ′)
µκκ′
]1/T
PRL 100, 078701

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

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

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

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

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

Coordinate inference
via maximum likelihood estimation
Goal: to infer ri, θi for every node i = 1...N in a given (real)
network
Approach: Maximum a posteriori probability estimation (MAP)
L(αij|{ri, θi}) =
∏
j<i
p(xij)αij
(1 − p(xij)(1−αij)
Inferring ri:
ri ≈ R − 2 ln ki
Inferring θi:
Numberical maximization of the Likelihood L(αij|{ri, θi})
See F. Papadopoulos et al., PRE 92, 022807 (2015)

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

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

Embedd real multiplex systems
into separate hyperbolic spaces
Internet Air and train Drosophila
C. Elegans Human brain arXiv
Rattus Physicians SacchPomb
Embeddings: PRE 92, 022807 (2015)

Embedd real multiplex systems
into separate hyperbolic spaces
Details and data references: Nat. Phys. 12, 1076–1081 (2016)
Networks and embeddings:
koljakleineberg.wordpress.com/materials/

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

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?

Reshuffling of node IDs destroys correlations
but preserves the single layer topologies
Real system
Reshuffled counterpart: Artificial multiplex that corresponds to
a random superposition of the constituent layer topologies of the
real system.

Reshuffling of node IDs destroys correlations
Real system Reshuffled
a random superposition of the constituent layer topologies of the
real system.

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
NMI = 0.34 NMI ≈ 0
NMI =
I(X; Y )
max{I(X; X), I(Y ; Y )}
I(X; Y ) =
∫
Y
∫
X
p(x, y) ln
(
p(x, y)
p(x)p(y)
)
dxdy
Kraskov el al., Phys. Rev. E69, 066138 (2004)

Geographic breakdown of the
2D communities in the IPv4/IPv6 Internet

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

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)

Arx36
Arx26
Arx23Arx612
Phys12
Arx56
Arx78
Arx35
Arx67
Internet
Arx12
CE23
Phys13
Phys23
Sac13
Sac37
Sac23
Sac12
Sac17
Dro12
CE13
Sac15
Sac25
Brain
Rattus
CE12
Sac35
AirTrain
0.5 0.6 0.7 0.8 0.9 1.0
0.5
0.6
0.7
0.8
0.9
1.0
AUC binary
AUChyperbolic

Arx36
Arx26
Arx23Arx612
Phys12
Arx56
Arx78
Arx35
Arx67
Internet
Arx12
CE23
Phys13
Phys23
Sac13
Sac37
Sac23
Sac12
Sac17
Dro12
CE13
Sac15
Sac25
Brain
Rattus
CE12
Sac35
AirTrain
0.5 0.6 0.7 0.8 0.9 1.0
0.5
0.6
0.7
0.8
0.9
1.0
AUC binary
AUChyperbolic
Geometric correlations enable precise trans-layer
link prediction.

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)

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

Mutual greedy routing
in the Internet
Greedy routing using only IPv4 network: 90% success rate
Mutual greedy routing (MGR) using both: 95% success rate

in the Internet
Questions:
1. How do radial and angular correlations affect the
performance of MGR?
2. Is MGR always better than single-layer GR?
3. Performance of MGR with more layers?

in the Internet
Questions:
1. How do radial and angular correlations affect the
performance of MGR?
2. Is MGR always better than single-layer GR?
3. Performance of MGR with more layers?
We need a realistic model to construct multiplexes
with geometric correlations.

The geometric multiplex model (GMM)
constructs realistic multiplexes with geometric correlations
General idea:
1. Single-layer topologies shall obey the S1 or H2 model (in
general with different N, γ, ¯k, T)
2. Allow for correlations between the radial and angular
coordinates of nodes that simultaneously exist in different
layers
3. Allow tuning correlation strengths
4. Constraint: marginal coordinate distributions in each layer
should be the ones prescribed by the S1/H2 model
5. We work with the S1 model (then map to H2)

The geometric multiplex model (GMM)
constructs realistic multiplexes with geometric correlations
First, draw coordinates for layer 1:
p(κ1) ∝ κ−γ
1
θ1 uniform
Then, draw coordinates for layer 2 conditioned on layer 1:
p(κ2) ∝ Fν(κ2|κ1)
θ2 ∝ fg(θ2|θ1)
We can satisfy the constraint and tune the correlations by using
copulas.
Copulas are multivariate probability distributions used to describe
the dependence between random variables.

