Structural measures for multiplex networks

Transcription

1 Structural measures for multiplex networks Federico Battiston, Vincenzo Nicosia and Vito Latora School of Mathematical Sciences, Queen Mary University of London Mathematics of Networks Imperial College London, Sept. 10, 2014 EU-FP7 LASAGNE Project Queen Mary University of London F. Battiston et al. Structural measures for multiplex networks 1/21

2 Outline General formalism for multiplex networks Measures to characterise the multiplexity of a system 1 Basic node and link properties 2 Local properties (clustering) 3 Global properties (transitivity, reachability, centrality) Validation of all measures on a genuine multi-layer dataset of Indonesian terrorists EU-FP7 LASAGNE Project Queen Mary University of London F. Battiston et al. Structural measures for multiplex networks 2/21

3 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 Layer index i = 1,..., N α = 1,..., M EU-FP7 LASAGNE Project Queen Mary University of London F. Battiston et al. Structural measures for multiplex networks 3/21

4 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 Layer index i = 1,..., N α = 1,..., M For each layer α: adjacency matrix A [α] = {a [α] ij } node degree k [α] i = j a[α] ij i k[α] i = 2K [α] K [α] is the size of layer α EU-FP7 LASAGNE Project Queen Mary University of London F. Battiston et al. Structural measures for multiplex networks 3/21

5 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 Layer index i = 1,..., N α = 1,..., M For each layer α: adjacency matrix A [α] = {a [α] ij } node degree k [α] i = j a[α] ij i k[α] i = 2K [α] K [α] is the size of layer α For the multiplex: vector of adjacency matrices A = {A [1],..., A [M] }. vector of degrees k i = (k [1] i,..., k [M] i ). Vectorial variables are necessary to store all the richness of multiplexes. EU-FP7 LASAGNE Project Queen Mary University of London F. Battiston et al. Structural measures for multiplex networks 3/21

6 General formalism: aggregated matrices Aggregated topological network: adjacency matrix A = {a ij }: a ij = { 1 if α : a [α] ij = 1 0 otherwise (1) node degree k i = j a ij i k i = 2K EU-FP7 LASAGNE Project Queen Mary University of London F. Battiston et al. Structural measures for multiplex networks 4/21

7 General formalism: aggregated matrices Aggregated topological network: adjacency matrix A = {a ij }: a ij = { 1 if α : a [α] ij = 1 0 otherwise (1) node degree k i = j a ij i k i = 2K Aggregated overlapping network: adjacency matrix O = {o ij }: o ij = α a [α] ij edge overlap (2) node degree o i = j o ij = α k[α] i, o i k i i o i = 2O EU-FP7 LASAGNE Project Queen Mary University of London F. Battiston et al. Structural measures for multiplex networks 4/21

8 General formalism: aggregated matrices Aggregated topological network: adjacency matrix A = {a ij }: a ij = { 1 if α : a [α] ij = 1 0 otherwise (1) node degree k i = j a ij i k i = 2K Aggregated overlapping network: adjacency matrix O = {o ij }: o ij = α a [α] ij edge overlap (2) node degree o i = j o ij = α k[α] i, o i k i i o i = 2O Scalar variables describing system s multiplexity cannot disregard the layer index [α]. Generalisation to the case of weighted layers: a [α] ij w [α] ij k [α] i s [α] i o ij oij w EU-FP7 LASAGNE Project Queen Mary University of London F. Battiston et al. Structural measures for multiplex networks 4/21

9 The multi-layer network of Indonesian terrorists 78 nodes 911 edges representing 4 social relationships: 1 Trust (weighted edges) 2 Operations (weighted edges) 3 Communications 4 Businness (only few information) EU-FP7 LASAGNE Project Queen Mary University of London F. Battiston et al. Structural measures for multiplex networks 5/21

10 The multi-layer network of Indonesian terrorists LAYER CODE N act K S O O w MULTIPLEX M / Trust T / / Operations O / / Communications C / / Businness B / / Figure : Coloured-edge representation of a subset of 10 nodes for the multiplex network of Indonesian terrorist: green edges represent trust, red edges communications and blue edges common operations. EU-FP7 LASAGNE Project Queen Mary University of London F. Battiston et al. Structural measures for multiplex networks 6/21

11 The multi-layer network of Indonesian terrorists LAYER CODE N act K S O O w MULTIPLEX M / Trust T / / Operations O / / Communications C / / Businness B / / EU-FP7 LASAGNE Project Queen Mary University of London F. Battiston et al. Structural measures for multiplex networks 6/21

12 Basic node properties A layer-by-layer exploration of node properties: the case of the degree distribution. Different layers show different patterns. EU-FP7 LASAGNE Project Queen Mary University of London F. Battiston et al. Structural measures for multiplex networks 7/21

