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- 1. 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].
- 2. 5VGRUVQ*CRRKPGUU • 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.
- 3. 1WVNKPG • 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
- 4. ň$WYQTF$KPIQŉ DWYQTFDKPIQ
- 5. “How Candide was multilayer networks were brought up in a magnificent castle and how he was they were driven thence” +06417%6+10
- 6. 'ZCORNG/WNVKRNGZ0GVYQTM • The concept of “multiplex network” has been around for many decades.!
- 7. 'ZCORNG'FIG%QNQTGF/WNVKITCRJ • Monster movement in the game “Munchkin Quest”
- 8. 'ZCORNG0GVYQTMQH0GVYQTMU • The notion (and terminology) “network of networks” is also several decades old.! (Craven and Wellman, 1973)
- 9. #0GVYQTMQH0GVYQTMUa 7-+PHTCUVTWEVWTG (Courtesy of Sco; Thacker, ITRC, University of Oxford)
- 10. 'ZCORNG%QIPKVKXG5QEKCN5VTWEVWTG (David Krackhardt, 1987)
- 11. /WNVKNC[GT0GVYQTM
- 12. ň CEJCT[-CTCVG%NWD%NWDŉ 0GVYQTM
- 13. 2KEVWTGUEQWTVGU[QH#CTQP%NCWUGV
- 14. 3WGUVKQPU!
- 15. “What befell Candide multilayer networks among the mathematicians” %10%'267#.#0 /#6*'/#6+%#.(4#/'914-5
- 16. 2.1. General Form
- 17. 5QHVYCTGHQT8KUWCNKCVKQPCPF#PCN[UKU • See h;p://www.plexmath.eu/?page_id=327 • M. De Domenico, M. A. Porter, A. Arenas, arXiv:1405.0843
- 18. 2.2. Tensorial Representaon • 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 disnguish between a node not present in a layer and nodes exisng but edges not present (use a supplementary tensor with labels for edges that could exist), as this is important for normalizaon in many quanes.
- 19. #FLCEGPE[6GPUQTYKVJF#URGEV • One can write a general (rank-‐4) mullayer 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 summaon convenon • Page 3 of De Domenico et al., PRX, 2013
- 20. /WNVKNKPGCT#NIGDTCCPF.CRNCEKCP6GPUQTU • 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) – Mullayer Laplacian tensors: De Domenico et al, Physical Review X, 2013 – Spectral properes of mullayer 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 arcle.
- 21. /WNVKNC[GT%QODKPCVQTKCN.CRNCEKCP • Mullayer combinatorial Laplacian: – First term: strength (i.e. weighted degree) tensor – A bit more on degree tensor later – Second term: mullayer adjacency tensor (recall) – U: tensor with all entries equal to 1 – E: canonical basis for tensors (recall) – δ: Kronecker delta
- 22. 'ZCORNGň/WNVKUNKEGŉ0GVYQTMU • P. J. Mucha, T. Richardson, K. Macon, MAP, J.-‐P. Onnela, “Community Structure in Time-‐Dependent, Mulscale, and Mulplex Networks”, Science, Vol. 328, No. 5980, 876–878 (2010) • Simple idea: Glue common nodes across “slices” (i.e. “layers”)
- 23. ň(NCVVGPGFŉ/WNVKUNKEG0GVYQTMU UWRTCCFLCEGPE[TGRTGUGPVCVKQP • Schematic from M. Bazzi, MAP, S. Williams, M. McDonald, D. J. Fenn, S. D. Howison, in preparation!
- 24. %NCUUKH[KPI/WNVKNC[GT0GVYQTMU • Special cases of mullayer networks include: mulplex networks, interdependent networks, networks of networks, node-‐colored networks, edge-‐colored mulgraphs, … • To obtain one of these special cases, we impose constraints on the general structure defined earlier. • See the review arcle for details.
- 25. %QPUVTCKPVU HTQOVJG6CDNG • 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 identy – 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 representaon of temporal networks • Example 1: Most –– but not all! –– “mul@plex networks” studied in the literature sasfy (1,3,4,5,6) and include d = 1 aspects. – Note: Many important situaons 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 invesgated thus far sasfy (3) and have addional constraints (which can be relaxed).
