1. Networks: from the Small World into the Real World Sang Hoon Lee (Ph.D. in physics) Department of Energy Science, Sungkyunkwan University http://sites.google.com/site/lshlj82 Special Lecture on Network Science, 7 May, 2015 2. statistical physics: micro interactions macro regular/random networks (interactions) microscale structure magnet macroscale properties microscale structure gas macroscale properties 3. irregular, or complex (partially random) networks How about this? Something new but ubiquitous topology ref) M. E. J. Newman, Phys. Rev. E 74, 036104 (2006). Albert Albert Nakarado Barabasi Jeong Aleksiejuk Holyst Stauffer Almaas Kovacs Vicsek Oltvai Krapivsky Redner Kulkarni Stroud Amaral Scala Barthelemy Stanley Meyers Newman Martin Schrag Antal Arena Cabrales DiazGuilera Guimera VegaRedondo DanonGleiser uski Bak Sneppen Bianconi Ravasz Neda Schubert B P Gondran Guichard BenNaim Frauenfelder Toroczkai Berlow Bernardes Costa Araujo Kertesz Capocci Ra Bucolo Fortuna Larosa Sole Caldarelli DeLosRios Coccetti Callaway Hopcroft Kleinberg Strogatz Watts Camacho Servedio Colaiori Caruso Latora Rapisarda Tadic ClausetMoore Cosenza Crucitti Frasca Stagni Usai Marchiori Porta DaFontouraCosta Dezso Dobrin Beg Dodds Muhamad Sabel Williams Martinez Ergun Rodgers Eriksen Simonsen Maslov Farkas Derenyi FerreriCanch Ja Kohler Lawrence Spata Fortunato Fronczak Gastner Girvan Goh Ghim Kahng Kim Lee O h Gorman Guardiola Llas Perez Giralt Mossa Turtschi Herrm Holme Edling Liljeros Ghoshal Huss Kim Yoon Han Trusina Minnhagen Hong Choi Park Mason Tombor Jin Jung Kim Park Kalapala Sanwalani Chung Kim Kinney Kumar Leyvraz SivKaski Aberg Lusseau Macdonald Zaliznyak Matthews Mirollo Montoya Moreira Andrade Forrest Balthrop Leicht Rho Onnela Kanto Jarisaramaki Bassler Corral Park Petermannn Pluchino Podani Szathmary Porter Mucha Warmbrand Somera Mongru DarbyDowman Rosvall Schwartz Salazarciudad Garciafernandez Aharony AdlerMeyerOrtmanns Thurner Tieri Valensin Castellani Remondini Franceschi Kozma Hengartner Korniss Cancho Vazquez Czirok Cohen L Wuchty Yeung Yook Tu a snapshot of network of network scientists 4. irregular, or complex (partially random) networks How about this? Something new but ubiquitous topology ref) M. E. J. Newman, Phys. Rev. E 74, 036104 (2006). Albert Albert Nakarado Barabasi Jeong Aleksiejuk Holyst Stauffer Almaas Kovacs Vicsek Oltvai Krapivsky Redner Kulkarni Stroud Amaral Scala Barthelemy Stanley Meyers Newman Martin Schrag Antal Arena Cabrales DiazGuilera Guimera VegaRedondo DanonGleiser uski Bak Sneppen Bianconi Ravasz Neda Schubert B P Gondran Guichard BenNaim Frauenfelder Toroczkai Berlow Bernardes Costa Araujo Kertesz Capocci Ra Bucolo Fortuna Larosa Sole Caldarelli DeLosRios Coccetti Callaway Hopcroft Kleinberg Strogatz Watts Camacho Servedio Colaiori Caruso Latora Rapisarda Tadic ClausetMoore Cosenza Crucitti Frasca Stagni Usai Marchiori Porta DaFontouraCosta Dezso Dobrin Beg Dodds Muhamad Sabel Williams Martinez Ergun Rodgers Eriksen Simonsen Maslov Farkas Derenyi FerreriCanch Ja Kohler Lawrence Spata Fortunato Fronczak Gastner Girvan Goh Ghim Kahng Kim Lee O h Gorman Guardiola Llas Perez Giralt Mossa Turtschi Herrm Holme Edling Liljeros Ghoshal Huss Kim Yoon Han Trusina Minnhagen Hong Choi Park Mason Tombor Jin Jung Kim Park Kalapala Sanwalani Chung Kim Kinney Kumar Leyvraz SivKaski Aberg Lusseau Macdonald Zaliznyak Matthews Mirollo Montoya Moreira Andrade Forrest Balthrop Leicht Rho Onnela Kanto Jarisaramaki Bassler Corral Park Petermannn Pluchino Podani Szathmary Porter Mucha Warmbrand Somera Mongru DarbyDowman Rosvall Schwartz Salazarciudad Garciafernandez Aharony AdlerMeyerOrtmanns Thurner Tieri Valensin Castellani Remondini Franceschi Kozma Hengartner Korniss Cancho Vazquez Czirok Cohen L Wuchty Yeung Yook Tu a snapshot of network of network scientists Educational projects The problem was posed in 1902 (in different terms, as a problem in group theory) by William Burnside, professor of mathematics at the Royal Naval College, Greenwich. We now know a lot about it, but much remains unknown: for exponent 2, B(m, 2) has size 2m; for exponent 3, Sharing the beauty of networks Mason A. Porter William Burnside During the past couple of years, with my Oxford students and some other collaborators and friends, I have invited several school students to Oxford and visited schools across the UK to teach pupils of ages 1316 about network science, the study of connectivity, closely related to graph theory. in to the study and beauty of mathematics. The students experience B(m, 3) has size 3(m3 + 5m)/6; for exponent 4 the list is finite and good estimates for its size have been developed (here in Oxford), of the form 4r m for suitable real numbers r. For exponent 5 the problem is wide open, though it is known that if B(2, 5) is finite, then its size is 534 . For exponent 6 we know that B(m, 6) is finite and we know its size precisely. For most exponents n > 100 it is known that B(m, n) is infinite. In the last few years Samson Adeleke has been developing a proof, which he was working to The group organisers complete during his stay in Oxford in 201213, that B(2, 11) is infinite. In this context, if correct, that is spectacular progress. What will be even more spectacular will be a solution of the Burnside Problem for exponent 5: is B(2, 5), the list of different words obtained from words in the alphabet {a, b} by insertion or deletion of strings uuuuu, finite or infinite? That way fame and fortune lie. Maths News 2014 [f]_Maths newsletter 1 10/04/2014 10:41 Page 10 5. They are everywhere, indeed. the Internet biochemical network brain ad infinitum 6. The most complicated system in the universe known to itself microscale structure: neuron macroscopic structure: brain or cognition 7. log(cumulativedistribution) 4 3 2 1 0 0.8 1.0 ersize Random 0.8 1.0 ersize Scale free 20 0 20 40 60 Anterior-Posterior 80 100 30 40 50 60 70 80 Ventral-Dorsal Parietal Occipital Inferior temporal Temporal pole Orbitofrontal Prefrontal Premotor Sensorimotor A B C D. S. Bassett and E. Bullmore, Small-World Brain Networks, The Neuroscientist 12, 512 (2006). system-level approach! (including mesoscale structures) The most complicated system in the universe known to itself microscale structure: neuron macroscopic structure: brain or cognition Ed BullmoreDanielle Bassett 8. Network terminology 9. Network terminology N = |V| = 7: number of n# of nodes 10. Network terminology M = |E| = 13: number of# of edges N = |V| = 7: number of n# of nodes 11. Network terminology 1 2 3 4 5 6 7 adjacency matrix W = 0 B B B B B B B B @ 0 8 12 0 0 0 0 0 0 5 0 0 0 0 12 0 0 0 0 0 2 0 0 16 0 8 10 0 0 0 0 0 0 8 2 0 0 0 1 0 0 3 0 0 0 0 6 0 0 1 C C C C C C C C A M = |E| = 13: number of# of edges N = |V| = 7: number of n# of nodes 12. Network terminology 1 2 3 4 5 6 7 adjacency matrix W = 0 B B B B B B B B @ 0 8 12 0 0 0 0 0 0 5 0 0 0 0 12 0 0 0 0 0 2 0 0 16 0 8 10 0 0 0 0 0 0 8 2 0 0 0 1 0 0 3 0 0 0 0 6 0 0 1 C C C C C C C C A 2 outgoing edges (out-degree) 1 incoming edges (in-degree) microscale structure 13. Network terminology 1 2 3 4 5 6 7 adjacency matrix W = 0 B B B B B B B B @ 0 8 12 0 0 0 0 0 0 5 0 0 0 0 12 0 0 0 0 0 2 0 0 16 0 8 10 0 0 0 0 0 0 8 2 0 0 0 1 0 0 3 0 0 0 0 6 0 0 1 C C C C C C C C A 2 outgoing edges (out-degree) 1 incoming edges (in-degree) microscale structure macroscale structure edge density = 13/(76) 0.31 14. some kind of nontrivial mesoscale structure? Network terminology 1 2 3 4 5 6 7 adjacency matrix W = 0 B B B B B B B B @ 0 8 12 0 0 0 0 0 0 5 0 0 0 0 12 0 0 0 0 0 2 0 0 16 0 8 10 0 0 0 0 0 0 8 2 0 0 0 1 0 0 3 0 0 0 0 6 0 0 1 C C C C C C C C A 15. clustering coefcient: how well my neighbors are connected to each other? neighborsnearestitsconnectingedgestheofumber totaltheisandneighborsnearestofnumbertheishere )1( 2 tCoefficienlustering yz zz y C = 3 1 6 2 ==C C(i) = 2yi ki(ki 1) where ki is the node is degree and yi is the number of edges connecting its neighbors to each other i 16. clustering coefcient: how well my neighbors are connected to each other? neighborsnearestitsconnectingedgestheofumber totaltheisandneighborsnearestofnumbertheishere )1( 2 tCoefficienlustering yz zz y C = 3 1 6 2 ==C C(i) = 2yi ki(ki 1) where ki is the node is degree and yi is the number of edges connecting its neighbors to each other i ! C(i) = 2 2 4 3 = 1 3 17. clustering coefcient: how well my neighbors are connected to each other? neighborsnearestitsconnectingedgestheofumber totaltheisandneighborsnearestofnumbertheishere )1( 2 tCoefficienlustering yz zz y C = 3 1 6 2 ==C C(i) = 2yi ki(ki 1) where ki is the node is degree and yi is the number of edges connecting its neighbors to each other i ! C(i) = 2 2 4 3 = 1 3 C = hC(i)i = X i Ci/N 18. clustering coefcient: how well my neighbors are connected to each other? neighborsnearestitsconnectingedgestheofumber totaltheisandneighborsnearestofnumbertheishere )1( 2 tCoefficienlustering yz zz y C = 3 1 6 2 ==C C(i) = 2yi ki(ki 1) where ki is the node is degree and yi is the number of edges connecting its neighbors to each other i ! C(i) = 2 2 4 3 = 1 3 C = hC(i)i = X i Ci/N real networks: much larger C than random networks! 19. average path length of a network path length l between i and j: the number of edges in the shortest path between i and j 1 2 3 4 5 6 7 20. average path length of a network path length l between i and j: the number of edges in the shortest path between i and j 1 2 3 4 5 6 7 21. average path length of a network path length l between i and j: the number of edges in the shortest path between i and j 1 2 3 4 5 6 7 l(1 ! 2) = 1 22. average path length of a network path length l between i and j: the number of edges in the shortest path between i and j 1 2 3 4 5 6 7 l(1 ! 2) = 1 23. average path length of a network path length l between i and j: the number of edges in the shortest path between i and j 1 2 3 4 5 6 7 l(1 ! 2) = 1 l(1 ! 7) = 2 24. average path length of a network path length l between i and j: the number of edges in the shortest path between i and j 1 2 3 4 5 6 7 l(1 ! 2) = 1 l(1 ! 7) = 2 25. average path length of a network path length l between i and j: the number of edges in the shortest path between i and j 1 2 3 4 5 6 7 l(1 ! 2) = 1 l(1 ! 7) = 2 l(1 ! 6) = 4 ... 26. average path length of a network path length l between i and j: the number of edges in the shortest path between i and j 1 2 3 4 5 6 7 average path length = l averaged over all of the node pairs l(1 ! 2) = 1 l(1 ! 7) = 2 l(1 ! 6) = 4 ... 27. average path length of a network path length l between i and j: the number of edges in the shortest path between i and j 1 2 3 4 5 6 7 average path length = l averaged over all of the node pairs real networks: much smaller average path length than regular networks! l(1 ! 2) = 1 l(1 ! 7) = 2 l(1 ! 6) = 4 ... 