13 Basic node properties: cartography of a multiplex Participation coefficient: P i = 1 ( ) M k [α] 2 i α=1 oi 1 Focused nodes 0 P i Mixed-pattern nodes 0.3 < P i Truly multiplex nodes P i > 0.6 Z-score of the overlapping degree: z i (o) = o i <o> σ o 1 Simple nodes 2 z i (o) 2 2 Hubs z i (o) > 2 EU-FP7 LASAGNE Project Queen Mary University of London F. Battiston et al. Structural measures for multiplex networks 8/21

14 Basic node properties: cartography of a multiplex EU-FP7 LASAGNE Project Queen Mary University of London F. Battiston et al. Structural measures for multiplex networks 9/21

15 Edge overlap and social reinforcement o ij Percentage of edges (%) EU-FP7 LASAGNE Project Queen Mary University of London F. Battiston et al. Structural measures for multiplex networks 10/21

16 Edge overlap and social reinforcement o ij Percentage of edges (%) Probability conditional overlap: F (a [α ] ij a [α] ij ) = ] ij a[α ij ij a[α] ij a [α] ij (3) EU-FP7 LASAGNE Project Queen Mary University of London F. Battiston et al. Structural measures for multiplex networks 10/21

17 Edge overlap and social reinforcement F (a [α ] ij a [α] ij ) F w (a [α ] ij w [α] ij ) α = O α = C α = B F w (α w [T] ij ) 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 Queen Mary University of London F. Battiston et al. Structural measures for multiplex networks 11/21

18 Triads and triangles EU-FP7 LASAGNE Project Queen Mary University of London F. Battiston et al. Structural measures for multiplex networks 12/21

19 Clustering C i,1 = α α α j i,m i (a[α] α a[α] mi ) j i,m i (a[α] ij a [α] (4) mi ) ij a [α ] jm EU-FP7 LASAGNE Project Queen Mary University of London F. Battiston et al. Structural measures for multiplex networks 13/21

20 Clustering C i,1 = α α α j i,m i (a[α] α a[α] mi ) j i,m i (a[α] ij a [α] (4) mi ) ij a [α ] jm C i,2 = α α α α α,α j i,m i (a[α] α α α j i,m i (a[α] ij a [α ] mi ) ij a [α ] ] jm a[α mi ) (5) EU-FP7 LASAGNE Project Queen Mary University of London F. Battiston et al. Structural measures for multiplex networks 13/21

21 Transitivity EU-FP7 LASAGNE Project Queen Mary University of London F. Battiston et al. Structural measures for multiplex networks 14/21

22 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 Queen Mary University of London F. Battiston et al. Structural measures for multiplex networks 15/21

23 Transitivity and clustering We call configuration model (CM) the set of multiplexes obtained from the original system by randomising edges and keeping fixed the sequence of degree vectors k 1, k 2,..., k N, i.e. keeping fixed the degree sequence at each layer α. Variable Real data Randomised data C C T T Measures of multi-clustering for real data are systematically higher than the ones obtained for randomised data, where edge correlations are washed out by randomisation EU-FP7 LASAGNE Project Queen Mary University of London F. Battiston et al. Structural measures for multiplex networks 16/21

24 Navigability: shortest paths and interdependence The interdependence λ i of node i is defined as: λ i = j i ψ ij σ ij (6) where σ ij is the total number of shortest paths between i and j and ψ ij is the number of interdependent shortest paths between node i and node j λ λi Rank EU-FP7 LASAGNE Project Queen Mary University of London F. Battiston et al. Structural measures for multiplex networks 17/21

25 Reachability: shortest paths and interdependence λi o i λi P i EU-FP7 LASAGNE Project Queen Mary University of London F. Battiston et al. Structural measures for multiplex networks 18/21

26 Centrality For a duplex we can construct the following adjacency matrix: M(b) = ba [1] + (1 b)a [2], M(b = 0.5) = O (7) T - O T - C O - C τk(ei(o), Ei(M)) b The symmetry/asymmetry of the curves tell us about the interplay between the layers in determining the centrality of the multi-layer system. Both layers T and O dominate C in determining the centrality of the multiplex. EU-FP7 LASAGNE Project Queen Mary University of London F. Battiston et al. Structural measures for multiplex networks 19/21

27 Summary We suggested a comprehensive formalism to deal with systems composed of several layers We also proposed a number of metrics to characterize multiplex systems with respect to: 1 Node degree 2 Node participation to different layers 3 Edge overlap 4 Clustering 5 Transitivity 6 Reachability 7 Eigenvector centrality 8 You can find more in the paper EU-FP7 LASAGNE Project Queen Mary University of London F. Battiston et al. Structural measures for multiplex networks 20/21

28 Structural measures for multiplex networks, Phys. Rev. E (2014). F. Battiston, V. Nicosia and V. Latora, the LASAGNE team at Queen Mary University of London. Acknowledge financial support from the EU-FP7 LASAGNE Project Queen Mary University of London F. Battiston et al. Structural measures for multiplex networks 21/21