- 26. The literature is messy. #makeitstop
- 27. #0QFG%QNQTGF0GVYQTM • Node-‐colored network: also known as interconnected network, network of networks, etc. • (three alternative representations)
- 28. /WNVKNC[GT0GVYQTM
- 29. ň CEJCT[-CTCVG%NWD%NWDŉ 0GVYQTM
- 30. 0QFG%QNQTGF0GVYQTMU
- 31. /WNVKRNGZ0GVYQTMU • 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?
- 32. *[RGTITCRJU • 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).
- 33. 1TFKPCN%QWRNKPIUCPF 6GORQTCN0GVYQTMU • 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
- 34. 1VJGT6[RGUQH /WNVKNC[GT0GVYQTMU GZCORNGU • 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
- 35. 3WGUVKQPU!
- 36. “What they saw in the Country of El Dorado real world” #6#
- 37. 5QOGCVC5GVU
- 38. 5QOG/QTGCVC5GVU
- 39. 2TCEVKECNKVKGUCPF/GUUCIGU • 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!
- 40. 3WGUVKQPU!
- 41. “Candide’s Our voyage to Constantinople Istanbul measuring and modeling” /1'.5/'6*15 +#)0156+%5#0;0#/+%5
- 42. #IITGICVKQPQH /WNVKNC[GT0GVYQTMU • 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
- 43. KCIPQUVKEU • 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.
- 44. GITGGUCPF0GKIJDQTJQQFU • 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
- 45. #FLCEGPE[6GPUQTYKVJF#URGEV TGECNNVJKUUNKFG • One can write a general (rank-‐4) mullayer 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 summaon convenon • Page 3 of De Domenico et al., PRX, 2013
- 46. 'ZCORNG6GPUQTKCN0QVCVKQPHQTGITGG GQOGPKEQGVCN24:R
- 47. 'ZCORNG/WNVKFGITGGEGPVTCNKV[a GQOGPKEQGVCN24:R
- 48. 9CNMU2CVJUCPFKUVCPEGU • 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
- 49. %NWUVGTKPI%QGHHKEKGPVU CPF6TCPUKVKXKV[ • 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.
- 50. 'ZCORNG%NWUVGTKPI%QGHHKEKGPV %QQGVCNCT:KX • Our perspective: multilayer walks, which can return to node i on different layers and traverse different numbers of layers!
- 51. %GPVTCNKV[/GCUWTGU • 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.
- 52. +PVGTNC[GTKCIPQUVKEU • 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
- 53. /QFGNUQH/WNVKRNGZ0GVYQTMU • 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
- 54. /QFGNUQH/WNVKRNGZ0GVYQTMU • 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.
- 55. /QFGNUQH +PVGTEQPPGEVGF0GVYQTMU • 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
- 56. %QOOWPKVKGUCPF1VJGT /GUQUECNG5VTWEVWTGU • 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.
- 57. ! 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”
- 58. 'ZCORNGň/WNVKUNKEGŉ0GVYQTMU • P. J. Mucha, T. Richardson, K. Macon, MAP, J.-‐P. Onnela, “Community Structure in Time-‐Dependent, Mulscale, and Mulplex Networks”, Science, Vol. 328, No. 5980, 876–878 (2010) • Simple idea: Glue common nodes across “slices” (i.e. “layers”) • “Diagonal” coupling
- 59. 'ZCORNGKCIPQUVKE/WNVKUNKEG/QFWNCTKV[ • Find communies algorithmically by opmizing “mulslice modularity” – We derived this funcon in Mucha et al, 2010 • Laplacian dynamics: find communies based on how long random walkers are trapped there. Exponenate and then linearize to derive modularity. • Generalizes derivaon 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
- 60. Example: Zachary Karate Club
- 61. 4QNN%CNN8QVKPI0GVYQTMUa GZCORNGVQKNNWUVTCVGGHHGEVQHRCTCOGVGTʩ • 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
- 62. P. J. Mucha M. A. Porter, Chaos, Vol. 20, No. 4, 041108 (2010)
- 63. Braiiiiiiiiiiiiins
- 64. Construcng Time-‐Dependent Networks
- 65. [PCOKE4GEQPHKIWTCVKQPQH*WOCP $TCKP0GVYQTMUWTKPI.GCTPKPIa $CUUGVVGVCN20#5 • 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
- 66. Staonarity and Flexibility • Community staonarity ζ (autocorrelaon over me of community membership): • Node flexibility: – fi = number of mes node i changed communies divided by total number of possible changes – Flexibility f = fi
- 67. [PCOKE%QOOWPKV[5VTWEVWTG • 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.
