Multi-relational social dynamics: interactions, opinion formation - - PowerPoint PPT Presentation

Multi-relational social dynamics:

interactions, opinion formation and the dissemination of cultures

Federico Battiston

School of Mathematical Sciences, Queen Mary University of London, UK CoSyDy @ QMUL - July 6, 2016 - London, UK EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 1/91

Many systems, one framework

adjacency matrix A = {aij}

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 2/91

Many systems, one framework

adjacency matrix A = {aij} node degree ki =

j aij

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 2/91

Towards a richer architecture: weighted networks

Weighted adjacency matrix W = {wij} Weights are used to represent strength, distance, cost, time, ...

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 3/91

General formalism for multiplex networks

A multiplex is a system whose basic units are connected through a variety of different

• relationships. Links of different kind are embedded in different layers.

Node index i = 1, . . . , N Layer index α = 1, . . . , M

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 4/91

General formalism for multiplex networks

A multiplex is a system whose basic units are connected through a variety of different

• relationships. Links of different kind are embedded in different layers.

Node index i = 1, . . . , N Layer index α = 1, . . . , M For each layer α: adjacency matrix A[α] = {a[α]

ij }

node degree k[α]

i

=

j a[α] ij

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 4/91

General formalism for multiplex networks

A multiplex is a system whose basic units are connected through a variety of different

• relationships. Links of different kind are embedded in different layers.

Node index i = 1, . . . , N Layer index α = 1, . . . , M For each layer α: adjacency matrix A[α] = {a[α]

ij }

node degree k[α]

i

=

j a[α] ij

For the multiplex: vector of adjacency matrices A = {A[1], ..., A[M]}. vector of degrees ki = (k[1]

i

, ..., k[M]

i

). Do we really need to preserve all this information?.

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 4/91

Multiplex networks: do we really care?

What are we losing collapsing all the information into a single network?

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 5/91

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 6/91

MULTIPLEX NETWORKS

STRUCTURE

Basic measures Community structure Core-periphery structure

DYNAMICS

Random walks Opinion formation Cultural dynamics Evolutionary game theory

APPLICATIONS

The human brain

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 7/91

The multi-layer network of Indonesian terrorists

LAYER CODE N K MULTIPLEX M 78 911 Trust T 70 259 Operations O 68 437 Communications C 74 200 Businness B 13 15

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 8/91

Basic node properties

A layer-by-layer exploration of node properties: the case of the degree distribution.

• verlapping degree:
• i = M

α=1 k[α] i

Different layers show different patterns.

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 9/91

Basic node properties: cartography of a multiplex

Z-score of the overlapping degree: zi(o) = oi −<o>

σo

• i = M

α=1 k[α] i

1 Simple nodes

−2 ≤ zi(o) ≤ 2

2 Hubs

zi(o) > 2 Participation coefficient: Pi =

M M−1

• 1 − M

α=1

• k[α]

i

• i

2

1 Focused nodes

0 ≤ Pi ≤ 1/3

2 Mixed-pattern nodes

1/3 < Pi ≤ 2/3

3 Truly multiplex nodes

2/3 < Pi ≤ 1

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 10/91

Basic node properties: cartography of a multiplex

Multiplex analysis successfully distinguishes node 16 from node 34.

• F. Battiston, V. Nicosia, V. Latora (2014)

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 11/91

Edge overlap

• ij

Percentage of edges (%) 1 46 2 27 3 23 4 4 Conditional probability to have overlap: P(a[α′]

ij

|a[α]

ij ) =

• ij a[α′]

ij

a[α]

ij

• ij a[α]

ij

(1)

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 12/91

Edge overlap and social reinforcement

P(a[α′]

ij

|a[α]

ij ) → Pw(a[α′] ij

|w[α]

ij

)

1 2 3

w[T]

ij

0.0 0.2 0.4 0.6 0.8 1.0

P w(α′|w[T]

ij )

α′ = O α′ = C α′ = B

The existence of strong connections in the Trust layer, which represents the strongest relationships between two people, actually fosters the creation of links in other layers.

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 13/91

Triads and triangles

• F. Battiston, V. Nicosia, V. Latora (2014)

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 14/91

Clustering

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 15/91

Clustering

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 15/91

Clustering

Ci,1 and Ci,2 show different patterns of multi-clustering and are not correlated with oi.

