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Mason A. Porter Mathematical Institute, University of Oxford (@masonporter, masonporter.blogspot.co.uk)
Mostly, we’ll be “following” (i.e. skimming through) our new review article:
M Kivelä, A. Arenas, M. Barthelemy, J. P. Gleeson, Y. Moreno, & MAP, “Multilayer Networks”, Journal of Complex Networks, Vol. 2, No. 3: 203–271 [2014].
• 1. Go to http://people.maths.ox.ac.uk/porterm/temp/netsci2014/ and download the .pdf file of this presentation.
• 2. Download the review article from http://comnet.oxfordjournals.org/content/2/3/203 and (just in case) download our earlier article on the tensorial formalism – http://people.maths.ox.ac.uk/porterm/papers/PhysRevX.3.041022.pdf
• 3. Use these materials and be happy.
• Browsing through the mega-‐review article – 1. Introduction – 2. Conceptual and Mathematical Framework – 3. Data – 4. Models, Methods, Diagnostics, and Dynamics
– 5. Conclusions and Outlook • Some Advertisements – Journals, workshops/conferences
“How Candide was multilayer networks were brought up in a magnificent castle and how he was they were driven thence”
• The concept of “multiplex network” has been around for many decades. !
• Monster movement in the game “Munchkin Quest”
• The notion (and terminology) “network of networks” is also several decades old. !
(Craven and Wellman, 1973)
(Courtesy of Sco; Thacker, ITRC, University of Oxford)
(David Krackhardt, 1987)
“What befell Candide multilayer networks among the mathematicians”
2.1. General Form
• See h;p://www.plexmath.eu/?page_id=327 • M. De Domenico, M. A. Porter, & A. Arenas, arXiv:1405.0843
2.2. Tensorial Representa\on
• Adjacency tensor for unweighted case:
• Elements of adjacency tensor: – Auvαβ = Auvα1β1 … αdβd = 1 iff ((u,α), (v,β)) is an element of EM (else
Auvαβ = 0)
• Important note: ‘padding’ layers with empty nodes – One needs to dis\nguish between a node not present in a layer
and nodes exis\ng but edges not present (use a supplementary tensor with labels for edges that could exist), as this is important for normaliza\on in many quan\\es.
• One can write a general (rank-‐4) mul\layer adjacency tensor M in terms of a tensor product between single-‐layer adjacency tensors [C(l) in upper right] and canonical basis tensors [see lower right]
• w: weights • E: canonical basis tensors • Weighted edge from node ni in
layer h to node nj in layer k • Note: Einstein summa\on
conven\on • Page 3 of De Domenico et al., PRX,
2013
• Explored in several papers. Examples: – Supra-‐Laplacian matrices: S. Gómez, A. Díaz-‐Guilera, J. Gómez-‐Gardeñes, C. J. Pérez-‐Vicent, Y. Moreno, & A. Arenas, Physical Review Le8ers, Vol. 110, 028701 (2013)
– Mul\layer Laplacian tensors: De Domenico et al, Physical Review X, 2013
– Spectral proper\es of mul\layer Laplacians: A. Solé-‐Ribalta, M. De Domenico, N. E. Kouvaris, A. Díaz-‐Guilera, S. Gómez, & A. Arenas, Physical Review E, Vol. 88, 032807 (2013)
– Also see summary in the review ar\cle.
• Mul\layer combinatorial Laplacian:
– First term: strength (i.e. weighted degree) tensor – A bit more on degree tensor later
– Second term: mul\layer adjacency tensor (recall) – U: tensor with all entries equal to 1 – E: canonical basis for tensors (recall) – δ: Kronecker delta
• P. J. Mucha, T. Richardson, K. Macon, MAP, & J.-‐P. Onnela, “Community Structure in Time-‐Dependent, Mul\scale, and Mul\plex Networks”, Science, Vol. 328, No. 5980, 876–878 (2010)
• Simple idea: Glue common nodes across “slices” (i.e. “layers”)
• Schematic from M. Bazzi, MAP, S. Williams, M. McDonald, D. J. Fenn, & S. D. Howison, in preparation !
