Structure and dynamics of multiplex networks Federico Battiston School of Mathematical Sciences, Queen Mary University of London, UK Brain & Spine Institute, CNRS, Paris, France Center for Network Science @ Central European University - February 13, 2017 - Budapest, Hungary EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 1/1
59
Embed
Structure and dynamics of multiplex networks · EU-FP7 LASAGNE Project j QMUL F. Battiston Structure and dynamics of multiplex networks 7/1. General formalism for multiplex networks
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Structure and dynamics of multiplex networks
Federico Battiston
School of Mathematical Sciences, Queen Mary University of London, UKBrain & Spine Institute, CNRS, Paris, France
Center for Network Science @ Central European University - February 13, 2017 - Budapest, Hungary
EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 1/1
A very short presentation...
ROME
Statistical mechanics / economic complexity
LONDON
Multi-layer networks
i jk
mn
jk
mn
i
i jk
mn
PARIS
Brain networks
EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 2/1
Towards richer architectures: multiplex networks
EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 3/1
Many systems, one framework
adjacency matrix A = {aij}
node degree ki =∑
j aij
EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 4/1
Many systems, one framework
adjacency matrix A = {aij}
node degree ki =∑
j aij
EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 4/1
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 Structure and dynamics of multiplex networks 5/1
Towards a richer architecture: temporal networks
Temporal networks: connections can change over time
EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 6/1
General formalism for multiplex networks
A multiplex is a system whose basic units are connected through a variety of differentrelationships. 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 Structure and dynamics of multiplex networks 7/1
General formalism for multiplex networks
A multiplex is a system whose basic units are connected through a variety of differentrelationships. 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 Structure and dynamics of multiplex networks 7/1
General formalism for multiplex networks
A multiplex is a system whose basic units are connected through a variety of differentrelationships. 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 Structure and dynamics of multiplex networks 7/1
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 Structure and dynamics of multiplex networks 8/1
Two early reviews...
EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 9/1
... and more recent material
BOOKS
INTRODUCTIONS
THEMATICREVIEWS
EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 10/1
MULTIPLEX NETWORKS
STRUCTURE DYNAMICSBasic measures
Motif analysisCommunity structure
Core-periphery structure
Random walksOpinion formationCultural dynamics
Evolutionary game theory
EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 11/1
Structure of multiplex networks
EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 12/1
The multi-layer network of Indonesian terrorists
LAYER CODE N K
MULTIPLEX M 78 911
Trust T 70 259Operations O 68 437
Communications C 74 200Businness B 13 15
EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 13/1
Basic node properties
A layer-by-layer exploration of node properties: the case of the degree distribution.
overlapping degree: oi =∑Mα=1 k
[α]i
Different layers show different patterns.
EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 14/1
Basic node properties: cartography of a multiplex
Z-score of the overlapping degree: zi (o) = oi−<o>σo
oi =∑Mα=1 k
[α]i
1 Simple nodes −2 ≤ zi (o) ≤ 2
2 Hubs zi (o) > 2
Participation coefficient: Pi = MM−1
[1−
∑Mα=1
(k
[α]ioi
)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 Structure and dynamics of multiplex networks 15/1
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 Structure and dynamics of multiplex networks 16/1
Edge overlap
oij Percentage of edges (%)1 462 273 234 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 Structure and dynamics of multiplex networks 17/1
Edge overlap and social reinforcement
P(a[α′]ij |a
[α]ij ) → Pw(a
[α′]ij |w
[α]ij )
1 2 3w
[T]ij
0.0
0.2
0.4
0.6
0.8
1.0
Pw(α′ |w
[T]
ij)
α′ = O
α′ = C
α′ = B
The existence of strong connections in the Trust layer, which represents the strongestrelationships between two people, actually fosters the creation of links in other layers.
EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 18/1
Social reinforcement and null models
1 2 3w
[T]ij
0.0
0.2
0.4
0.6
0.8
1.0
Pw(O|w
[T]
ij)
real data
randomised data: fixed P (k[O])
randomised data: fixed K [O]
Social reinforcement obtained in real data can not simply be explained by inter-layerdegree-degree correlation.
EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 19/1
Triads and triangles
F. Battiston, V. Nicosia, V. Latora (2014)
EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 20/1
Clustering
EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 21/1
Clustering
EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 21/1
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 Structure and dynamics of multiplex networks 22/1
Our structural measures for multiplex networks have been used in different disciplines
ECOLOGY
MEDICINE
MOBILITY
SOCIOLOGYENERGY
ECONOMICS
SYSTEMIC RISK
NEUROSCIENCE
EPIDEMIOLOGYLINGUISTIC
EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 23/1
Not only local node and edge properties!
Real-world systems are characterised by non-trivial structures at the micro- and themeso-scale, such as motifs, communities and cores
EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 24/1
Communities and triadic closure
At each time step a new node attaches with 2 links:a) the first link is at randomb) the second link closes a triangle with probability p
Figure from G. Bianconi et al., Physical Review E (2014)
EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 25/1
Communities and triadic closure
G. Bianconi et al., Physical Review E (2014)
EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 26/1
Different layers may have more or less similar community structureEU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 27/1
EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 49/1
Our handbook for the analysis of multiplex network datasets [EMPIRICAL FOCUS]:measures of multiplexity, models to reproduce empirical findings and to assess theirstatistical significance (EPJST 2017)
EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 50/1
Our open-source software library for the analysis of multiplex networks on GitHubV. Nicosia & F. Battiston
Still in the making of...
EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 51/1
Dynamics of multiplex networks
EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 52/1
DIFFUSION
Multiplex networks can be superdiffusivethe time scale associated to the whole system is smaller than that of the single layers consideredindependently
EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 53/1
DIFFUSION
EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 54/1
DIFFUSION
REACTION
EU-FP7 LASAGNE Project | QMUL F. Battiston Structure and dynamics of multiplex networks 55/1