28. Watts-Strogatz Small World Network D. J.Watts and S. H. Strogatz, Nature 393, 440 (1998). Duncan Watts Steven Strogatz 29. Watts-Strogatz Small World Network D. J.Watts and S. H. Strogatz, Nature 393, 440 (1998). Duncan Watts Steven Strogatz existence of the intermediate regime: small world with clustering l / log N small world criterion: 30. degree: the number of neighboring nodes & its distribution ab = c ! b = loga c 31. p(k) / k p(k) / e (k kaverage)2 / 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7 8 9 p(k) k normal distribution power-law distribution degree: the number of neighboring nodes & its distribution 32. p(k) / k p(k) / e (k kaverage)2 / 2 hubs with large degree 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7 8 9 p(k) k normal distribution power-law distribution degree: the number of neighboring nodes & its distribution 33. p(k) / k p(k) / e (k kaverage)2 / 2 hubs with large degree 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7 8 9 p(k) k normal distribution power-law distribution degree: the number of neighboring nodes & its distribution ubiquitous topology, in fact! 34. Really? (physical) Internet 35. Really? World-Wide-Web 36. metabolic network Really? 37. Implication of the power-law degree distribution p(k) / k 38. Implication of the power-law degree distribution p(k) / k many real networks: 2 < < 3 39. Implication of the power-law degree distribution p(k) / k many real networks: 2 < < 3 nite mean, diverging variance! hki = Z 1 kmin dk kp(k) / Z 1 kmin dk k1 = nite value hk2 i = Z 1 kmin dk k2 p(k) / Z 1 kmin dk k2 ! 1 40. Implication of the power-law degree distribution p(k) / k many real networks: 2 < < 3 nite mean, diverging variance! hki = Z 1 kmin dk kp(k) / Z 1 kmin dk k1 = nite value hk2 i = Z 1 kmin dk k2 p(k) / Z 1 kmin dk k2 ! 1 in general, for n + 1hkn i = Z 1 kmin dk kn p(k) / Z 1 kmin dk kn ! 1 41. Barabsi-Albert Scale-Free Network A.-L. Barabsi and R.Albert, Science 286, 509 (1999). attaching a new node to existing ones with the probability (ki) = ki/ P j kj Albert-Lszl Barabsi Rka Albert k = 2 k = 2 k = 2 (fully connected) initial seed nodes attaching a new node to the existing node i with the probability 42. Barabsi-Albert Scale-Free Network A.-L. Barabsi and R.Albert, Science 286, 509 (1999). attaching a new node to existing ones with the probability (ki) = ki/ P j kj Albert-Lszl Barabsi Rka Albert k = 2 k = 2 k = 2 attaching a new node to the existing node i with the probability 43. Barabsi-Albert Scale-Free Network A.-L. Barabsi and R.Albert, Science 286, 509 (1999). attaching a new node to existing ones with the probability (ki) = ki/ P j kj Albert-Lszl Barabsi Rka Albert k = 2 k = 2 k = 3 k = 1 attaching a new node to the existing node i with the probability 44. Barabsi-Albert Scale-Free Network A.-L. Barabsi and R.Albert, Science 286, 509 (1999). attaching a new node to existing ones with the probability (ki) = ki/ P j kj Albert-Lszl Barabsi Rka Albert k = 2 k = 2 k = 3 k = 1 attaching a new node to the existing node i with the probability 45. Barabsi-Albert Scale-Free Network A.-L. Barabsi and R.Albert, Science 286, 509 (1999). attaching a new node to existing ones with the probability (ki) = ki/ P j kj Albert-Lszl Barabsi Rka Albert k = 2 k = 2 k = 4 k = 1 k = 1 attaching a new node to the existing node i with the probability 46. Barabsi-Albert Scale-Free Network A.-L. Barabsi and R.Albert, Science 286, 509 (1999). attaching a new node to existing ones with the probability (ki) = ki/ P j kj Albert-Lszl Barabsi Rka Albert k = 2 k = 2 k = 4 k = 1 k = 1 attaching a new node to the existing node i with the probability 47. Barabsi-Albert Scale-Free Network A.-L. Barabsi and R.Albert, Science 286, 509 (1999). attaching a new node to existing ones with the probability (ki) = ki/ P j kj Albert-Lszl Barabsi Rka Albert k = 2 k = 3 k = 4 k = 1 k = 1k = 1 attaching a new node to the existing node i with the probability 48. Barabsi-Albert Scale-Free Network A.-L. Barabsi and R.Albert, Science 286, 509 (1999). attaching a new node to existing ones with the probability (ki) = ki/ P j kj Albert-Lszl Barabsi Rka Albert k = 2 k = 3 k = 4 k = 1 k = 1k = 1 attaching a new node to the existing node i with the probability 49. Barabsi-Albert Scale-Free Network A.-L. Barabsi and R.Albert, Science 286, 509 (1999). attaching a new node to existing ones with the probability (ki) = ki/ P j kj Albert-Lszl Barabsi Rka Albert k = 2 k = 3 k = 5 k = 1 k = 1 k = 1k = 1 attaching a new node to the existing node i with the probability 50. Barabsi-Albert Scale-Free Network A.-L. Barabsi and R.Albert, Science 286, 509 (1999). attaching a new node to existing ones with the probability (ki) = ki/ P j kj Albert-Lszl Barabsi Rka Albert k = 2 k = 3 k = 5 k = 1 k = 1 k = 1k = 1 ! p(k) / k 3 k p(k) attaching a new node to the existing node i with the probability 51. Community structures in networks modularity (the objective function to be maximized) M.A. Porter, J.-P. Onnela, and P. J. Mucha, Not.Am. Math. Soc. 56, 1082 (2009); S. Fortunato, Phys. Rep. 486, 75 (2010). Q = 1 2m X ij Aij kikj 2m (gi, gj) where the adjacency matrix Aij 6= 0 if nodes i and j are connected and Aij = 0 otherwise, ki is the degree (number of neighboring nodes of i) or strength (sum of weights around i), gi is the community to which i belongs, and m is the total number of edges or sum of weights in the network resolution parameter: controlling the characteristic size of communities importing network data identifying community structure visualizing smaller communities TAXONOMIESOFNETWORKSFROMCOMMUNITYSTRUCTUREPHYSICALREVIEWE86,036104(2012) thatalledgesareantiferromagneticatresolution=maxand therebyforceseachnodeintoitsowncommunity. III.MESOSCOPICRESPONSEFUNCTIONS(MRFS) Todescribehowanetworkdisintegratesintocommunities asthevalueofisincreasedfrommintomax[seeFig.1(a) foraschematic],oneneedstoselectsummarystatistics.There aremanypossiblewaystosummarizesuchadisintegration process,andwefocusonthreediagnosticsthatcharacterize fundamentalpropertiesofnetworkcommunities. First,weusethevalueoftheHamiltonianH()(1),which isascalarquantitycloselyrelatedtonetworkmodularity andquantiestheenergyofthesystem[13,14].Second, wecalculateapartitionentropyS()tocharacterizethe communitysizedistribution.Todothis,letnkdenotethe numberofnodesincommunitykanddenepk=nk/N tobetheprobabilitytochooseanodefromcommunityk uniformlyatrandom.Thisyieldsa(Shannon)partitionentropy ofS()= () k=1pklogpk,whichquantiesthedisorderin theassociatedcommunitysizedistribution.Third,weusethe numberofcommunities(). (a) (b) BecauseweneedtonormalizeH,S,andtocomparethem acrossnetworks,wedeneaneffectiveenergy Heff()= H()Hmin HmaxHmin =1 H() Hmin ,(4) whereHmin=H(min)andHmax=H(max);aneffective entropy Seff()= S()Smin SmaxSmin = S() logN ,(5) whereSmin=S(min)andSmax=S(max);andaneffective numberofcommunities eff()= ()min maxmin = ()1 N1 ,(6) wheremin=(min)=1andmax=(max)=N. Somenetworkscontainasmallnumberofentriesij thatareordersofmagnitudelargerthanmostotherentries. Forexample,inthenetworkofFacebookfriendshipsat Caltech[21,22],98%oftheijentriesarelessthan100, but0.02%ofthemarelargerthan8000.Theselargeij valuesarisewhentwolow-strengthnodesbecomeconnected. UsingthenullmodelPij=kikj/(2m),theinteractionbetween twonodesiandjbecomesantiferromagneticwhen> Aij/Pij=2mAij/(kikj).Ifanetworkhasalargetotaledge weightbutbothiandjhavesmallstrengthscompared toothernodesinthenetwork,thenneedstobelarge tomaketheinteractionantiferromagnetic.Inpriorstudies, networkcommunitystructurehasbeeninvestigatedatdifferent mesoscopicscalesbyconsideringplotsofvariousdiagnostics asafunctionoftheresolutionparameter[13,14,17].In J.-P. Onnela et al., Phys. Rev. E 86, 036104 (2012). 52. Community structures in networks modularity (the objective function to be maximized) M.A. Porter, J.-P. Onnela, and P. J. Mucha, Not.Am. Math. Soc. 56, 1082 (2009); S. Fortunato, Phys. Rep. 486, 75 (2010). Q = 1 2m X ij Aij kikj 2m (gi, gj) where the adjacency matrix Aij 6= 0 if nodes i and j are connected and Aij = 0 otherwise, ki is the degree (number of neighboring nodes of i) or strength (sum of weights around i), gi is the community to which i belongs, and m is the total number of edges or sum of weights in the network resolution parameter: controlling the characteristic size of communities importing network data identifying community structure visualizing smaller communities TAXONOMIESOFNETWORKSFROMCOMMUNITYSTRUCTUREPHYSICALREVIEWE86,036104(2012) thatalledgesareantiferromagneticatresolution=maxand therebyforceseachnodeintoitsowncommunity. III.MESOSCOPICRESPONSEFUNCTIONS(MRFS) Todescribehowanetworkdisintegratesintocommunities asthevalueofisincreasedfrommintomax[seeFig.1(a) foraschematic],oneneedstoselectsummarystatistics.There aremanypossiblewaystosummarizesuchadisintegration process,andwefocusonthreediagnosticsthatcharacterize fundamentalpropertiesofnetworkcommunities. First,weusethevalueoftheHamiltonianH()(1),which isascalarquantitycloselyrelatedtonetworkmodularity andquantiestheenergyofthesystem[13,14].Second, wecalculateapartitionentropyS()tocharacterizethe communitysizedistribution.Todothis,letnkdenotethe numberofnodesincommunitykanddenepk=nk/N tobetheprobabilitytochooseanodefromcommunityk uniformlyatrandom.Thisyieldsa(Shannon)partitionentropy ofS()= () k=1pklogpk,whichquantiesthedisorderin theassociatedcommunitysizedistribution.Third,weusethe numberofcommunities(). (a) (b) BecauseweneedtonormalizeH,S,andtocomparethem acrossnetworks,wedeneaneffectiveenergy Heff()= H()Hmin HmaxHmin =1 H() Hmin ,(4) whereHmin=H(min)andHmax=H(max);aneffective entropy Seff()= S()Smin SmaxSmin = S() logN ,(5) whereSmin=S(min)andSmax=S(max);andaneffective numberofcommunities eff()= ()min maxmin = ()1 N1 ,(6) wheremin=(min)=1andmax=(max)=N. Somenetworkscontainasmallnumberofentriesij thatareordersofmagnitudelargerthanmostotherentries. Forexample,inthenetworkofFacebookfriendshipsat Caltech[21,22],98%oftheijentriesarelessthan100, but0.02%ofthemarelargerthan8000.Theselargeij valuesarisewhentwolow-strengthnodesbecomeconnected. UsingthenullmodelPij=kikj/(2m),theinteractionbetween twonodesiandjbecomesantiferromagneticwhen> Aij/Pij=2mAij/(kikj).Ifanetworkhasalargetotaledge weightbutbothiandjhavesmallstrengthscompared toothernodesinthenetwork,thenneedstobelarge tomaketheinteractionantiferromagnetic.Inpriorstudies, networkcommunitystructurehasbeeninvestigatedatdifferent mesoscopicscalesbyconsideringplotsofvariousdiagnostics asafunctionoftheresolutionparameter[13,14,17].In J.-P. Onnela et al., Phys. Rev. E 86, 036104 (2012). doi: 10.1038/nature09182 SUPPLEMENTARY INFORMATION Gavroche Valjean Bossuet Mabeuf Bahorel Grantaire Gervais Fauchelevent Gribier Fameuil Listolier Thenardier Bamatabois Champmathieu MmeHucheloup Montparnasse Courfeyrac Enjolras Gillenormand Fantine Tholomyes MariusJoly Brujon Gueulemer Favourite Zephine Eponine MmeMagloire Myriel MmeThenardier Cosette LtGillenormand MlleGillenormand Feuilly MlleBaptistine Blacheville Claquesous Combeferre Javert Woman1 Dahlia Child1 Child2 Perpetue Simplice Babet Pontmercy Chenildieu Napoleon Cravatte Champtercier Scaufflaire Boulatruelle Labarre Judge BaronessT CountessDeLo Isabeau Marguerite Brevet Cochepaille MmePontmercy MlleVaubois Magnon Woman2 Prouvaire MmeDeR Toussaint Count MotherPlutarch MmeBurgon MotherInnocent Anzelma OldMan Jondrette Geborand 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 3: Link communities for the coappearance network of characters in the novel Les Miserables [9]. (Top) the network with link colors indicating the clustering, with grey indicating single-link clusters. Each node is depicted as a pie-chart representing its membership distribution. The main characters have more diverse community membership. (Bottom) the full link dendrogram (left) and partition density (right). Note the internal blue community in the large blue and red clique Y.