• Special cases of mul\layer networks include: mul\plex networks, interdependent networks, networks of networks, node-‐colored networks, edge-‐colored mul\graphs, …
• To obtain one of these special cases, we impose constraints on the general structure defined earlier. • See the review ar\cle for details.
• 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 be the
same ones). • 4. Diagonal coupling: Inter-‐layer edges only can exist between nodes and their
counterparts. • 5. Layer coupling: coupling between layers is independent of node iden\ty
– Note: special case of “diagonal coupling” • 6. Categorical coupling: diagonal couplings in which inter-‐layer edges can be
present between any pair of layers – Contrast: “ordinal” coupling for tensorial representa\on of temporal networks
• Example 1: Most –– but not all! –– “mul@plex networks” studied in the literature sa\sfy (1,3,4,5,6) and include d = 1 aspects. – Note: Many important situa\ons need (1,3) to be relaxed. (E.g. Some people have Facebook
accounts but not Twi;er accounts.) • Example 2: The “networks of networks” that have been inves\gated thus far sa\sfy
(3) and have addi\onal constraints (which can be relaxed).
The literature is messy. #makeitstop
• Node-‐colored network: also known as interconnected network, network of networks, etc.
• (three alternative representations)
• Networks with multiple types of edges – Also known as multirelational networks, edge-‐colored multigraphs, etc.
• Many studies in practice use the same sets of nodes in each layer, but this isn’t required. – Challenge for tensorial representation: need to keep track of lack of presence of a tie versus a node not being present in a layer (relevant e.g. for normalization of multiplex clustering coefficients)
• Question: When should you include inter-‐layer edges and when should you ignore them?
• Hyperedges generalize edges. A hyperedge can include any (nonzero) number of nodes.
• Example: A k-‐uniform hypergraph has cardinality k for each hyperedge (e.g. a folksonomy like Flickr). – One can represent a k-‐uniform hypergraph using adjacency tensors, and there have been some studies of multiplex networks by mapping them into k-‐uniform hypergraphs.
– A nice paper: Michoel & Nachtergaele, PRE, 2012 • Note that multilayer networks are still formulated for pairwise connections (but a more general type of pairwise connections than usual).
• Ordinal coupling: diagonal inter-‐layer edges among consecutive layers (e.g. multilayer representation of a temporal network)
• Categorical coupling: diagonal inter-‐layer edges between all pairs of edges
• Both can be present in a multilayer network, and both can be generalized
• k-‐partite graphs – Bipartite networks are most commonly studied
• Coupled-‐cell networks – Associate a dynamical system with each node of a
multigraph. Network structure through coupling terms.
• Multilevel networks – Very popular in social statistics literature
(upcoming special issue of Social Networks) – Each level is a layer – Think ‘hierarchical’ situations. Example: ‘micro-‐
level’ social network of researchers and a ‘macro-‐level’ for a research-‐exchange network between laboratories to which the researchers belong
“What they saw in the Country of El Dorado real world”
• Lots of reliable data on intra-‐layer relations (i.e. the usual kind of edges)
• It’s much more challenging to collect reliable data for inter-‐layer edges. We need more data. – E.g. Transportation data should be a very good resource.
Think about the amount of time to change gates during a layover in an airport.
– E.g. Transition probabilities of a person using different social media (each medium is a layer).
• Most empirical multilayer-‐network studies thus far have tended to be multiplex networks.
• Determining inter-‐layer edges as a problem in trying to reconcile node identities across networks. (Can you figure out that a Twitter account and Facebook account belong to the same person?) – Major implications for privacy issues
• Take-‐home message: Be creative about how you construct multilayer networks and define layers!
“Candide’s Our voyage to Constantinople Istanbul measuring and modeling”
• Construct single-‐layer (i.e. “monoplex”) networks and apply the usual tools. – Obtain edge weights as weighted average of
connections in different layers. You get a different weighted network with a different weighting vector. • E.g. Zachary Karate Club
– Information loss • Is there a way to do this to minimize information loss?
