Multilayer networks! Mikko Kivelä Assistant professor @bolozna www.mkivela.com C&)*$+, S-"%+)" Tutorial @ The 9th International Conference on @!".##$%&.'( Complex Networks and their Applications Outline 1. Why multilayer networks 2. Conceptual and mathematical framework 3. Multilayer network systems and data 4. How to analyse multilayer networks 5. Dynamics and multilayer networks 6. Tools an packages Why multilayer networks? Networks are everywhere Nodes Links Neurons, Synapses, brain areas axons Friendships, People phys. contacts, kinships, … Species, Genetic similarity, populations trophic interactions, individuals competition Genes, Regulatory proteins relationships Network representations – are simple graphs enough? vs Example: Sociograms G. C. Homans. ”Human Group”, Routledge 1951 F. Roethlisberger, W. Dickson. ”Management and the worker”, Cambridge University Press 1939 Example: Multivariate social networks S. Wasserman, K. Faust. ”Social Network Analysis”, Cambridge University Press 1994 Example: Cognitive social structures D. Krackhardt 1987 Example: Temporal networks M. Kivelä, R. K. Pan, K. Kaski, J. Kertész, J. Saramäki, M. Karsai: Multiscale analysis of spreading in a large communication network, J. Stat. Mech. 3 P03005 (2012) Example: Interdependent infrastructure networks S. V. Buldyrev, R. Parshani, G. Paul, H. E. Stanley, S. Havlin. ”Catastrophic cascade of failures in interdependent networks”, Nature 464:1025 2010 Example: UK infrastructure networks (Courtesy of Scott Thacker, ITRC, University of Oxford) More realistic network representations Interacting networks Temporal networks Multiplex networks Multidimensional Overlay networks networks Networks of networks Interdependent networks More realistic network representations Interacting networks Temporal networks Multiplex networks Multilayer networks Multidimensional Overlay networks networks Networks of networks Interdependent networks Conceptual and mathematical framework Review article on multilayer networks M. Kivelä, A. Arenas, M. Barthelemy, J. P. Gleeson, Y. Moreno, & MAP, Journal of Complex Networks 2(3): 203-271 (2014), Multilayer network M. Kivelä, A. Arenas, M. Barthelemy, J.P. Gleeson, Y. Moreno, M.A. Porter: Multilayer networks, J. Complex Netw. 2(3): 203-271 (2014) Multilayer network, formal definition V : Set of nodes ● As in ordinary graphs L : Sequence of sets of elementary layers, one set for each aspect La : Set of elementary layers for aspect a VM : Set of node-layer tuples that are present in the network EM : Set of edges M. Kivelä, A. Arenas, M. Barthelemy, J.P. Gleeson, Y. Moreno, M.A. Porter: Multilayer networks, J. Complex Netw. 2(3): 203-271 (2014) Multilayer network, example V = {1,2,3,4} L = [L1,L2] L1 = {A, B} ; L2 = {X, Y} VM = {(1,A,X), (2,A,X), (3,A,X), (2,A,Y), (3,A,Y), (1,B,X), (3,B,X), (4,B,X), (1,B,Y)} EM = { [(1,A,X),(2,A,X)], [(1,A,X),(1,B,X)], [(1,A,X),(4,B,X)], [(3,A,X),(3,A,Y)], … } M. Kivelä, A. Arenas, M. Barthelemy, J.P. Gleeson, Y. Moreno, M.A. Porter: Multilayer networks, J. Complex Netw. 2(3): 203-271 (2014) Underlying graph - Information about layers is lost (if node labels are not considered) + You can now use any tools and theory for graphs! M. Kivelä, A. Arenas, M. Barthelemy, J.P. Gleeson, Y. Moreno, M.A. Porter: Multilayer networks, J. Complex Netw. 2(3): 203-271 (2014) Generalized network representations Interacting networks Temporal networks Multiplex networks Multilayer networks Multidimensional Overlay networks networks Networks of networks Interdependent networks Example: multiplex networks -> multilayer networks Examples of multiplex networks F. Buccafurri et al. “Bridge analysis in a R. Gallotti, M. Barthelemy. “The multilayer social internetworking scenario“, Inf. Sci. temporal network of public transportation 2013 in Great Britain“, Sci. Data 2015 Multilayer network representations Example: node-colored networks -> multilayer networks Interconnected network, colored graph, multiplex network or multilayer network? Edge-colored multigraph Multilayer network Interconnected network • Node-colored graph in which “any path whose edges are between nodes of a different color cannot contain more than one node of any given color” can always be represented as a multiplex network Example: temporal networks Event between nodes A and B at time t1 t1 t2 Time Multilayer network representations Multilayer networks in literature M. Kivelä, A. Arenas, M. Barthelemy, J.P. Gleeson, Y. Moreno, M.A. Porter: Multilayer Networks, J. Complex Netw. 2(3): 203-271 (2014) Constraints (from the table) 1. Node-aligned (or fully interconnected): All layers contain all nodes 2.Layer disjoint: Each node exists in at most one layer 3.Equal size: Each layer has the same number of nodes (but they need not to be the same