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 16/91

Communities and triadic closure

At each time step a new node attaches with 2 links: a) the first link is at random b) the second link closes a triangle with probability p

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 17/91

Communities and triadic closure

• G. Bianconi et al., Physical Review E (2014)

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 18/91

Community structure

APS: Particle (P), Nuclear (N), Condensed Matter (CM) and Interdisciplinary (I) physics IMDb: Action (A), Crime (C), Thriller (T) and Romance (R) genres

APS IMDb

NMI

0.64 0.82

N I CM P

0.81 0.77 . 7 5 0.75 . 7 2 0.71

T R A C

0.66 0.74 0.72 0.76 0.74 0.65 0.76 0.70

Different layers may have more or less similar community structure

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 19/91

Community structure

APS: Particle (P), Nuclear (N), Condensed Matter (CM) and Interdisciplinary (I) physics IMDb: Action (A), Crime (C), Thriller (T) and Romance (R) genres NMI(Pα, Pβ) = −2 Mα

m=1

Mβ

m′=1 Nmm′ log

Nmm′ N

NmNm′

• Mα

m=1 Nm log

• Nm

N

• + Mβ

m′=1 Nm′ log

Nm′

N

• L. Danon et al., Journal of Statistical Mechanics: Theory and Applications (2015)

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 20/91

Growing models for multiplexes with communities

Real mechanisms by which collaborations grow: 1) ’intra-layer’ triadic closure (with prob. p) 2) ’inter-layer’ proximity bias (with prob. p∗)

1-p* p*/2 p*/2 1-p p 1-p p

a) b) c)

• F. Battiston, J. Iacovacci et al., (2016)

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 21/91

Growing models for multiplexes with communities p=0.9 p*=0.9 p=0.9 p*=0.1

By tuning the strength of the ’inter-layer’ proximity bias mechanism we can obtain similar (p∗ = 0.9) or different (p∗ = 0.1) community structures

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 22/91

Growing models with multiplex communities

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 23/91

General model

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 24/91

TOPIC 1

• F. Battiston, A. Cairoli, et al. (2016)

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 25/91

TOPIC 1 TOPIC 2

• F. Battiston, A. Cairoli, et al. (2016)

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 26/91

TOPIC 1 TOPIC 2

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 27/91

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 28/91

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 29/91

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 30/91

peer pressure

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 31/91

media peer (intra-layer) pressure

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 32/91

coupling (inter-layer) media peer (intra-layer) pressure

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 33/91

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 34/91

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 35/91

maximize

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 36/91

maximize two coupled Ising models

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 37/91

maximize

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 38/91

maximize

consensus

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 39/91

maximize

consensus coherence

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 40/91

15 10 5 5 10 15 1.0 0.5 0.0 0.5 1.0

C

(2,-1)

ABRUPT TRANSITION

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 41/91

15 10 5 5 10 15 1.0 0.5 0.0 0.5 1.0

M [2]

(2,-1)

15 10 5 5 10 15 1.0 0.5 0.0 0.5 1.0

C

(2,-1)

media with different intensities

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 42/91

15 10 5 5 10 15 1.0 0.5 0.0 0.5 1.0

M [2]

(2,-1)

15 10 5 5 10 15 1.0 0.5 0.0 0.5 1.0

C

(2,-1)

media with different intensities

15 10 5 5 10 15 1.0 0.5 0.0 0.5 1.0

C

(1,-1)

15 10 5 5 10 15 1.0 0.5 0.0 0.5 1.0

M [2]

(1,-1)

same intensities media with

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 43/91

15 10 5 5 10 15 1.0 0.5 0.0 0.5 1.0

M [2]

(2,-1)

15 10 5 5 10 15 1.0 0.5 0.0 0.5 1.0

C

(2,-1)

media with different intensities

15 10 5 5 10 15 1.0 0.5 0.0 0.5 1.0

C

(1,-1)

15 10 5 5 10 15 1.0 0.5 0.0 0.5 1.0

M [2]

(1,-1)

same intensities media with full consensus no consensus

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 44/91

15 10 5 5 10 15 1.0 0.5 0.0 0.5 1.0

C

(0,0) (9,-9) (-9,-9)

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 45/91

• 5
• 4
• 3
• 2
• 1

+1 +2 +3 +4

15 10 5 5 10 15 1.0 0.5 0.0 0.5 1.0

C

(0,0) (9,-9) (-9,-9)

• 5

5

h[1]

• 5

5

h[2]

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 46/91

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 47/91

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 48/91

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 49/91

qualitatively same behavior below a critical noise

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 50/91

Abrupt transition and hysteresis loop for the coherence C as a function of γ Empirical formula for the critical points γ+ and γ−. Heteroneous agents (values of χi) are needed to obtain non-trivial consensus Media are responsible for the level of consensus of the system Results are robust up to a critical level of noise

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 51/91

Why are societies inherently multicultural?

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 52/91

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 53/91

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 54/91

SOCIAL INFLUENCE (IMITATION)

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 55/91

SOCIAL INFLUENCE (IMITATION)

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 56/91

SOCIAL INFLUENCE (IMITATION) GLOBALIZATION (FULL CONSENSUS)

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 57/91

F

features

q

t r a i t s ( c

• l
• r

s ) (cilinders)

• R. Axelrod, Journal of Conflict Resolutions (1997)

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 58/91

cultural overlap

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 59/91

cultural overlap

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 60/91

cultural overlap social influence

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 61/91

cultural overlap homophily

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 62/91

cultural overlap homophily

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 63/91

cultural overlap homophily frozen bonds

full similarity full diversity

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 64/91

globalization fragmentation

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 65/91

• C. Castellano, M. Marsili, A. Vespignani, PRL (2000)

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 66/91

small-world networks p=0 p=1 small-world connectivity promotes globalization

• K. Klemm et al., Physical Review E (2003a)

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 67/91

drift: spontaneous mutation of cultural traits

constant noise with rate r

• K. Klemm et al., Physical Review E (2003b)

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 68/91

no noise

(order-disorder transition)

drift: spontaneous mutation of cultural traits

constant noise with rate r

• K. Klemm et al., Physical Review E (2003b)

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 69/91

low noise

multiculturality is unstable!

drift: spontaneous mutation of cultural traits

constant noise with rate r

multiculturality is unstable under cultural drift!