-Y.Ahn, J. P. Bagrow, and S. Lehmann, Nature 466, 761 (2010). 53. Temporal networks P. Holme and J. Saramki, Phys. Rep. 519, 97 (2012). Petter Holme Jari Saramki P. Holme, J. Saramki / Physics Reports 519 (2012) 97125 the reachability issue and the intransitivity of temporal networks (more specifically a contact sequence). In ( 54. Spatial networks M. Barthlemy, Phys. Rep. 499, 1 (2011). Marc Barthlemy M. Barthlemy / Physics Reports 499 (2011) 1101 10 as crow flies distance between the nodes A and B is dE (A, B) = p 10, while the route distance o equal to Q (A, B) = 4/ p 10 ' 1.265. degree k. If this ratio is small, the number of dead ends and unfinished cro r of regular crossings (k = 4). In the opposite case of large rN , there is a domi ganized city. ompactness of a city, which measures how much a city is filled with roads he total length of roads, the compactness 2 [0, 1] can be defined in terms 18 M. Barthlemy / Physics Reports 499 (2011) 1101 Fig. 12. Physical map of the Swiss railway network. Source: From [83]. (a) (b) (c) (d) Fig. 13. EU rail dataset. (a) Physical layout of the network showing node of size proportional to (a) the degree, (b) the node betweenness centrality, (c) t real load. In (d, top) the degree versus the centrality is shown and in (d, bottom) the betweenness versus the real load. [96]. 55. Multilayer networks M. Kivel,A.Arenas, M. Barthlemy, J. P. Gleeson,Y. Moreno, and M.A. Porter, J. Complex Networks 2, 203 (2014). Mikko Kivel Mason Porter 212 M. KIVEL ET AL. (a) (b) Fig. 3. (a) Visualization of the Zachary Karate Club Club (ZKCC) network as a multilayer network. Nodes (i.e. elements of V) in the network are the four network scientists who have held the coveted karate trophy for a period of time and have been awarded the associated membership in the ZKCC [142]. The current members of the ZKCC are Cris Moore (CM), Mason A. Porter (MAP), Yong-Yeol Ahn (YYA) and Marin Bogu (MB). In the gure, Aaron Clauset (AC) is standing in for CM, as the former awarded the karate trophy to MAP at NetSci13 on behalf of the latter (who did not attend any of the conferences in the gure). The ZKCC network has two aspects: the rst one is the type of relationship between the scientists (talked to each other, went to a talk by the other) and the second one represents a conference in which the trophy was awarded (and thereby passed from one recipient to the next). We assume for simplicity that all of the edges are undirected, even though the relationship of attending a talk obviously need not be a reciprocal one. Each layer includes one elementary layer from each of the two aspects. We represent intra-layer edges using solid curves and inter-layer edges using dotted curves. All of the inter-layer edges are coupling edges because nodes are adjacent only to themselves (and not to other nodes) in other layers, and the inter-layer edges are therefore diagonal. The coupling edges in the rst aspect are categorical (because such edges exist between corresponding nodes in all possible pairs of layers), and the coupling edges in the second aspect are ordinal (see Section 2.7) because there exist such edges only in conferences that are contiguous to each other in time. Additionally, there is a coupling edge between a node and its counterpart in two different layers only when the layers differ from each other in exactly one aspect. [For example, MAP in layer (ECCS13, Talked to each other) is not adjacent to MAP in layer (Workshop in Oxford, Went to a talk by the other).] That is, the ZKCC network does not include inter-aspect coupling (see Section 2.2.1). Additionally, the ZKCC network is not node-aligned because some of the byguestonAugust21http://comnet.oxfordjournals.org/Downloadedfrom 224 M. KIVEL ET AL. (a) (b) Fig. 6. Visualization of two multilayer data sets. (a) Air-transportation network from Ref. [ from a different airline. We drew this network using a recently developed visualization tool fo Bank-wiring room network from Ref. [26]. We represent each individual using a different nod ties from a different type of a relationship. We drew this network using a recently developed (and visualization) of multilayer networks [138]. In both visualizations, the layout of the nod network (which is monoplex). It is thus identical in all layers. for example, there is also a large data set that covers relationships via mu that yields layers that are not node-aligned [202]. Thus far, most empirical studies of multilayer networks have used da network framework. In Table 2, we give a sample of data sets that haveS. Boccaletti et al., Phys. Rep. 544, 1 (2014). Alex Arenas Marc Barthlemy James Gleeson Yamir Moreno 56. note: i and j are node indices, and s and r are layer indices. The adjacency tensor Aijs 6= 0 if nodes i and j are connected in layer s, and Aijs = 0 otherwise. kis is the degree (or strength) of node i in layer s, ms is the number of edges (or sum of weights) in layer s, and s = is the resolution parameter in layer s. Cjsr = ! 6= 0 if layers s and r are connected via node j, and Cjsr = 0 otherwise. The normalization factor 2 = P ijs Aijs + P jsr Cjsr for Qmultilayer 2 [ 1, 1]. Qmultilayer = 1 2 X ijsr Aijs s kiskjs 2ms sr + ijCjsr (gis, gjr) different scales. However, the usual procedure for establishing a quality function as a direct count of the intracommunity edge weight minus that 1 2 3 4 Fig. 1. Schematic of a multislice network. Four slices s = {1, 2, 3, 4} represented by adjacencies Aijs encode intraslice connections (solid lines). Interslice con- nections (dashedlines) are encodedbyCjrs, specifying the coupling of node j to itself between slices r and s. For clarity, interslice couplings are shown for only two nodes and depict two different types of couplings: (i) coupling between neighboring slices, appropriate for ordered slices; and (ii) all-to-all interslice coupling, appropriate for categorical slices. resolution parameters 1 2 3 4 nodes resolution parameters coupling = 1 1 2 3 4 5 10 15 20 25 30 Fig. 2. Multislice community detection of the Zachary Karate Club network (22) across multiple resolutions. Colors depict community assignments of the 34 nodes (renumbered vertically to group similarly assigned nodes) in each of the 16 slices (with resolution parameters gs = {0.25, 0.5, , 4}), for w = 0 (top), w = 0.1 (middle), and w = 1 (bottom). Dashed lines bound the communities obtained using the default resolution (g = 1). Download different slices:time series or categories nodes in individual slices (weighted) edges P. J. Mucha,T. Richardson, K. Macon, M.A. Porter, and J.-P. Onnela, Science 328, 876 (2010). bipartite, directed, and signed networks (14). First, we obtained the resolution-parameter generaliza- ing the Laplacian dynamics to include motion along different kinds of connectionsin this case, for sign We models existing an add betwee by adj interslic r to itse attentio (Aijs = incorpo couplin single-s each no and ac multisli time La respect interslic probab jrkjr, terms o slice s tureallo for intr risj jr where which motion of the probab an inter and it i selectin slice s. from t expone obtaine (14): Qmultisli where probab in each rameter nitude we pre the abs 1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 2000 40PA, 24F, 8AA 151DR, 30AA, 14PA, 5F 141F, 43DR 44D, 2R 1784R, 276D, 149DR, 162J, 53W, 84other 176W, 97AJ, 61DR, 49A, 24D, 19F, 13J, 37other 3168D, 252R, 73other 222D, 6W, 11other 1490R, 247D, 19other Year Senator 10 20 30 40 50 60 70 80 90 100 110 CT ME MA NH RI VT DE NJ NY PA IL IN MI OH WI IA KS MN MO NE ND SD VA AL AR FL GA LA MS NC SC TX KY MD OK TN WV AZ CO ID MT NV NM UT WY CA OR WA AK HI Congress # A B Fig. 3. Multislice community detection of U.S. Senate roll call vote similarities (23) with w = 0.5 coupling of 110 slices (i.e., the number of 2-year Congresses from 1789 to 2008) across time. (A) Colors indicate assignments to nine communities of the 1884 unique senators (sorted vertically and connected across Congresses by dashed lines) in each Congress in which they appear. The dark blue and red communities correspond closely to the modern Democratic and Republican parties, respectively. Horizontal bars indicate the historical period of each community, with accompanying text enumerating nominal party affiliations of the single-slice nodes (each representing a senator in a Congress): PA, pro-administration; AA, anti-administration; F, Federalist; DR, Democratic-Republican; W, Whig; AJ, anti-Jackson; A, Adams; J, Jackson; D, Democratic; R, Republican. Vertical gray bars indicate Congresses in which three communities appeared simultaneously. (B) The same assignments according to state affiliations. www.sciencemag.org SCIENCE VOL 328 14 MAY 2010 JUKKA-PEKKA ONNELA et al. Social Facebook Political: voting Political: cosponsorship Political: committee Protein interaction Metabolic Brain Fungal Financial Language Collaboration effeffeff FIG. 7. (Color online) MRFs for all of the network categories containing at least eight networks (see Table I). At each value of , the upper curve shows the maximum value of Heff (magenta, left panel in each category), Seff (blue, center panels), and eff (black, right panels) for all networks in the category and the lower curve shows the mini- mum value. The dashed curves show the corresponding mean MRFs. A. Voting in the United States Senate Our rst example deals with roll-call voting in the US Senate [3134,48]. Establishing a taxonomy of networks detailing the voting similarities of individual legislators com- plements previous studies of these data, and it facilitates the comparison of voting similarity networks across time. We consider Congresses 1110, which cover the period 17892008. As in Ref. [34], we construct networks from the roll-call data [31,32] for each two-year Congress such that the adjacency matrix element Aij [0,1] represents the number of times Senators i and j voted the same way on a bill (either both in favor of it or both against it) divided by the total number of bills on which both of them voted. Following the approach of Ref. [32], we consider only nonunanimous roll-call votes, which are dened as votes in which at least 3% of the Senators were in the minority. Much research on the US Congress has been devoted to the ebb and ow of partisan polarization over time and the inuence of parties on roll-call voting [33,34]. In highly polarized legislatures, representatives tend to vote along party lines, so there are strong similarities in the voting patterns of members of the same party and strong differences Modularity(Q)Modularity(Q) (a) (b) FI works repres dendr modu color simila time, stem modu of eac polari the in a netw polar [34]. by Q In to re have demo US senators community structures in multilayer networks 57. note: i and j are node indices, and s and r are layer indices. The adjacency tensor Aijs 6= 0 if nodes i and j are connected in layer s, and Aijs = 0 otherwise. kis is the degree (or strength) of node i in layer s, ms is the number of edges (or sum of weights) in layer s, and s = is the resolution parameter in layer s. Cjsr = ! 