Constructing a two layer multiplex system
with geometric correlations
bivariate Gumbel-Hougaard copula
Cη(F1(κ1), F2(κ2)) = e−[(− ln F1(κ1))η+(− ln F2(κ2))η]1/η
.
with F1(κ1) = 1 − κ
(1−γ1)
1 κmin(γ1−1)
1 and F2 analogously.
Conditional CDF (η ≡ 1/(1 − ν)):
Fν(κ2|κ1, {γ1, γ2, κmin
1 , κmin
2 }) =
∂Cη(F1(κ1), F2(κ2))
∂κ1
1
ρi(κ1)
= e−(φ
1/(1−ν)
1 +φ
1/(1−ν)
2 )1−ν
[
φ
1/(1−ν)
1 + φ
1/(1−ν)
2
]−ν
×
φ
ν/(1−ν)
1 κmin
1 κγ1
1
κmin
1 κγ1
1 − κminγ1
1 κ1
with φi = − ln
[
1 − (κmin
i /κi)γi−1
]
, for i = 1, 2.

Constructing a two layer multiplex system
with geometric correlations
θ2,i = mod
[
θ1,i +
2πli
N
, 2π
]
,
fσ(l) =
1
σ ϕ
( l
σ
)
Φ
( N
2σ
)
− Φ
(
− N
2σ
), −
N
2
≤ l ≤
N
2
,
σ ≡ σ0
(
1
g
− 1
)
,
where ϕ(x) = 1√
2π
e− 1
2
x2
, Φ(x) =
∫
dx ϕ(x), σ ∈ (0, ∞), and
g ∈ [0, 1] is the angular correlation strength parameter.

We can generate multiplexes
with any number of layers
Assign node coordinates in layer j = 1
Then work in layer pairs: assign coordinates to nodes in layer
j ≥ 2 conditioned on the values of the node coordinates in
layer j − 1
Can have different correlation strengths νj+1,j, gj+1,j
between layer pairs

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 (independent of
temperature parameter)
Details: Nat. Phys. 12,
1076–1081 (2016)
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

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

Self-organization into clusters of cooperators
only occurs if angular correlations are present

Correlations can be even more important
than the individual layer topologies
a) b) c) d)
e) f) g) h)

Interdependent systems
Robustness

Interdependencies make systems particularly vulnerable
and can lead to cascading failures
[Nature 464, 1025–8]

Mutual percolation is a proxy of the vulnerability
of the system against random failures
Order parameter: Mutually connected component (MCC) is
largest fraction of nodes connected by a path in every layer using
only nodes in the component

Mutual percolation is a proxy of the vulnerability
of the system against random failures
Degree (radial) correlations mitigate vulnerability: Reis et al.,
Nature Physics 10, 762–767 (2014); Serrano et al., New Journal
of Physics 17, 053033 (2015)

Robustness of multiplexes against targeted attacks:
percolation properties as a proxy
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

Robustness of multiplexes against targeted attacks:
percolation properties as a proxy
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
Are real systems more robust than a random
superposition of their constituent layer topologies?

Racall: Reshuffling of node IDs destroys correlations
Real system Reshuffled
a random superposition of the individual layer topologies of the
real system.

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

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?

Geometric (similarity) correlations mitigate failures cascades
and can lead to a smooth transition
a) b) c) d)
e) f) g) h) i)

Geometric (similarity) correlations mitigate failures cascades
and can lead to a smooth transition
a) b) c) d)
e) f) g) h) i)
Does the strength of similarity correlations predict
the robustness of real systems?

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

Targeted attacks lead to catastrophic cascades
even with degree correlations

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

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

Constituent network layers of real multiplexes
exhibit significant hidden geometric correlations
FrameworkResultBasis
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 PRL 118, 218301

Marian Boguñá M. Angeles Serrano Fragkiskos Papadopoulos
Lubos Buzna Roberta Amato

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«
PRL 118, 218301 (2017)
K-K. Kleineberg, L. Buzna, F. Papadopoulos, M. Boguñá, M. A. Serrano
»Interplay between social influence and competitive strategical games
in multiplex networks«
Scientific Reports 7, 7087 (2017)
R. Amato, A. Díaz-Guilera, K-K. Kleineberg
Kaj Kolja Kleineberg:
• kkleineberg@ethz.ch
• @KoljaKleineberg
• koljakleineberg.wordpress.com

References:
Nat. Phys. 12, 1076–1081 (2016)
PRL 118, 218301 (2017)
• @KoljaKleineberg ← Slides
• koljakleineberg.wordpress.com

References:
Nat. Phys. 12, 1076–1081 (2016)
PRL 118, 218301 (2017)
• @KoljaKleineberg ← Slides
• koljakleineberg.wordpress.com ← Data Model

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