- 68. Dynamic Reconfiguraon 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 idenfying dynamic changes in modular organizaon over mulple me scales • Main result: flexibility, as measured by allegiance of nodes to communies, in one session predicts amount of learning in subsequent session
- 69. Development of Null Models for Mullayer Networks • D. S. Basse;, M. A. Porter, N. F. Wymbs, S. T. Graon, J. M. Carlson, P. J. Mucha, Chaos, 23(1): 013142 (2013) • Addional structure in adjacency tensors gives more freedom (and responsibility) for choosing null models. • Null models that incorporate informaon 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 resoluon 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 funconal brain networks.
- 70. /GVJQFU$CUGFQP 6GPUQTGEQORQUKVKQP • 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
- 71. [PCOKECN5[UVGOUQP /WNVKNC[GT0GVYQTMU • 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?
- 72. %QPPGEVGF%QORQPGPVU CPF2GTEQNCVKQP • 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
- 73. 2GTEQNCVKQP%CUECFGU • Example (from Buldyrev et al, Nature, 2010)
- 74. 2GTEQNCVKQP%CUECFGU
- 75. 2GTEQNCVKQP%CUECFGU • 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
- 76. %QORCTVOGPVCN5RTGCFKPI /QFGNUCPFKHHWUKQP • 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)
- 77. %QORCTVOGPVCN5RTGCFKPI /QFGNUCPFKHHWUKQP
- 78. %QORCTVOGPVCN5RTGCFKPI /QFGNUCPFKHHWUKQP
- 79. %QORCTVOGPVCN5RTGCFKPI /QFGNUCPFKHHWUKQP • 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)
- 80. %QWRNGFEGNN0GVYQTMU • 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.
- 81. 1VJGT[PCOKECN5[UVGOU • The usual suspects. Pick your favorite. :) • Kuramoto model • Threshold models of social influence – Percolation-‐like – E.g. Watts model • Games on networks • Sandpiles • Others
- 82. %QPVTQNCPF[PCOKEU • 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.
- 83. 3WGUVKQPU!
- 84. 4GOKPFGT)NQUUCT[
- 85. “What befell Candide us at the end of his our journey” %10%.75+105#0 176.11-
- 86. %QPENWUKQPU • 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”!
- 87. 1WVNQQM • All is for the best in this best of all possible worlds. • (Also: The future’s so bright, we gotta to wear shades.)
- 88. 3WGUVKQPU!
- 89. “What befell Candide us after the story ended” #8'46+5'/'065
- 90. #FXGTVKUGOGPVa ,QWTPCNUGFKECVGFVQ0GVYQTM5EKGPEG • 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)!
- 91. #FXGTVKUGOGPV.CMG%QOQ5EJQQNQP %QORNGZ0GVYQTMU /C[ł • 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!
- 92. #FXGTVKUGOGPV9QTMUJQRQPVJG/CVJGOCVKEUCPF2J[UKEU QH/WNVKNC[GT%QORNGZ0GVYQTMU /#2%1/ • 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!
- 93. #FXGTVKUGOGPV/$+5GOGUVGT2TQITCO QP0GVYQTMU 5RTKPI • Mathematical Biosciences Institute, The Ohio State University, USA! • Semester program on “Dynamics of Biologically Inspired Networks”! – http://mbi.osu.edu/programs/ emphasis-programs/future-programs/ spring-2016-dynamics-biologically-inspired- networks/! • Focuses on theoretical questions on networks that arise from biology!
- 94. #FXGTVKUGOGPV/$+92a ň)GPGTCNKGF0GVYQTM5VTWEVWTGUCPF[PCOKEUŉ • March 21–25, 2016! • http://mbi.osu.edu/event/?id=898!

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