• Important: Loss of “Markovianity” (a la temporal networks) – Processes that are Markovian on a multilayer network
may yield non-‐Markovian processes after aggregating the network
• Generalizations of the usual suspects – Degree/strength – Neighborhood
• Which layers should you consider? – Centralities – Walks, paths, and distances – Transitivity and local clustering
• Important note: Sometimes you want to define different values for different node-‐layers (e.g. a vector of centralities for each entity) and sometimes you want a scalar.
• Need to be able to consider different subsets of the layers
• Need more genuinely multilayer diagnostics – It is important to go beyond “bigger and better”
versions of the usual concepts.
• Simplest way: Use aggregation and then measure degree, strength, and neighborhoods on a monoplex network obtained from aggregation. – Possibly only consider a subset of the layers
• More sophisticated: Define a multi-‐edge as a vector to track the information in each layer. With weighted multilayer networks, you can keep track of different weights in intra-‐layer versus inter-‐layer edges.
• Towards multilayer measures: overlap multiplicity for a multiplex network can track how often an edge between entities i and j occurs in multiple layers
• One can write a general (rank-‐4) mul\layer adjacency tensor M in terms of a tensor product between single-‐layer adjacency tensors [C(l) in upper right] and canonical basis tensors [see lower right]
• w: weights • E: canonical basis tensors • Weighted edge from node ni in
layer h to node nj in layer k • Note: Einstein summa\on
conven\on • Page 3 of De Domenico et al., PRX,
2013
• To define a walk (or a path) on a multilayer network, we need to consider the following: – Is changing layers considered to be a step? Is there a
“cost” to changing layers? How do you measure this cost? • E.g. transportation networks vs social networks
– Are intra-‐layer steps different in different layers? • Example: labeled walks (i.e. compound relations) are
walks in a multiplex network that are associated with a sequence of layer labels
• Generalizing walks and paths is necessary to develop generalizations for ideas like clustering coefficients, transitivity, communicability, random walks, graph distance, connected components, betweenness centralities, motifs, etc.
• Towards multilayer measures: Interdependence is the ratio of the number of shortest paths that traverse more than one layer to the number of shortest paths
• Our approach: Cozzo et al., 2013 – Use the idea of multilayer walks. Keep track of returning to entity i (possibly in a different layer from where we started) separately for 1 total layer, 2 total layers, 3 total layers (and in principle more).
• Insight: Need different types of transitivity for different types of multiplex networks. – Example (again): transportation vs social networks
– There are several different clustering coefficients for monoplex weighted networks, and this situation is even more extreme for multilayer networks.
• Our perspective: multilayer walks, which can return to node i on different layers and traverse different numbers of layers !
• In studies of networks, people compute a crapload of centralities.
• The common ones have been generalized in various ways for multilayer networks. – Again, one needs to ask whether you want a centrality for
a node-‐layer or for a given entity (across all layers or a subset of layers).
• Eigenvector centralities and related ideas can be derived from random walks on multilayer networks. – Consider different spreading weights for different types
of edges (e.g. intra-‐layer vs inter-‐layer edges; or different in different layers)
• Betweenness centralities can be calculated for different generalizations of short paths.
• A point of caution: “What the world needs now is another centrality measure.” – I.e. although they can be very useful, please don’t go too
crazy with them.