6= 0 if layers s and r are connected via node j, and Cjsr = 0 otherwise. The normalization factor 2 = P ijs Aijs + P jsr Cjsr for Qmultilayer 2 [ 1, 1]. Qmultilayer = 1 2 X ijsr Aijs s kiskjs 2ms sr + ijCjsr (gis, gjr) different scales. However, the usual procedure for establishing a quality function as a direct count of the intracommunity edge weight minus that 1 2 3 4 Fig. 1. Schematic of a multislice network. Four slices s = {1, 2, 3, 4} represented by adjacencies Aijs encode intraslice connections (solid lines). Interslice con- nections (dashedlines) are encodedbyCjrs, specifying the coupling of node j to itself between slices r and s. For clarity, interslice couplings are shown for only two nodes and depict two different types of couplings: (i) coupling between neighboring slices, appropriate for ordered slices; and (ii) all-to-all interslice coupling, appropriate for categorical slices. resolution parameters 1 2 3 4 nodes resolution parameters coupling = 1 1 2 3 4 5 10 15 20 25 30 Fig. 2. Multislice community detection of the Zachary Karate Club network (22) across multiple resolutions. Colors depict community assignments of the 34 nodes (renumbered vertically to group similarly assigned nodes) in each of the 16 slices (with resolution parameters gs = {0.25, 0.5, , 4}), for w = 0 (top), w = 0.1 (middle), and w = 1 (bottom). Dashed lines bound the communities obtained using the default resolution (g = 1). Download different slices:time series or categories nodes in individual slices (weighted) edges P. J. Mucha,T. Richardson, K. Macon, M.A. Porter, and J.-P. Onnela, Science 328, 876 (2010). bipartite, directed, and signed networks (14). First, we obtained the resolution-parameter generaliza- ing the Laplacian dynamics to include motion along different kinds of connectionsin this case, for sign We models existing an add betwee by adj interslic r to itse attentio (Aijs = incorpo couplin single-s each no and ac multisli time La respect interslic probab jrkjr, terms o slice s tureallo for intr risj jr where which motion of the probab an inter and it i selectin slice s. from t expone obtaine (14): Qmultisli where probab in each rameter nitude we pre the abs 1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 2000 40PA, 24F, 8AA 151DR, 30AA, 14PA, 5F 141F, 43DR 44D, 2R 1784R, 276D, 149DR, 162J, 53W, 84other 176W, 97AJ, 61DR, 49A, 24D, 19F, 13J, 37other 3168D, 252R, 73other 222D, 6W, 11other 1490R, 247D, 19other Year Senator 10 20 30 40 50 60 70 80 90 100 110 CT ME MA NH RI VT DE NJ NY PA IL IN MI OH WI IA KS MN MO NE ND SD VA AL AR FL GA LA MS NC SC TX KY MD OK TN WV AZ CO ID MT NV NM UT WY CA OR WA AK HI Congress # A B Fig. 3. Multislice community detection of U.S. Senate roll call vote similarities (23) with w = 0.5 coupling of 110 slices (i.e., the number of 2-year Congresses from 1789 to 2008) across time. (A) Colors indicate assignments to nine communities of the 1884 unique senators (sorted vertically and connected across Congresses by dashed lines) in each Congress in which they appear. The dark blue and red communities correspond closely to the modern Democratic and Republican parties, respectively. Horizontal bars indicate the historical period of each community, with accompanying text enumerating nominal party affiliations of the single-slice nodes (each representing a senator in a Congress): PA, pro-administration; AA, anti-administration; F, Federalist; DR, Democratic-Republican; W, Whig; AJ, anti-Jackson; A, Adams; J, Jackson; D, Democratic; R, Republican. Vertical gray bars indicate Congresses in which three communities appeared simultaneously. (B) The same assignments according to state affiliations. www.sciencemag.org SCIENCE VOL 328 14 MAY 2010 JUKKA-PEKKA ONNELA et al. Social Facebook Political: voting Political: cosponsorship Political: committee Protein interaction Metabolic Brain Fungal Financial Language Collaboration effeffeff FIG. 7. (Color online) MRFs for all of the network categories containing at least eight networks (see Table I). At each value of , the upper curve shows the maximum value of Heff (magenta, left panel in each category), Seff (blue, center panels), and eff (black, right panels) for all networks in the category and the lower curve shows the mini- mum value. The dashed curves show the corresponding mean MRFs. A. Voting in the United States Senate Our rst example deals with roll-call voting in the US Senate [3134,48]. Establishing a taxonomy of networks detailing the voting similarities of individual legislators com- plements previous studies of these data, and it facilitates the comparison of voting similarity networks across time. We consider Congresses 1110, which cover the period 17892008. As in Ref. [34], we construct networks from the roll-call data [31,32] for each two-year Congress such that the adjacency matrix element Aij [0,1] represents the number of times Senators i and j voted the same way on a bill (either both in favor of it or both against it) divided by the total number of bills on which both of them voted. Following the approach of Ref. [32], we consider only nonunanimous roll-call votes, which are dened as votes in which at least 3% of the Senators were in the minority. Much research on the US Congress has been devoted to the ebb and ow of partisan polarization over time and the inuence of parties on roll-call voting [33,34]. In highly polarized legislatures, representatives tend to vote along party lines, so there are strong similarities in the voting patterns of members of the same party and strong differences Modularity(Q)Modularity(Q) (a) (b) FI works repres dendr modu color simila time, stem modu of eac polari the in a netw polar [34]. by Q In to re have demo US senators multilayer community index: for node i on layer s community structures in multilayer networks 58. note: i and j are node indices, and s and r are layer indices. The adjacency tensor Aijs 6= 0 if nodes i and j are connected in layer s, and Aijs = 0 otherwise. kis is the degree (or strength) of node i in layer s, ms is the number of edges (or sum of weights) in layer s, and s = is the resolution parameter in layer s. Cjsr = ! 6= 0 if layers s and r are connected via node j, and Cjsr = 0 otherwise. The normalization factor 2 = P ijs Aijs + P jsr Cjsr for Qmultilayer 2 [ 1, 1]. Qmultilayer = 1 2 X ijsr Aijs s kiskjs 2ms sr + ijCjsr (gis, gjr) different scales. However, the usual procedure for establishing a quality function as a direct count of the intracommunity edge weight minus that 1 2 3 4 Fig. 1. Schematic of a multislice network. Four slices s = {1, 2, 3, 4} represented by adjacencies Aijs encode intraslice connections (solid lines). Interslice con- nections (dashedlines) are encodedbyCjrs, specifying the coupling of node j to itself between slices r and s. For clarity, interslice couplings are shown for only two nodes and depict two different types of couplings: (i) coupling between neighboring slices, appropriate for ordered slices; and (ii) all-to-all interslice coupling, appropriate for categorical slices. resolution parameters 1 2 3 4 nodes resolution parameters coupling = 1 1 2 3 4 5 10 15 20 25 30 Fig. 2. Multislice community detection of the Zachary Karate Club network (22) across multiple resolutions. Colors depict community assignments of the 34 nodes (renumbered vertically to group similarly assigned nodes) in each of the 16 slices (with resolution parameters gs = {0.25, 0.5, , 4}), for w = 0 (top), w = 0.1 (middle), and w = 1 (bottom). Dashed lines bound the communities obtained using the default resolution (g = 1). Download all of the layers are connected to each other:categorical multilayer communities only the adjacent layers are connected to each other: ordered multilayer communities different slices:time series or categories nodes in individual slices (weighted) edges P. J. Mucha,T. Richardson, K. Macon, M.A. Porter, and J.-P. Onnela, Science 328, 876 (2010). bipartite, directed, and signed networks (14). First, we obtained the resolution-parameter generaliza- ing the Laplacian dynamics to include motion along different kinds of connectionsin this case, for sign We models existing an add betwee by adj interslic r to itse attentio (Aijs = incorpo couplin single-s each no and ac multisli time La respect interslic probab jrkjr, terms o slice s tureallo for intr risj jr where which motion of the probab an inter and it i selectin slice s. from t expone obtaine (14): Qmultisli where probab in each rameter nitude we pre the abs 1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 2000 40PA, 24F, 8AA 151DR, 30AA, 14PA, 5F 141F, 43DR 44D, 2R 1784R, 276D, 149DR, 162J, 53W, 84other 176W, 97AJ, 61DR, 49A, 24D, 19F, 13J, 37other 3168D, 252R, 73other 222D, 6W, 11other 1490R, 247D, 19other Year Senator 10 20 30 40 50 60 70 80 90 100 110 CT ME MA NH RI VT DE NJ NY PA IL IN MI OH WI IA KS MN MO NE ND SD VA AL AR FL GA LA MS NC SC TX KY MD OK TN WV AZ CO ID MT NV NM UT WY CA OR WA AK HI Congress # A B Fig. 3. Multislice community detection of U.S. Senate roll call vote similarities (23) with w = 0.5 coupling of 110 slices (i.e., the number of 2-year Congresses from 1789 to 2008) across time. (A) Colors indicate assignments to nine communities of the 1884 unique senators (sorted vertically and connected across Congresses by dashed lines) in each Congress in which they appear. The dark blue and red communities correspond closely to the modern Democratic and Republican parties, respectively. Horizontal bars indicate the historical period of each community, with accompanying text enumerating nominal party affiliations of the single-slice nodes (each representing a senator in a Congress): PA, pro-administration; AA, anti-administration; F, Federalist; DR, Democratic-Republican; W, Whig; AJ, anti-Jackson; A, Adams; J, Jackson; D, Democratic; R, Republican. Vertical gray bars indicate Congresses in which three communities appeared simultaneously. (B) The same assignments according to state affiliations. www.sciencemag.org SCIENCE VOL 328 14 MAY 2010 JUKKA-PEKKA ONNELA et al. Social Facebook Political: voting Political: cosponsorship Political: committee Protein interaction Metabolic Brain Fungal Financial Language Collaboration effeffeff FIG. 7. (Color online) MRFs for all of the network categories containing at least eight networks (see Table I). At each value of , the upper curve shows the maximum value of Heff (magenta, left panel in each category), Seff (blue, center panels), and eff (black, right panels) for all networks in the category and the lower curve shows the mini- mum value. The dashed curves show the corresponding mean MRFs. A. Voting in the United States Senate Our rst example deals with roll-call voting in the US Senate [3134,48]. Establishing a taxonomy of networks detailing the voting similarities of