ones) 4.Diagonal coupling: Inter-layer edges can only exist between nodes and their counterparts 5.Layer coupling: coupling between layers is independent of node identity 6.Categorical coupling: diagonal couplings in which inter- layer edges are always present between any pair of layers 7.Number of layers: Often only fixed number of layers is allowed M. Kivelä, A. Arenas, M. Barthelemy, J.P. Gleeson, Y. Moreno, M.A. Porter: Multilayer Networks, J. Complex Netw. 2(3): 203-271 (2014) Multilayer networks in literature ~multiplex networks ~networks of networks M. Kivelä, A. Arenas, M. Barthelemy, J.P. Gleeson, Y. Moreno, M.A. Porter: Multilayer Networks, J. Complex Netw. 2(3): 203-271 (2014) Multilayer networks in literature ~networks with colored edges ~networks with colored nodes M. Kivelä, A. Arenas, M. Barthelemy, J.P. Gleeson, Y. Moreno, M.A. Porter: Multilayer Networks, J. Complex Netw. 2(3): 203-271 (2014) Tensorial representation • Adjacency matrices is a common and powerful way of representing normal networks: 1 if link from i to j Aij = {0 otherwise • Equivalent concept for multilayer networks is adjacency tensors: iγ˜ 1 if link from i at layer γ˜ to j at layer δ˜ M = jδ˜ {0 otherwise • With d aspects there are 2(d+1) indices Tensorial notation conventions • Tensor indices with Greek letters: αγ˜ 1 if link from node α at layer γ˜ to node β at layer δ˜ M = βδ˜ {0 otherwise • Einstein notation for summation, if index appears twice in a term, then it is summed over: Mαγ˜V = Mαγ˜V βδ˜ αγ˜ ∑ βδ˜ αγ˜ α,γ˜ M. De Domenico, A. Solé-Ribalta, E. Cozzo, M. Kivelä, Y. Moreno, M. A. Porter, S. Gómez, A. Arenas.: Mathematical formulation of multilayer networks, Phys. Rev. X 3, 041022 (2013) Example: multilayer “eigentensors” • Multilayer eigenvalue problem: αγ˜ M V = λV ˜ βδ˜ αγ˜ βδ • Example: replace M by combinatorial Laplacian tensor L: Lαγ˜ = Mηϵ˜U Eρσ˜(βδ˜)δαγ˜ − Mαγ˜ βδ˜ ρσ˜ ηϵ˜ βδ˜ βδ˜ M. De Domenico, A. Solé-Ribalta, E. Cozzo, M. Kivelä, Y. Moreno, M. A. Porter, S. Gómez, A. Arenas.: Mathematical formulation of multilayer networks, Phys. Rev. X 3, 041022 (2013) Combinatorial Laplacian tensor Lαγ˜ = Mηϵ˜U Eρσ˜(βδ˜)δαγ˜ − Mαγ˜ βδ˜ ρσ˜ ηϵ˜ βδ˜ βδ˜ Tensor with all elements equal to 1 Kronecker delta Canonical basis for tensors • Example: diffusion Xβδ˜(t) = − Lαγ˜X (t) dt βδ˜ αγ˜ • Compare to combinatorial Laplacian matrix: Lij = (AU)ijδij − Aij = Dij − Aij M. De Domenico, A. Solé-Ribalta, E. Cozzo, M. Kivelä, Y. Moreno, M. A. Porter, S. Gómez, A. Arenas.: Mathematical formulation of multilayer networks, Phys. Rev. X 3, 041022 (2013) Tensors and missing nodes Warning! ● Adjacency tensors cannot represent nodes missing from layers! ● One needs to do 'padding' of layers with empty nodes ● Be careful with normalization and interpretation of results after such padding process M. De Domenico, A. Solé-Ribalta, E. Cozzo, M. Kivelä, Y. Moreno, M. A. Porter, S. Gómez, A. Arenas.: Mathematical formulation of multilayer networks, Phys. Rev. X 3, 041022 (2013) Supra-adjacency matrices ● "Supra-adjacency matrix" is adjacency matrix of the underlying graph GM. ● (AM)ij = 1 if node-layer tuples i and j connected, 0 otherwise ● Useful to separate intra- and inter-layer edges to matrix A and matrix C, such that AM = A + C Linear algebraic representations Graphs: Multilayer networks: Adjacency tensor Supra-adjacency matrix M. De Domenico, A. Solé-Ribalta, E. Cozzo, M. Kivelä, Y. Moreno, M. A. Porter, S. Gómez, A. Arenas.: Mathematical formulation of multilayer networks, Phys. Rev. X 3, 041022 (2013) Linear algebraic representations Multilayer network Underlying graph Graphical representations Adjacency tensor Supra-adjacency matrix Tensor flattening representations Linear algebraic M. De Domenico, A. Solé-Ribalta, E. Cozzo, M. Kivelä, Y. Moreno, M. A. Porter, S. Gómez, A. Arenas.: Mathematical formulation of multilayer networks, Phys. Rev. X 3, 041022 (2013) Summary: Concepts and mathematical framework ● Multilayer networks offer a framework for working with various types of networks ● Many graph generalisations can be mapped to multilayer networks ● Multiple ways of representing multilayer networks ● Graphical representation ● Tensor representation ● Set of supra-adjacency matrices Multilayer network systems and data Data • Most multilayer networks data is multiplex networks (unless you count temporal network and bipartite/hypergraphs) • That is, most data on intra-layer links • In general it is more difficult to collect data on inter-layer links (links between different systems) • Problem: matching the node identities across layers • E.g., in social network platforms privacy issues Social networks • Most common: multiplex networks with same set of people, different types of interactions/ relationships • Other examples: • Layers =