• The community needs to construct genuinely multilayer diagnostics and go beyond ‘bigger and better’ versions of the concepts we know and (presumably) love. – Not very many yet
• Correlations of network structures between layers – E.g. interlayer degree-‐degree correlations (or any other diagnostic) • ! Interpreting communities as layers, quantities like assortativity can be construed as inter-‐layer diagnostics
• Interdependence is the ratio of the number of shortest paths that traverse more than one layer to the number of shortest paths
• Straightforward: Use your favorite monoplex model for intra-‐layer connections and then construct inter-‐layer edges in some way. – E.g. random-‐graph models like Erdös-‐Rényi, network growth models like preferential attachment
• Correlated layers: Include correlations between properties in different intra-‐layer networks in the construction of random-‐graph ensembles. – E.g. Include intra-‐layer degree-‐degree correlations ρ in [-‐1,1]
• Exponential Random Graph Models (ERGMs) for multiplex networks – Used a lot for multilevel networks
• Statistical-‐mechanical ensembles of multiplex networks
• Generalize growth mechanisms like preferential attachment – Again, one can include inter-‐layer correlations in designing a model
• It would be good to go beyond “bigger and better” versions of the usual ideas. – Including simple inter-‐layer correlations (especially between intra-‐layer degrees) has been the main approach so far.
• Straightforward: Construct different layers separately using your favorite model (or even one that you hate) and then add inter-‐layer edges uniformly at random.
• More sophisticated: Be more strategic in adding inter-‐layer edges.
• Some random-‐graph modules with community structure can be useful, where we think of each community as a separate layer (i.e. as a separate network in a network of networks) – E.g. Melnik et al’s paper (Chaos, 2014) on random
graphs with heterogeneous degree assortativity • The homophily is different in different layers and there is a mixing matrix for inter-‐layer connections
• Communities are dense sets of nodes in a network (typically relative to some null model). – One can use these ideas for multilayer networks (e.g.
multislice modularity). • Interpreting communities as roadblocks to some dynamical process
(e.g. starting from some initial condition), one can have such a process on a multilayer network—with different spreading rates in different types of edges—to algorithmically find communities in multilayer networks.
• Most work thus far on multilayer representation of temporal networks. – One exception is recent work on “Kantian fractionalization” in
international relations. • Challenge: Develop multilayer null models for community detection
(different for ordinal vs. categorical coupling)
• Blockmodels • Spectral clustering (e.g. Michoel & Nachtergaele) • Note: Because I have done a lot of work in this area, I
will go through a bit in some detail to help illustrate some general points that are also relevant in other studies of multilayer networks.
! Communities = Cohesive groups/modules/mesoscopic structures › In stat phys, you try to
derive macroscopic and mesoscopic insights from microscopic information
! Community structure consists of complicated interactions between modular (horizontal) and hierarchical (vertical) structures
! communities have denser set of Internal edges relative to some null model for what edges are present at random › “Modularity”
• P. J. Mucha, T. Richardson, K. Macon, MAP, & J.-‐P. Onnela, “Community Structure in Time-‐Dependent, Mul\scale, and Mul\plex Networks”, Science, Vol. 328, No. 5980, 876–878 (2010)
• Simple idea: Glue common nodes across “slices” (i.e. “layers”) • “Diagonal” coupling
• Find communi\es algorithmically by op\mizing “mul\slice modularity” – We derived this func\on in Mucha et al, 2010
• Laplacian dynamics: find communi\es based on how long random walkers are trapped there. Exponen\ate and then linearize to derive modularity.
• Generalizes deriva\on of monoplex modularity from R. Lambio;e, J.-‐C. Delvenne, &. M Barahona, arXiv:0812.1770 • Different spreading weights on different types of edges
– Node x in layer r is a different node-‐layer from node x in layer s
Example: Zachary Karate Club
• A. S. Waugh, L. Pei, J. H. Fowler, P. J. Mucha, & M. A. Porter [2012], arXiv:0907.3509 (without multilayer formulation)
• Modularity Q as a measure of polarization • Can calculate how closely each legislator is tied to their community
(e.g. by looking at magnitude of corresponding component of leading eigenvector of modularity matrix if using a spectral optimization method)
• Medium levels of optimized modularity as a predictor of majority turnover – By contrast, leading political science measure doesn’t give statistically
significant indication • One network slice for each two-‐year Congress
P. J. Mucha & M. A. Porter, Chaos, Vol. 20, No. 4, 041108 (2010)
Braiiiiiiiiiiiiins
Construc\ng Time-‐Dependent Networks
• fMRI data: network from correlated time series
• Examine role of modularity in human learning by identifying dynamic changes in modular organization over multiple time scales
• Main result: flexibility, as measured by allegiance of nodes to communities, in one session predicts amount of learning in subsequent session
Sta\onarity and Flexibility
• Community sta\onarity ζ (autocorrela\on over \me of community membership):
• Node flexibility: – fi = number of \mes node i changed communi\es divided by total number of possible changes
– Flexibility f = <fi>
• Investigating community structure in a multilayer framework requires consideration of new null models
• Many more details! – E.g., Robustness of
results to choice of size of time window, size of inter-‐slice coupling, particular definition of flexibility, complicated modularity landscape, definition of ‘similarity’ of time series, etc.