individual legislators com- plements previous studies of these data, and it facilitates the comparison of voting similarity networks across time. We consider Congresses 1110, which cover the period 17892008. As in Ref. [34], we construct networks from the roll-call data [31,32] for each two-year Congress such that the adjacency matrix element Aij [0,1] represents the number of times Senators i and j voted the same way on a bill (either both in favor of it or both against it) divided by the total number of bills on which both of them voted. Following the approach of Ref. [32], we consider only nonunanimous roll-call votes, which are dened as votes in which at least 3% of the Senators were in the minority. Much research on the US Congress has been devoted to the ebb and ow of partisan polarization over time and the inuence of parties on roll-call voting [33,34]. In highly polarized legislatures, representatives tend to vote along party lines, so there are strong similarities in the voting patterns of members of the same party and strong differences Modularity(Q)Modularity(Q) (a) (b) FI works repres dendr modu color simila time, stem modu of eac polari the in a netw polar [34]. by Q In to re have demo US senators multilayer community index: for node i on layer s parameter space = [ (intralayer resolution), ! (interlayer coupling strength)] community structures in multilayer networks 59. Lagrangian Coherent Structures (LCSs) time-evolving surfaces that shape trajectory patterns in non- autonomous dynamical systems, such as turbulent fluid flows from Mohammad Farazmands Ph.D. thesis 60. Lagrangian Coherent Structures (LCSs) time-evolving surfaces that shape trajectory patterns in non- autonomous dynamical systems, such as turbulent fluid flows from Mohammad Farazmands Ph.D. thesis I n April 2010, fine, airborne ash from a volcanic eruption in Iceland caused chaos throughout European airspace. The same month, the explo- sion at the Deepwater Horizon drilling rig in the Gulf of Mexico left a gushing oil well on the sea flowand thus hides key organizing structures of that flow. Furthermore, traditional trajectory analysis fo- cuses on full trajectory histories that yield convoluted spaghetti plots that are hard to interpret. Improved understanding and forecasting therefore requires New techniques promise better forecasting of where damaging contaminants in the ocean or atmosphere will end up. Thomas Peacock and George Haller Lagrangian coherent structures The hidden skeleton of fluid flows T. Peacock and G. Haller, Physics Today (February, 2013), pp. 41-47. 61. Lagrangian Coherent Structures (LCSs) time-evolving surfaces that shape trajectory patterns in non- autonomous dynamical systems, such as turbulent fluid flows from Mohammad Farazmands Ph.D. thesis I n April 2010, fine, airborne ash from a volcanic eruption in Iceland caused chaos throughout European airspace. The same month, the explo- sion at the Deepwater Horizon drilling rig in the Gulf of Mexico left a gushing oil well on the sea flowand thus hides key organizing structures of that flow. Furthermore, traditional trajectory analysis fo- cuses on full trajectory histories that yield convoluted spaghetti plots that are hard to interpret. Improved understanding and forecasting therefore requires New techniques promise better forecasting of where damaging contaminants in the ocean or atmosphere will end up. Thomas Peacock and George Haller Lagrangian coherent structures The hidden skeleton of fluid flows T. Peacock and G. Haller, Physics Today (February, 2013), pp. 41-47. https://vimeo.com/68802165 Density field and 3D Lagrangian coherent structures obtained from 7 million particle releases in a transitional multi-scale flow in which surface buoyancy driven frontal instabilities trigger deeper baroclinic instabilities. Dispersion characteristics of pollutants in such oceanic flows has been explored. 62. The importance of pollution transport on the ocean surface and on sur- faces of constant density in the atmosphere. Generally speaking, the LCS approach pro- vides a means of identifying key material lines that organize fluid-flow transport. Such material lines account for the linear shape of the ash cloud in figure 1a, the structure of the oil spill in 1b, and the tendrils in the spread of radioactive contamination considering flow transport because patterns such as those in figure 1 arise from material advection. Lagrangian structures a b c Figure 1. Large-scale contaminant flows. (a) A 150-km-wide view of the ash cloud from the 2010 Icelandic volcano eruption. (b) A 300-km-wide view of the 2010 Deepwater Horizon oil spill in the Gulf of Mexico. (c) A prediction of the eastward spread of radioactive contamination into the Pacific Ocean from the 2011 Fukushima reactor disaster in Japan. NASA NASA ASR Lagrangian Coherent Structures (LCSs) T. Peacock and G. Haller, Physics Today (February, 2013), pp. 41-47. 63. The importance of pollution transport on the ocean surface and on sur- faces of constant density in the atmosphere. Generally speaking, the LCS approach pro- vides a means of identifying key material lines that organize fluid-flow transport. Such material lines account for the linear shape of the ash cloud in figure 1a, the structure of the oil spill in 1b, and the tendrils in the spread of radioactive contamination considering flow transport because patterns such as those in figure 1 arise from material advection. Lagrangian structures a b c Figure 1. Large-scale contaminant flows. (a) A 150-km-wide view of the ash cloud from the 2010 Icelandic volcano eruption. (b) A 300-km-wide view of the 2010 Deepwater Horizon oil spill in the Gulf of Mexico. (c) A prediction of the eastward spread of radioactive contamination into the Pacific Ocean from the 2011 Fukushima reactor disaster in Japan. NASA NASA ASR Lagrangian Coherent Structures (LCSs) T. Peacock and G. Haller, Physics Today (February, 2013), pp. 41-47. Maria Antonova @mashant RETWEETS 2,635 FAVORITES 1,468 What happens when you flush a bunch of GPS trackers down a St. Petersburg toilet "@leprasorium: " 2:51 AM - 17 Nov 2014 Follow Related headlines Sddeutsche Zeitung @SZ 18. November 2014 Reply to @mashant @leprasorium Home Notifications Messages Discover Search Twitter 64. Lagrangian vs Eulerian viewpoint on fluid Lagrangian Eulerian from S.Takagi, K. Sugiyama, S. Ii, andY. Matsumoto, J.Appl. Mech. 79, 010911 (2011). Joseph-Louis Lagrange (1736-1813) Leonhard Euler (1707-1783) 65. uid-element network 66. Community structure in network modularity (the objective function to be maximized) M.A. Porter, J.-P. Onnela, and P. J. Mucha, Not.Am. Math. Soc. 56, 1082 (2009); S. Fortunato, Phys. Rep. 486, 75 (2010). Q = 1 2m X ij Aij kikj 2m (gi, gj) where the adjacency matrix Aij 6= 0 if nodes i and j are connected and Aij = 0 otherwise, ki is the degree (number of neighboring nodes of i) or strength (sum of weights around i), gi is the community to which i belongs, and m is the total number of edges or sum of weights in the network importing network data identifying community structure visualizing resolution parameter: controlling the characteristic size of communities smaller communities TAXONOMIESOFNETWORKSFROMCOMMUNITYSTRUCTUREPHYSICALREVIEWE86,036104(2012) thatalledgesareantiferromagneticatresolution=maxand therebyforceseachnodeintoitsowncommunity. III.MESOSCOPICRESPONSEFUNCTIONS(MRFS) Todescribehowanetworkdisintegratesintocommunities asthevalueofisincreasedfrommintomax[seeFig.1(a) foraschematic],oneneedstoselectsummarystatistics.There aremanypossiblewaystosummarizesuchadisintegration process,andwefocusonthreediagnosticsthatcharacterize fundamentalpropertiesofnetworkcommunities. First,weusethevalueoftheHamiltonianH()(1),which isascalarquantitycloselyrelatedtonetworkmodularity andquantiestheenergyofthesystem[13,14].Second, wecalculateapartitionentropyS()tocharacterizethe communitysizedistribution.Todothis,letnkdenotethe numberofnodesincommunitykanddenepk=nk/N tobetheprobabilitytochooseanodefromcommunityk uniformlyatrandom.Thisyieldsa(Shannon)partitionentropy ofS()= () k=1pklogpk,whichquantiesthedisorderin theassociatedcommunitysizedistribution.Third,weusethe numberofcommunities(). (a) (b) BecauseweneedtonormalizeH,S,andtocomparethem acrossnetworks,wedeneaneffectiveenergy Heff()= H()Hmin HmaxHmin =1 H() Hmin ,(4) whereHmin=H(min)andHmax=H(max);aneffective entropy Seff()= S()Smin SmaxSmin = S() logN ,(5) whereSmin=S(min)andSmax=S(max);andaneffective numberofcommunities eff()= ()min maxmin = ()1 N1 ,(6) wheremin=(min)=1andmax=(max)=N. Somenetworkscontainasmallnumberofentriesij thatareordersofmagnitudelargerthanmostotherentries. Forexample,inthenetworkofFacebookfriendshipsat Caltech[21,22],98%oftheijentriesarelessthan100, but0.02%ofthemarelargerthan8000.Theselargeij valuesarisewhentwolow-strengthnodesbecomeconnected. UsingthenullmodelPij=kikj/(2m),theinteractionbetween twonodesiandjbecomesantiferromagneticwhen> Aij/Pij=2mAij/(kikj).Ifanetworkhasalargetotaledge weightbutbothiandjhavesmallstrengthscompared toothernodesinthenetwork,thenneedstobelarge tomaketheinteractionantiferromagnetic.Inpriorstudies, networkcommunitystructurehasbeeninvestigatedatdifferent mesoscopicscalesbyconsideringplotsofvariousdiagnostics asafunctionoftheresolutionparameter[13,14,17].In J.-P. Onnela et al., Phys. Rev. E 86, 036104 (2012). 67. note: i and j are node indices, and s and r are layer indices. The adjacency tensor Aijs 6= 0 if nodes i and j are connected in layer s, and Aijs = 0 otherwise. kis is the degree (or strength) of node i in layer s, ms is the number of edges (or sum of weights) in layer s, and s = is the resolution parameter in layer s. Cjsr = ! 6= 0 if layers s and r are connected via node j, and Cjsr = 0 otherwise. The normalization factor 2 = P ijs Aijs + P jsr Cjsr for Qmultilayer 2 [ 1, 1]. Qmultilayer = 1 2 X ijsr Aijs s kiskjs 2ms sr + ijCjsr (gis, gjr) Community structure in time-dependent or multilayer network different scales. However, the usual procedure for establishing a quality function as a direct count of the intracommunity edge weight minus that 1 2 3 4 Fig. 1. Schematic of a multislice network. Four slices s = {1, 2, 3, 4} represented by adjacencies Aijs encode intraslice connections (solid lines). Interslice con- nections (dashedlines) are encodedbyCjrs, specifying the coupling of node j to itself between slices r and s. For clarity, interslice couplings are shown for only two nodes and depict two different types of couplings: (i) coupling between neighboring slices, appropriate for ordered slices; and (ii) all-to-all interslice coupling, appropriate for categorical slices. resolution parameters 1 2 3 4 nodes resolution parameters coupling = 1 1 2 3 4 5 10 15 20 25 30 Fig. 2. Multislice community detection of the Zachary Karate Club network (22) across multiple resolutions. Colors depict community assignments of the 34 nodes (renumbered vertically to group similarly assigned nodes) in each of the 16 slices (with resolution parameters gs = {0.25, 0.5, , 4}), for w = 0 (top), w = 0.1 (middle), and w = 1 (bottom). Dashed lines bound the communities obtained using the default resolution (g = 1). Download P. J. Mucha,T. Richardson, K. Macon, M.A. Porter, and J.