1. The Architecture of Complexity: From the WWW to network biology www.nd.edu/~networks title

2. protein-gene interactions protein-protein interactions PROTEOME GENOME Citrate Cycle METABOLISM Bio-chemical reactions Bio-Map

3. Erdös-Rényi model (1960) - Democratic - Random Pál Erdös Pál Erdös (1913-1996) Connect with probability p p=1/6 N=10  k  ~ 1.5 Poisson distribution

4. World Wide Web Over 3 billion documents ROBOT: collects all URL’s found in a document and follows them recursively Nodes: WWW documents Links: URL links R. Albert, H. Jeong, A-L Barabasi, Nature, 401 130 (1999). WWW Expected P(k) ~ k -  Found

5. What does it mean? Poisson distribution Random Network Power-law distribution Scale-free Network Airlines

6. INTERNET BACKBONE (Faloutsos, Faloutsos and Faloutsos, 1999) Nodes: computers, routers Links: physical lines Internet

7. Internet-Map

8. Swedish sex-web Nodes: people (Females; Males) Links: sexual relationships Liljeros et al. Nature 2001 4781 Swedes; 18-74; 59% response rate.

9. Many real world networks have the same architecture: Scale-free networks WWW, Internet (routers and domains), electronic circuits, computer software, movie actors, coauthorship networks, sexual web, instant messaging, email web, citations, phone calls, metabolic, protein interaction, protein domains, brain function web, linguistic networks, comic book characters, international trade, bank system, encryption trust net, energy landscapes, earthquakes, astrophysical network…

10. Scale-free model Barabási & Albert, Science 286, 509 (1999) P(k) ~k -3 BA model (1) Networks continuously expand by the addition of new nodes WWW : addition of new documents GROWTH: add a new node with m links PREFERENTIAL ATTACHMENT: the probability that a node connects to a node with k links is proportional to k. (2) New nodes prefer to link to highly connected nodes. WWW : linking to well known sites

11. protein-gene interactions protein-protein interactions PROTEOME GENOME Citrate Cycle METABOLISM Bio-chemical reactions Bio-Map

12. Citrate Cycle METABOLISM Bio-chemical reactions

13. Boehring-Mennheim

14. Metabolic Network Nodes : chemicals (substrates) Links : bio-chemical reactions Metab-movie

15. Metabolic network Organisms from all three domains of life are scale-free networks! H. Jeong, B. Tombor, R. Albert, Z.N. Oltvai, and A.L. Barabasi, Nature, 407 651 (2000) ArchaeaBacteriaEukaryotes Meta-P(k)

16. protein-gene interactions protein-protein interactions PROTEOME GENOME Citrate Cycle METABOLISM Bio-chemical reactions Bio-Map

17. protein-protein interactions PROTEOME

18. Topology of the protein network H. Jeong, S.P. Mason, A.-L. Barabasi, Z.N. Oltvai, Nature 411, 41-42 (2001) Prot P(k) Nodes : proteins Links : physical interactions (binding)

19. Origin of the scale-free topology: Gene Duplication Perfect copy Mistake: gene duplication Wagner (2001); Vazquez et al. 2003; Sole et al. 2001; Rzhetsky & Gomez (2001); Qian et al. (2001); Bhan et al. (2002). Proteins with more interactions are more likely to get a new link: Π(k)~k (preferential attachment).

20. Robustness Complex systems maintain their basic functions even under errors and failures (cell  mutations; Internet  router breakdowns) node failure fcfc 01 Fraction of removed nodes, f 1 S Robustness

21. Robustness of scale-free networks 1 S 0 1 f fcfc Attacks   3 : f c =1 (R. Cohen et al PRL, 2000) Failures Robust-SF Albert, Jeong, Barabasi, Nature 406 378 (2000)

22. Yeast protein network - lethality and topological position - Highly connected proteins are more essential (lethal)... Prot- robustness H. Jeong, S.P. Mason, A.-L. Barabasi, Z.N. Oltvai, Nature 411, 41-42 (2001)

23.  Hypothesis: Biological function are carried by discrete functional modules.  Hartwell, L.-H., Hopfield, J. J., Leibler, S., & Murray, A. W. (1999).  Question: Is modularity a myth, or a structural property of biological networks? (are biological networks fundamentally modular?) Modularity in Cellular Networks  Traditional view of modularity:

24. Modular vs. Scale-free Topology Scale-free (a) Modular (b)

25. Hierarchical Networks 3. Clustering coefficient scales C(k)= # links between k neighbors k(k-1)/2

26. Scaling of the clustering coefficient C(k) The metabolism forms a hierachical network. Ravasz, Somera, Mongru, Oltvai, A-L. B, Science 297, 1551 (2002).

27. Can we identify the modules? l i,j is 1 if there is a direct link between i and j, 0 otherwise

28. Modules in the E. coli metabolism

29. The structure of pyrimidine metabolism

30. System level experimental analysis of essentiality in E. coli Whole-Genome Essentiality by Transposomics Aerobic growth: 620 essential 3,126 dispensable genes Gerdes et al. J Bact. 185, 5673-5684 (2003).

31. Pyrimidine metabolism

32. Essentiality: Red: highly essential Green: dispensable Evolutionary conservation: Red: highly conserved Green: non-conserved (reference: 32 bacteria) System level analysis of the full E coli metabolism Gerdes et al. J Bact. 185, 5673-5684 (2003).

33. Characterizing the links Metabolism: Flux Balance Analysis (Palsson) Metabolic flux for each reaction Edwards, J. S. & Palsson, B. O, PNAS 97, 5528 (2000). Edwards, J. S., Ibarra, R. U. & Palsson, B. O. Nat Biotechnol 19, 125 (2001). Ibarra, R. U., Edwards, J. S. & Palsson, B. O. Nature 420, 186 (2002).

34. Global flux organization in the E. coli metabolic network E. Almaas, B. Kovács, T. Vicsek, Z. N. