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 = { a ij } EU-FP7 LASAGNE Project | QMUL F. Battiston et al. Structure and dynamics of multiplex networks 2/91

Many systems, one framework adjacency matrix A = { a ij } node degree k i = � j a ij 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 = { w ij } 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 ] j a [ α ] = � i 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 ] j a [ α ] = � i ij For the multiplex: vector of adjacency matrices A = { A [1] , ..., A [ M ] } . vector of degrees k i = ( k [1] , ..., k [ M ] ). i 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 DYNAMICS STRUCTURE Random walks Basic measures Opinion formation Community structure Cultural dynamics Core-periphery structure 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. α =1 k [ α ] o i = � M overlapping degree: 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 i ( o ) = o i − < o > α =1 k [ α ] o i = � M Z-score of the overlapping degree: σ o i 1 Simple nodes − 2 ≤ z i ( o ) ≤ 2 2 Hubs z i ( o ) > 2 � 2 � � � k [ α ] M 1 − � M Participation coefficient: P i = i α =1 M − 1 o i 1 Focused nodes 0 ≤ P i ≤ 1 / 3 2 Mixed-pattern nodes 1 / 3 < P i ≤ 2 / 3 3 Truly multiplex nodes 2 / 3 < P i ≤ 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 o ij Percentage of edges (%) 1 46 2 27 3 23 4 4 Conditional probability to have overlap: ij a [ α ′ ] a [ α ] � P ( a [ α ′ ] | a [ α ] ij ij ij ) = (1) ij ij a [ α ] � ij 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 ) → P w ( a [ α ′ ] | a [ α ] | w [ α ] ) ij ij ij 1.0 α ′ = O α ′ = C α ′ = B 0.8 ij ) 0.6 P w ( α ′ | w [T] 0.4 0.2 0.0 1 2 3 w [T] ij 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 C i , 1 and C i , 2 show different patterns of multi-clustering and are not correlated with o i . 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 0.82 0.71 0.74 A T CM P 0.76 NMI 0.77 0.81 0.72 0.74 0.65 5 0 0.76 0.70 7 . 7 . 0 2 R I N C 0.66 0.75 0.64 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 � N mm ′ N � M β � − 2 � M α m ′ =1 N mm ′ log m =1 N m N m ′ NMI ( P α , P β ) = � N m ′ + � M β � � � � M α N m m =1 N m log m ′ =1 N m ′ log N 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 ∗ ) a) b) c) 1-p p 1-p* p p*/2 p*/2 1-p 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

peer media pressure (intra-layer) EU-FP7 LASAGNE Project | QMUL F. Battiston et al. Structure and dynamics of multiplex networks 32/91