• K. Klemm et al., Physical Review E (2003b)

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 70/91

drift: spontaneous mutation of cultural traits

constant noise with rate r

multiculturality is unstable under cultural drift!

• K. Klemm et al., Physical Review E (2003b)

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 71/91

drift: spontaneous mutation of cultural traits

constant noise with rate r

multiculturality is unstable under cultural drift!

• K. Klemm et al., Physical Review E (2003b)

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 72/91

drift: spontaneous mutation of cultural traits

constant noise with rate r

multiculturality is unstable under cultural drift!

• K. Klemm et al., Physical Review E (2003b)

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 73/91

drift: spontaneous mutation of cultural traits

constant noise with rate r

multiculturality is unstable under cultural drift!

• K. Klemm et al., Physical Review E (2003b)

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 74/91

• F. Battiston et al., (2015)

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 75/91

• F. Battiston et al., (2015)

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 76/91

• F. Battiston et al., (2015)

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 77/91

p=0 p=1 classical layered

= =

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 78/91

Rewire tunes the edge overlap

log scale linear scale

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 79/91

0.0 0.2 0.4 0.6 0.8 1.0 100 200 300 1 1/F 3/F 5/F 7/F

q

• 0.0

0.2 0.4 0.6 0.8 1.0 100 200 300 1 1/F 3/F 5/F 7/F

q

• CLASSICAL

LAYERED

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 80/91

0.0 0.2 0.4 0.6 0.8 1.0 100 200 300 1 1/F 3/F 5/F 7/F

q

• 0.0

0.2 0.4 0.6 0.8 1.0 100 200 300 1 1/F 3/F 5/F 7/F

q

• CLASSICAL

LAYERED

• c

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 81/91

0.0 0.2 0.4 0.6 0.8 1.0 100 200 300 1 1/F 3/F 5/F 7/F

q

• LAYERED

C U L T U R A L D R I F T

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 82/91

0.0 0.2 0.4 0.6 0.8 1.0 100 200 300 1 1/F 3/F 5/F 7/F

q

• LAYERED

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 83/91

0.0 0.2 0.4 0.6 0.8 1.0 100 200 300 1 1/F 3/F 5/F 7/F

q

• 1

r

LAYERED

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 84/91

0.0 0.2 0.4 0.6 0.8 1.0 100 200 300 1 1/F 3/F 5/F 7/F

q

• 2

r

1

r

LAYERED

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 85/91

0.0 0.2 0.4 0.6 0.8 1.0 100 200 300 1 1/F 3/F 5/F 7/F

q

• 2

r

3

r

1

r

LAYERED

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 86/91

0.0 0.2 0.4 0.6 0.8 1.0 100 200 300 1 1/F 3/F 5/F 7/F

q

• 2

r

3

r

4

r

1

r

LAYERED

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 87/91

0.0 0.2 0.4 0.6 0.8 1.0 100 200 300 1 1/F 3/F 5/F 7/F

q

• 2

r

3

r

4

r

1

r

LAYERED

MULTICULTURALITY UNSTABLE STABLE MULTICULTURALITY

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 88/91

0.0 0.2 0.4 0.6 0.8 1.0 100 200 300 1 1/F 3/F 5/F 7/F

q

• 2

r

3

r

4

r

1

r

LAYERED

MULTICULTURALITY UNSTABLE STABLE MULTICULTURALITY multiculturality as a natural consequence

• f realistic layered interaction patterns

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 89/91

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 90/91

Structural measures for multiplex networks, Physical Review E, 89 (3) (2014)

• F. Battiston, V. Nicosia and V. Latora

Emergence of multiplex communities in collaboration networks, Plos One, 11 (1) e0147451 (2016)

• F. Battiston, J. Iacovacci, V. Nicosia, G. Bianconi, V. Latora

Interplay between consensus and coherence in a model of interacting opinions, Physica D, 323, 12-19 (2016)

• F. Battiston, A. Cairoli, V. Nicosia, A. Baule, V. Latora

Robust multiculturality emerges from layered social influence, arXiv:1606.05641, (2016)

• F. Battiston, V. Nicosia, V. Latora and M. San Miguel

Email: f.battiston@qmul.ac.uk Twitter: @fede7j Acknowledge financial support from the

EU-FP7 LASAGNE Project | QMUL

• F. Battiston et al.

Structure and dynamics of multiplex networks 91/91