Dynamic Reconfigura\on of Human Brain Networks During Learning
(Basse; et al, PNAS, 2011)
• fMRI data: network from correlated \me series
• Examine role of modularity in human learning by iden\fying dynamic changes in modular organiza\on over mul\ple \me scales
• Main result: flexibility, as measured by allegiance of nodes to communi\es, in one session predicts amount of learning in subsequent session
Development of Null Models for Mul\layer Networks
• D. S. Basse;, M. A. Porter, N. F. Wymbs, S. T. Gra�on, J. M. Carlson, & P. J. Mucha, Chaos, 23(1): 013142 (2013)
• Addi\onal structure in adjacency tensors gives more freedom (and responsibility) for choosing null models.
• Null models that incorporate informa\on about a system • E.g. chain null model fixes network topology but
randomizes network “geometry” (edge weights)
• Also: Examine null models from shuffling \me series directly (before turning into a network)
• Structural (γ) versus temporal resolu\on parameter (ω) • More generally, how to choose inter-‐layer (off-‐
diagonal) terms Cjrs • Time series from experiments as well as output of a
dynamical system (e.g. Kuramoto model). Analogous to structural vs func\onal brain networks.
• Many different generalizations of singular value decomposition (SVD) to tensors – Every matrix has a unique SVD, but we have to relax this for tensors.
– See Kolda and Bader, SIAM Review, 2009 – Tensor rank vs matrix rank: hard to determine that rank of tensors of order 3+ • Note: “rank” is also used as a synonym for “order” (see earlier). Here, “rank” is the generalization of matrix rank: the minimum number of column vectors needed to span the range of a matrix. The tensor rank is the minimum number of rank-‐1 tensors with which one can express a tensor as a sum. The purpose of an SVD (and generalizations) is to find a low-‐rank approximation.
• Non-‐negative tensor factorization
• Basic question: How do multilayer structures affect dynamical systems on networks? – Effects of multiplexity? (edge colorings) – Effects of interconnectedness? (node colorings)
• Important goal: Find new phenomena that cannot occur without multilayer structures. – Example: Speeding up vs slowing down spreading?
– Example: Multiplexity-‐induced correlations in dynamics?
– Example: Effect of different costs for changing layers?
• Connected component defined as in monoplex networks, except that multiple types of edges can occur in a path.
• In multilayer networks, one again uses branching-‐process approximations that allow the use of generating function technology. – Same fundamental idea (and limitations) as in monoplex networks, but the calculations are more intricate
• More flavors of giant connected components (GCCs) that can be defined
• Example (from Buldyrev et al, Nature, 2010)
• Numerous papers for both multiplex networks and interconnected networks
• A few interesting ideas – Localized attack • More generally, multilayer networks allow more creativity in targeted attacks. Why in Hell is it almost always by degree (even for monoplex networks)? Be creative!
– Viable cluster: mutually connected giant component
• Random walks and Laplacians – Different spreading rates on different types of edges • See earlier discussions of multislice community structure
• Strong vs weak inter-‐layer coupling – Examine generic properties of phase transitions (e.g. as a function of weights of inter-‐layer edges)
• Competing (toy models of) biological contagions – Your favorite toy models (SI, SIS, SIR, SIRS, etc.)