-P. Onnela, Science 328, 876 (2010). different slices:time series nodes in individual slices (weighted) edges multilayer community index: for node i on layer s we have generalized to recover the null models for bipartite, directed, and signed networks (14). First, we obtained the resolution-parameter generaliza- (again with a resolution parameter) by gener ing the Laplacian dynamics to include mo along different kinds of connectionsin this c 1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 2000 40PA, 24F, 8AA 151DR, 30AA, 14PA, 5F 141F, 43DR 44D, 2R 1784R, 276D, 149DR, 162J, 53W, 84other 176W, 97AJ, 61DR, 49A, 24D, 19F, 13J, 37other 3168D, 252R, 73other 222D, 6W, 11other 1490R, 247D, 19other Year Senator 10 20 30 40 50 60 70 80 90 100 110 CT ME MA NH RI VT DE NJ NY PA IL IN MI OH WI IA KS MN MO NE ND SD VA AL AR FL GA LA MS NC SC TX KY MD OK TN WV AZ CO ID MT NV NM UT WY CA OR WA AK HI Congress # A B Fig. 3. Multislice community detection of U.S. Senate roll call vote similarities (23) with w = 0.5 coup of 110 slices (i.e., the number of 2-year Congresses from 1789 to 2008) across time. (A) Colors indi assignments to nine communities of the 1884 unique senators (sorted vertically and connected ac Congresses by dashed lines) in each Congress in which they appear. The dark blue and red commun correspond closely to the modern Democratic and Republican parties, respectively. Horizontal indicate the historical period of each community, with accompanying text enumerating nominal p affiliations of the single-slice nodes (each representing a senator in a Congress): PA, pro-administrat AA, anti-administration; F, Federalist; DR, Democratic-Republican; W, Whig; AJ, anti-Jackson; A, Adam Jackson; D, Democratic; R, Republican. Vertical gray bars indicate Congresses in which three commun appeared simultaneously. (B) The same assignments according to state affiliations. 68. An Example of LCS: simulated ow nary Results of Community Detection in Flow Maps (last updated: January 1, 2014) MAP DATA FIG. 1. Original ow map.M. Farazmand and G. Haller, e-print arXiv:1402.4835. ulent ow. Figure 1 shows LCSs from a direct numerical ulation of the forced NavierStokes equation @u @t + u ru = rp + r2 u + f, r u = 0, (1) the domain [0, 2] [0, 2] with doubly periodic bound- conditions. The Lagrangian analysis is carried out over a eddy turn-over times after the ow has reached its fully ulent state (see Ref. [11] for a detailed analysis). he repelling and attracting LCSs (red and blue curves, re- tively, in gure 1) are the main drivers of mixing through nsive stretching and folding of nearby material elements. green islands, in contrast, represent elliptic LCSs that bit mixing by preserving their shape over relatively long scales. etwork Representation.A fresh way to look at those (gre syst cret scri dete twe cuss netw in R the whi poin them tal s simulated ow from the forced Navier-Stokes equation repelling LCSs attracting LCSs elliptic LCSs pressure external forceviscosity u(x, t) is the velocity eld dened on the two-dimensional domain U as x 2 U = [0, 2] [0, 2] at time t with doubly periodic boundary conditions 69. An Example of LCS: simulated ow nary Results of Community Detection in Flow Maps (last updated: January 1, 2014) MAP DATA FIG. 1. Original ow map.M. Farazmand and G. Haller, e-print arXiv:1402.4835. ulent ow. Figure 1 shows LCSs from a direct numerical ulation of the forced NavierStokes equation @u @t + u ru = rp + r2 u + f, r u = 0, (1) the domain [0, 2] [0, 2] with doubly periodic bound- conditions. The Lagrangian analysis is carried out over a eddy turn-over times after the ow has reached its fully ulent state (see Ref. [11] for a detailed analysis). he repelling and attracting LCSs (red and blue curves, re- tively, in gure 1) are the main drivers of mixing through nsive stretching and folding of nearby material elements. green islands, in contrast, represent elliptic LCSs that bit mixing by preserving their shape over relatively long scales. etwork Representation.A fresh way to look at those (gre syst cret scri dete twe cuss netw in R the whi poin them tal s simulated ow from the forced Navier-Stokes equation repelling LCSs attracting LCSs elliptic LCSs pressure external forceviscosity u(x, t) is the velocity eld dened on the two-dimensional domain U as x 2 U = [0, 2] [0, 2] at time t with doubly periodic boundary conditions 70. Network community analysis of LCS esults of Community Detection in Flow Maps (last updated: January 1, 2014) DATA FIG. 1. Original ow map. ow map: Fig. 1 [512 512 uniform grid points corresponding to [0, 2) [0, 2) eriodic boundary condition (PBC)all the metrics such as distance between two sider the PBC, as presented in Sec. II]. ow map: sampling every fourth (n = 4) element for x and y axes, which yields 8 grid points = 16 384 nodes and their interactions) N OF WEIGHTS W(1) AB = |ri(A, B)| |rf (A, B)| , (1) 1 Lagrangian coherent structures (LCSs) on an example: a bulent ow. Figure 1 shows LCSs from a direct numerical mulation of the forced NavierStokes equation @u @t + u ru = rp + r2 u + f, r u = 0, (1) er the domain [0, 2] [0, 2] with doubly periodic bound- y conditions. The Lagrangian analysis is carried out over a w eddy turn-over times after the ow has reached its fully bulent state (see Ref. [11] for a detailed analysis). The repelling and attracting LCSs (red and blue curves, re- ectively, in gure 1) are the main drivers of mixing through ensive stretching and folding of nearby material elements. e green islands, in contrast, represent elliptic LCSs that hibit mixing by preserving their shape over relatively long me scales. Network Representation.A fresh way to look at those FIG. 1. (color online). Repelling (red), attracting (blue) and elliptic (green) Lagrangian coherent structures (LCSs). systems for that purpose is to consider such systems as dis- crete interacting objects, as in the force-chain networks de- scribing granular material systems [12, 13] or the plume detection problem in uid [14]. General relationships be- tween community nding, transport, and partition are dis- cussed in Refs. [15, 16]. Another example of using the network-theory tools to analyze the ow network is presented in Refs. [17, 18], where the mass transport is represented as the directed edges between geographical sub-areas (nodes), which in fact is rather in line with the spirit of the Eulerian point of view. We, by contrast, consider the uid elements themselves as the nodes, so we can highlight more fundamen- tal structural properties of LCSs. 71. Network community analysis of LCS esults of Community Detection in Flow Maps (last updated: January 1, 2014) DATA FIG. 1. Original ow map. ow map: Fig. 1 [512 512 uniform grid points corresponding to [0, 2) [0, 2) eriodic boundary condition (PBC)all the metrics such as distance between two sider the PBC, as presented in Sec. II]. ow map: sampling every fourth (n = 4) element for x and y axes, which yields 8 grid points = 16 384 nodes and their interactions) N OF WEIGHTS W(1) AB = |ri(A, B)| |rf (A, B)| , (1) 1 Lagrangian coherent structures (LCSs) on an example: a bulent ow. Figure 1 shows LCSs from a direct numerical mulation of the forced NavierStokes equation @u @t + u ru = rp + r2 u + f, r u = 0, (1) er the domain [0, 2] [0, 2] with doubly periodic bound- y conditions. The Lagrangian analysis is carried out over a w eddy turn-over times after the ow has reached its fully bulent state (see Ref. [11] for a detailed analysis). The repelling and attracting LCSs (red and blue curves, re- ectively, in gure 1) are the main drivers of mixing through ensive stretching and folding of nearby material elements. e green islands, in contrast, represent elliptic LCSs that hibit mixing by preserving their shape over relatively long me scales. Network Representation.A fresh way to look at those FIG. 1. (color online). Repelling (red), attracting (blue) and elliptic (green) Lagrangian coherent structures (LCSs). systems for that purpose is to consider such systems as dis- crete interacting objects, as in the force-chain networks de- scribing granular material systems [12, 13] or the plume detection problem in uid [14]. General relationships be- tween community nding, transport, and partition are dis- cussed in Refs. [15, 16]. Another example of using the network-theory tools to analyze the ow network is presented in Refs. [17, 18], where the mass transport is represented as the directed edges between geographical sub-areas (nodes), which in fact is rather in line with the spirit of the Eulerian point of view. We, by contrast, consider the uid elements themselves as the nodes, so we can highlight more fundamen- tal structural properties of LCSs. resented in Sec. II]. every fourth (n = 4) element for x and y axes, which yields 84 nodes and their interactions) W(1) AB = |ri(A, B)| |rf (A, B)| , (1) 1 p {Fxx(A)[xi(B) xi(A)] + Fxy(A)[yi(B) yi(A)]}2 + {Fyx(A)[xi(B) The distance measures such as ri(A, B) and the coordinates such as r among all the possible distances considering the PBC: x itself and y 2 for y (9 combinations in total). III. COMMUNITY DETECTION METHODS W(1) AB(= W(1) BA) in Eq. (1): the Louvain method [1, 2] with the G and the resolution parameter [4] is used for Fig. 3, where th the values of quality measure QGN are specied for four diere here describe the (mutually exclusive for nowwe can extend communities into account by using other methods) groups of interactions are signicantly stronger than the inter-group in 3 nln h maxB2(A)|rf(A,B)|/|ri(A,B)| i forthesamegridelementsasinFig.1ofthemaintext.The nodeA. A, B: discretized 512512 grid cell indices 72. Network community analysis of LCS esults of Community Detection in Flow Maps (last updated: January 1, 2014) DATA FIG. 1. Original ow map. ow map: Fig. 1 [512 512 uniform grid points corresponding to [0, 2) [0, 2) eriodic boundary condition (PBC)all the metrics such as distance between two sider the PBC, as presented in Sec. II]. ow map: sampling every fourth (n = 4) element for x and y axes, which yields 8 grid points = 16 384 nodes and their interactions) N OF WEIGHTS W(1) AB = |ri(A, B)| |rf (A, B)| , (1) 1 Lagrangian coherent structures (LCSs) on an example: a bulent ow. Figure 1 shows LCSs from a direct numerical mulation of the forced NavierStokes equation @u @t + u ru = rp + r2 u + f, r u = 0, (1) er the domain [0, 2] [0, 2] with doubly periodic bound- y conditions. The Lagrangian analysis is carried out over a w eddy turn-over times after the ow has reached its fully bulent state (see Ref. [11] for a detailed analysis). The repelling and attracting LCSs (red and blue curves, re- ectively, in gure 1) are the main drivers of mixing through ensive stretching and folding of nearby material elements. e green islands, in contrast, represent elliptic LCSs that hibit mixing by preserving their shape over relatively long me scales. Network Representation.A fresh way to look at those FIG. 1. (color online). Repelling (red), attracting (blue) and elliptic (green) Lagrangian coherent structures (LCSs). systems for that purpose is to consider such systems as dis- crete interacting objects, as in the force-chain networks de- scribing granular material systems [12, 13] or the plume detection problem in uid [14]. General relationships be- tween community nding, transport, and partition are dis- cussed in Refs. [15, 16]. Another example of using the network-theory tools to analyze the ow network is presented in Refs. [17, 18], where the mass transport is represented as the directed edges between geographical sub-areas (nodes), which in fact is rather in line with the spirit of the Eulerian point of view. We, by contrast, consider the uid elements themselves as the nodes, so we can highlight more fundamen- tal structural properties of LCSs. resented in Sec. II]. every fourth (n = 4) element for x and y axes, which yields 84 nodes and their interactions) W(1) AB = |ri(A, B)| |rf (A, B)| , (1) 1 p {Fxx(A)[xi(B) xi(A)] + Fxy(A)[yi(B) yi(A)]}2 + {Fyx(A)[xi(B) The distance measures such as ri(A, B) and the coordinates such as r among all the possible distances considering the PBC: x itself and y 2 for y (9 combinations in total). III. COMMUNITY DETECTION METHODS W(1) AB(= W(1) BA) in Eq. (1): the Louvain method [1, 2] with the G and the resolution parameter [4] is used for Fig. 3, where th the values of quality measure QGN are specied for four diere here describe the (mutually exclusive for nowwe can extend communities into account by using other methods) groups of interactions are signicantly stronger than the inter-group in 3 nln h maxB2(A)|rf(A,B)|/|ri(A,B)| i forthesamegridelementsasinFig.1ofthemaintext.The nodeA. A, B: discretized 512512 grid cell indices A B C D 73. Network community analysis of LCS esults of Community Detection in Flow Maps (last updated: January 1, 2014) DATA FIG. 1. Original ow map. ow map: Fig. 1 [512 512 uniform grid points corresponding to [0, 2) [0, 2) eriodic boundary condition (PBC)all the metrics such as distance between two sider the PBC, as presented in Sec. II]. ow map: sampling every fourth (n = 4) element for x and y axes, which yields 8 grid points = 16 384 nodes and their interactions) N OF WEIGHTS W(1) AB = |ri(A, B)| |rf (A, B)| , (1) 1 Lagrangian coherent structures (LCSs) on an example: a bulent ow. Figure 1 shows LCSs from a direct numerical mulation of the forced NavierStokes equation @u @t + u ru = rp + r2 u + f, r u = 0, (1) er the domain [0, 2] [0, 2] with doubly periodic bound- y conditions. The Lagrangian analysis is carried out over a w eddy turn-over times after the ow has reached its fully bulent state (see Ref. [11] for a detailed analysis). The repelling and attracting LCSs (red and blue curves, re- ectively, in gure 1) are the main drivers of mixing through ensive stretching and folding of nearby material elements. e green islands, in contrast, represent elliptic LCSs that hibit mixing by preserving their shape over relatively long me scales. Network Representation.A fresh way to look at those FIG. 1. (color online). Repelling (red), attracting (blue) and elliptic (green) Lagrangian coherent structures (LCSs). systems for that purpose is to consider such systems as dis- crete interacting objects, as in the force-chain networks de- scribing granular material systems [12, 13] or the plume detection problem in uid [14]. General relationships be- tween community nding, transport, and partition are dis- cussed in Refs. [15, 16]. Another example of using the network-theory tools to analyze the ow network is presented in Refs. [17, 18], where the mass transport is represented as the directed edges between geographical sub-areas (nodes), which in fact is rather in line with the spirit of the Eulerian point of view. We, by contrast, consider the uid elements themselves as the nodes, so we can highlight more fundamen- tal structural properties of LCSs. resented in Sec. II]. every fourth (n = 4) element for x and y axes, which yields 84 nodes and their interactions) W(1) AB = |ri(A, B)| |rf (A, B)| , (1) 1 p {Fxx(A)[xi(B) xi(A)] + Fxy(A)[yi(B) yi(A)]}2 + {Fyx(A)[xi(B) The distance measures such as ri(A, B) and the coordinates such as r among all the possible distances considering the PBC: x itself and y 2 for y (9 combinations in total). III. COMMUNITY DETECTION METHODS W(1) AB(= W(1) BA) in Eq. (1): the Louvain method [1, 2] with the G and the resolution parameter [4] is used for Fig. 3, where th the values of quality measure QGN are specied for four diere here describe the (mutually exclusive for nowwe can extend communities into account by using other methods) groups of interactions are signicantly stronger than the inter-group in 3 nln h maxB2(A)|rf(A,B)|/|ri(A,B)| i forthesamegridelementsasinFig.1ofthemaintext.The nodeA. A, B: discretized 512512 grid cell indices |ri(A, B)| A B C D |ri(C, D)| 74. Network community analysis of LCS esults of Community Detection in Flow Maps (last updated: January 1, 2014) DATA FIG. 1. Original ow map. ow map: Fig. 1 [512 512 uniform grid points corresponding to [0, 2) [0, 2) eriodic boundary condition (PBC)all the metrics such as distance between two sider the PBC, as presented in Sec. II]. ow map: sampling every fourth (n = 4) element for x and y axes, which yields 8 grid points = 16 384 nodes and their interactions) N OF WEIGHTS W(1) AB = |ri(A, B)| |rf (A, B)| , (1) 1 Lagrangian coherent structures (LCSs) on an example: a bulent ow. Figure 1 shows LCSs from a direct numerical mulation of the forced NavierStokes equation @u @t + u ru = rp + r2 u + f, r u = 0, (1) er the domain [0, 2] [0, 2] with doubly periodic bound- y conditions. The Lagrangian analysis is carried out over a w eddy turn-over times after the ow has reached its fully bulent state (see Ref. [11] for a detailed analysis). The repelling and attracting LCSs (red and blue curves, re- ectively, in gure 1) are the main drivers of mixing through ensive stretching and folding of nearby material elements. e green islands, in contrast, represent elliptic LCSs that hibit mixing by preserving their shape over relatively long me scales. Network Representation.A fresh way to look at those FIG. 1. (color online). Repelling (red), attracting (blue) and elliptic (green) Lagrangian coherent structures (LCSs). systems for that purpose is to consider such systems as dis- crete interacting objects, as in the force-chain networks de- scribing granular material systems [12, 13] or the plume detection problem in uid [14]. General relationships be- tween community nding, transport, and partition are dis- cussed in Refs. [15, 16]. Another example of using the network-theory tools to analyze the ow network is presented in Refs. [17, 18], where the mass transport is represented as the directed edges between geographical sub-areas (nodes), which in fact is rather in line with the spirit of the Eulerian point of view. We, by contrast, consider the uid elements themselves as the nodes, so we can highlight more fundamen- tal structural properties of LCSs. resented in Sec. II]. every fourth (n = 4) element for x and y axes, which yields 84 nodes and their interactions) W(1) AB = |ri(A, B)| |rf (A, B)| , (1) 1 p {Fxx(A)[xi(B) xi(A)] + Fxy(A)[yi(B) yi(A)]}2 + {Fyx(A)[xi(B) The distance measures such as ri(A, B) and the coordinates such as r among all the possible distances considering the PBC: x itself and y 2 for y (9 combinations in total). III. COMMUNITY DETECTION METHODS W(1) AB(= W(1) BA) in Eq. (1): the Louvain method [1, 2] with the G and the resolution parameter [4] is used for Fig. 3, where th the values of quality measure QGN are specied for four diere here describe the (mutually exclusive for nowwe can extend communities into account by using other methods) groups of interactions are signicantly stronger than the inter-group in 3 nln h maxB2(A)|rf(A,B)|/|ri(A,B)| i forthesamegridelementsasinFig.1ofthemaintext.The nodeA. A, B: discretized 512512 grid cell indices |rf (A, B)| |ri(A, B)| A BA B C D |ri(C, D)| 75. Network community analysis of LCS esults of Community Detection in Flow Maps (last updated: January 1, 2014) DATA FIG. 1. Original ow map. ow map: Fig. 1 [512 512 uniform grid points corresponding to [0, 2) [0, 2) eriodic boundary condition (PBC)all the metrics such as distance between two sider the PBC, as presented in Sec. II]. ow map: sampling every fourth (n = 4) element for x and y axes, which yields 8 grid points = 16 384 nodes and their interactions) N OF WEIGHTS W(1) AB = |ri(A, B)| |rf (A, B)| , (1) 1 Lagrangian coherent structures (LCSs) on an example: a bulent ow. Figure 1 shows LCSs from a direct numerical mulation of the forced NavierStokes equation @u @t + u ru = rp + r2 u + f, r u = 0, (1) er the domain [0, 2] [0, 2] with doubly periodic bound- y conditions. The Lagrangian analysis is carried out over a w eddy turn-over times after the ow has reached its fully bulent state (see Ref. [11] for a detailed analysis). The repelling and attracting LCSs (red and blue curves, re- ectively, in gure 1) are the main drivers of mixing through ensive stretching and folding of nearby material elements. e green islands, in contrast, represent elliptic LCSs that hibit mixing by preserving their shape over relatively long me scales. Network Representation.A fresh way to look at those FIG. 1. (color online). Repelling (red), attracting (blue) and elliptic (green) Lagrangian coherent structures (LCSs). systems for that purpose is to consider such systems as dis- crete interacting objects, as in the force-chain networks de- scribing granular material systems [12, 13] or the plume detection problem in uid [14]. General relationships be- tween community nding, transport, and partition are dis- cussed in Refs. [15, 16]. Another example of using the network-theory tools to analyze the ow network is presented in Refs. [17, 18], where the mass transport is represented as the directed edges between geographical sub-areas (nodes), which in fact is rather in line with the spirit of the Eulerian point of view. We, by contrast, consider the uid elements themselves as the nodes, so we can highlight more fundamen- tal structural properties of LCSs. resented in Sec. II]. every fourth (n = 4) element for x and y axes, which yields 84 nodes and their interactions) W(1) AB = |ri(A, B)| |rf (A, B)| , (1) 1 p {Fxx(A)[xi(B) xi(A)] + Fxy(A)[yi(B) yi(A)]}2 + {Fyx(A)[xi(B) The distance measures such as ri(A, B) and the coordinates such as r among all the possible distances considering the PBC: x itself and y 2 for y (9 combinations in total). III. COMMUNITY DETECTION METHODS W(1) AB(= W(1) BA) in Eq. (1): the Louvain method [1, 2] with the G and the resolution parameter [4] is used for Fig. 3, where th the values of quality measure QGN are specied for four diere here describe the (mutually exclusive for nowwe can extend communities into account by using other methods) groups of interactions are signicantly stronger than the inter-group in 3 nln h maxB2(A)|rf(A,B)|/|ri(A,B)| i forthesamegridelementsasinFig.1ofthemaintext.The nodeA. A, B: discretized 512512 grid cell indices |rf (A, B)| |ri(A, B)| A BA B C D D C |ri(C, D)| |rf (C, D)| 76. belonging to the different coherent structures (small )W (1) CD Network community analysis of LCS esults of Community Detection in Flow Maps (last updated: January 1, 2014) DATA FIG. 1. Original ow map. ow map: Fig. 1 [512 512 uniform grid points corresponding to [0, 2) [0, 2) eriodic boundary condition (PBC)all the metrics such as distance between two sider the PBC, as presented in Sec. II]. ow map: sampling every fourth (n = 4) element for x and y axes, which yields 8 grid points = 16 384 nodes and their interactions) N OF WEIGHTS W(1) AB = |ri(A, B)| |rf (A, B)| , (1) 1 Lagrangian coherent structures (LCSs) on an example: a bulent ow. Figure 1 shows LCSs from a direct numerical mulation of the forced NavierStokes equation @u @t + u ru = rp + r2 u + f, r u = 0, (1) er the domain [0, 2] [0, 2] with doubly periodic bound- y conditions. The Lagrangian analysis is carried out over a w eddy turn-over times after the ow has reached its fully bulent state (see Ref. [11] for a detailed analysis). The repelling and attracting LCSs (red and blue curves, re- ectively, in gure 1) are the main drivers of mixing through ensive stretching and folding of nearby material elements. e green islands, in contrast, represent elliptic LCSs that hibit mixing by preserving their shape over relatively long me scales. Network Representation.A fresh way to look at those FIG. 1. (color online). Repelling (red), attracting (blue) and elliptic (green) Lagrangian coherent structures (LCSs). systems for that purpose is to consider such systems as dis- crete interacting objects, as in the force-chain networks de- scribing granular material systems [12, 13] or the plume detection problem in uid [14]. General relationships be- tween community nding, transport, and partition are dis- cussed in Refs. [15, 16]. Another example of using the network-theory tools to analyze the ow network is presented in Refs. [17, 18], where the mass transport is represented as the directed edges between geographical sub-areas (nodes), which in fact is rather in line with the spirit of the Eulerian point of view. We, by contrast, consider the uid elements themselves as the nodes, so we can highlight more fundamen- tal structural properties of LCSs. resented in Sec. II]. every fourth (n = 4) element for x and y axes, which yields 84 nodes and their interactions) W(1) AB = |ri(A, B)| |rf (A, B)| , (1) 1 p {Fxx(A)[xi(B) xi(A)] + Fxy(A)[yi(B) yi(A)]}2 + {Fyx(A)[xi(B) The distance measures such as ri(A, B) and the coordinates such as r among all the possible distances considering the PBC: x itself and y 2 for y (9 combinations in total). III. COMMUNITY DETECTION METHODS W(1) AB(= W(1) BA) in Eq. (1): the Louvain method [1, 2] with the G and the resolution parameter [4] is used for Fig. 3, where th the values of quality measure QGN are specied for four diere here describe the (mutually exclusive for nowwe can extend communities into account by using other methods) groups of interactions are signicantly stronger than the inter-group in 3 nln h maxB2(A)|rf(A,B)|/|ri(A,B)| i forthesamegridelementsasinFig.1ofthemaintext.The nodeA. A, B: discretized 512512 grid cell indices belonging to the same coherent structure (large )W (1) AB |rf (A, B)| |ri(A, B)| A BA B C D D C |ri(C, D)| |rf (C, D)| 77. Network community analysis of LCS esults of Community Detection in Flow Maps (last updated: January 1, 2014) DATA FIG. 1. Original ow map. ow map: Fig. 1 [512 512 uniform grid points corresponding to [0, 2) [0, 2) eriodic boundary condition (PBC)all the metrics such as distance between two sider the PBC, as presented in Sec. II]. ow map: sampling every fourth (n = 4) element for x and y axes, which yields 8 grid points = 16 384 nodes and their interactions) N OF WEIGHTS W(1) AB = |ri(A, B)| |rf (A, B)| , (1) 1 Lagrangian coherent structures (LCSs) on an example: a bulent ow. Figure 1 shows LCSs from a direct numerical mulation of the forced NavierStokes equation @u @t + u ru = rp + r2 u + f, r u = 0, (1) er the domain [0, 2] [0, 2] with doubly periodic bound- y conditions. The Lagrangian analysis is carried out over a w eddy turn-over times after the ow has reached its fully bulent state (see Ref. [11] for a detailed analysis). The repelling and attracting LCSs (red and blue curves, re- ectively, in gure 1) are the main drivers of mixing through ensive stretching and folding of nearby material elements. e green islands, in contrast, represent elliptic LCSs that hibit mixing by preserving their shape over relatively long me scales. Network Representation.A fresh way to look at those FIG. 1. (color online). Repelling (red), attracting (blue) and elliptic (green) Lagrangian coherent structures (LCSs). systems for that purpose is to consider such systems as dis- crete interacting objects, as in the force-chain networks de- scribing granular material systems [12, 13] or the plume detection problem in uid [14]. General relationships be- tween community nding, transport, and partition are dis- cussed in Refs. [15, 16]. Another example of using the network-theory tools to analyze the ow network is presented in Refs. [17, 18], where the mass transport is represented as the directed edges between geographical sub-areas (nodes), which in fact is rather in line with the spirit of the Eulerian point of view. We, by contrast, consider the uid elements themselves as the nodes, so we can highlight more fundamen- tal structural properties of LCSs. resented in Sec. II]. every fourth (n = 4) element for x and y axes, which yields 84 nodes and their interactions) W(1) AB = |ri(A, B)| |rf (A, B)| , (1) 1 p {Fxx(A)[xi(B) xi(A)] + Fxy(A)[yi(B) yi(A)]}2 + {Fyx(A)[xi(B) The distance measures such as ri(A, B) and the coordinates such as r among all the possible distances considering the PBC: x itself and y 2 for y (9 combinations in total). III. COMMUNITY DETECTION METHODS W(1) AB(= W(1) BA) in Eq. (1): the Louvain method [1, 2] with the G and the resolution parameter [4] is used for Fig. 3, where th the values of quality measure QGN are specied for four diere here describe the (mutually exclusive for nowwe can extend communities into account by using other methods) groups of interactions are signicantly stronger than the inter-group in 3 nln h maxB2(A)|rf(A,B)|/|ri(A,B)| i forthesamegridelementsasinFig.1ofthemaintext.The nodeA. A, B: discretized 512512 grid cell indices without involving differential terms such as the deformation-gradient tensor in ow problems 3] and the analysis 4, 15]. Network- tion, were used to 18]. Note, how- transport GH:How edges between ge- ok what is essen- lar type of vortex is investigated in n technique in the number of vortex- type of coherent y contrast, we ex- neral types of dy- more work; also, section? I made taneous displacement instead of the nal position. T W(2) AB = |ri(A, B)| |F(A)ri(A, B)| , where F(A) is the deformation-gradient tensor appl location of node A dened as the Jacobian @(xf , yf of the ow map. The relative dispersions W(1) AB an Eqs. (1) and (3) give the weighted adjacency-matrix between nodes A and B [7]. Larger values of these m ments indicate stronger connections between the tw ements A and B. For each of the measures, distance such as ri(A, B) and coordinates such as ri(A) use th distance among all of the possible distances. (Note th periodic boundary conditions for the simulation dat trast to the real drifter data, GH:This does not appea a generally usable approach to ocean data SHL:I bit of comments to avoid the possible confusion. and x 2 both yield x; and y and y 2 both yield ems ysis ork- d to ow- How ge- sen- rtex d in the tex- rent ex- dy- lso, ade eed we taneous displacement instead of the nal position. This yields W(2) AB = |ri(A, B)| |F(A)ri(A, B)| , (3) where F(A) is the deformation-gradient tensor applied to the location of node A dened as the Jacobian @(xf , yf )/@(xi, yi) of the ow map. The relative dispersions W(1) AB and W(2) AB in Eqs. (1) and (3) give the weighted adjacency-matrix elements between nodes A and B [7]. Larger values of these matrix ele- ments indicate stronger connections between the two uid el- ements A and B. For each of the measures, distance measures such as ri(A, B) and coordinates such as ri(A) use the shortest distance among all of the possible distances. (Note that we use periodic boundary conditions for the simulation data, in con- trast to the real drifter data, GH:This does not appear to give a generally usable approach to ocean data SHL:I added a bit of comments to avoid the possible confusion. so x itself and x 2 both yield x; and y and y 2 both yield y.) Community Detection.There are several methods to de- tect communities in networks [8]. In our paper, we will ap- : the Jacobian of the ow map, so that ding, transport, . The Jacobian lyzed explicitly mesoscale struc- ow problems and the analysis 15]. Network- n, were used to ]. Note, how- nsport GH:How ges between ge- what is essen- type of vortex investigated in echnique in the mber of vortex- pe of coherent ontrast, we ex- ral types of dy- of A. If the uid elements are initially given on a rectangular grid, then (A) is given by the four closest neighbors consti- tute. In Fig. S1 of the SM [5], we show the relative dispersions for the same simulation as in Fig. 1. An alternative way to dene the relative dispersion is to use the deformation-gradient tensor [19] to obtain the instan- taneous displacement instead of the nal position. This yields W(2) AB = |ri(A, B)| |F(A)ri(A, B)| , (3) where F(A) is the deformation-gradient tensor applied to the location of node A dened as the Jacobian @(xf , yf )/@(xi, yi) of the ow map. The relative dispersions W(1) AB and W(2) AB in Eqs. (1) and (3) give the weighted adjacency-matrix elements between nodes A and B [7]. Larger values of these matrix ele- ments indicate stronger connections between the two uid el- ements A and B. For each of the measures, distance measures such as ri(A, B) and coordinates such as ri(A) use the shortest distance among all of the possible distances. (Note that we use periodic boundary conditions for the simulation data, in con- trast to the real drifter data, GH:This does not appear to give 78. Network community analysis of LCS esults of Community Detection in Flow Maps (last updated: January 1, 2014) DATA FIG. 1. Original ow map. ow map: Fig. 1 [512 512 uniform grid points corresponding to [0, 2) [0, 2) eriodic boundary condition (PBC)all the metrics such as distance between two sider the PBC, as presented in Sec. II]. ow map: sampling every fourth (n = 4) element for x and y axes, which yields 8 grid points = 16 384 nodes and their interactions) N OF WEIGHTS W(1) AB = |ri(A, B)| |rf (A, B)| , (1) 1 Lagrangian coherent structures (LCSs) on an example: a bulent ow. Figure 1 shows LCSs from a direct numerical mulation of the forced NavierStokes equation @u @t + u ru = rp + r2 u + f, r u = 0, (1) er the domain [0, 2] [0, 2] with doubly periodic bound- y conditions. The Lagrangian analysis is carried out over a w eddy turn-over times after the ow has reached its fully bulent state (see Ref. [11] for a detailed analysis). The repelling and attracting LCSs (red and blue curves, re- ectively, in gure 1) are the mai