• Layers with biological contagions interacting with layers of information diffusion (e.g. of awareness)
• Metapopulation models as biological epidemics on networks of networks – E.g. Melnik et al. random-‐graph model (different degree assortativities in different layers), similar model by Joel Miller and collaborators (explicitly in a metapopulation context)
• Each node is associated with a dynamical system, and two nodes have the same color if they have the same state space and an identical dynamical system.
• The couplings between dynamical systems are the edges (or hyperedges). Two edges have the same color if the couplings are equivalent
• There exist many nice results for generic bifurcations in small coupled-‐celled networks. – Spiritually similar results for generic phase
transitions in random walks and Laplacians, but for very low-‐dimensional systems instead of high-‐dimensional ones
• Surgeon General’s warning: The papers on coupled-‐cell networks (many by Marty Golubitsky and company) are very mathematical.
• The usual suspects. Pick your favorite. :)
• Kuramoto model • Threshold models of social influence – Percolation-‐like – E.g. Watts model
• Games on networks • Sandpiles • Others
• It’s important to consider feedback loops. • Maybe one is only allowed to apply controls to a
subset of the layers? • Layer decompositions: Start with a network and try
to infer layers – Reminiscent of community detection, but with layers
instead of dense modules – E.g. research by Prescott and Papachristodoulou on
biochemical networks – Similar problem in social networks
• “Control network” used to influence an “open-‐loop network” (which doesn’t include feedback)
• “Pinning control”, in which one controls a small fraction of nodes to try to influence the dynamics of other nodes, in the context of interconnected networks.
“What befell Candide us at the end of his our journey”
• Multilayer networks are interesting and important objects to study. !• We have developed a unified framework that allows a
classification of different types of multilayer networks. !• Many real networks have multilayer structures. !• Multilayer networks make it possible to throw away less data.
Additionally, they have interesting structural features and have interesting effects in dynamical processes. !
• Adjacency tensors: their time has come !– We need to use tools from multilinear algebra. Tensors generalize
matrices, but there are important differences to consider. !• Challenge: Need to collect good data, especially w.r.t. realiable
quantitative values for inter-layer edges !• Challenge: Need more genuinely interlayer diagnostics !
• Not just “bigger and better” version of monoplex objects !• Challenge: Need additional general results on dynamical processes
(bifurcations, phase transitions). There are some, but we need more. !
• Challenge: Need to move farther beyond the usual percolation-like models !• Not just “bigger and better” versions of monoplex processes !
• Review article of multilayer networks: Journal of Complex Networks, in press (arXiv:1309.7233) !
• Code for visualization and analysis of multilayer networks: http://www.plexmath.eu/?page_id=327!
• Thanks: James S. McDonnell Foundation, EPSRC, FET-Proactive project “PLEXMATH” !
• All is for the best in this best of all possible worlds.
• (Also: The future’s so bright, we gotta to wear shades.)
“What befell Candide us after the story ended”
• Ones with me on the editorial board: !– Journal of Complex Networks (OUP) !– IEEE Transactions on Network Science and Engineering (IEEE) !
• Without me: !– Network Science (CUP)!
• Lake Como School of Advanced Studies: !– http://lakecomoschool.org/ !
• School on Complex Networks !– The Boss: Carlo Piccardi !– Scientific Board: Stefano Battiston, Vittoria Colizza, Peter Holme, Yamir Moreno, Mason Porter !
• Organizers: Alex Arenas, Mason Porter !
• July 6–8, 2015, Max Planck Institute for the Physics of Complex Systems, Dresden, Germany !
• Watch this space: !– http://www.mpipks-dresden.mpg.de/pages/veranstaltungen/frames_veranst_en.html !
• Mathematical Biosciences Institute, The Ohio State University, USA !
• Semester program on “Dynamics of Biologically Inspired Networks” !– http://mbi.osu.edu/programs/emphasis-programs/future-programs/spring-2016-dynamics-biologically-inspired-networks/!
• Focuses on theoretical questions on networks that arise from biology !
• March 21–25, 2016 !• http://mbi.osu.edu/event/?id=898 !
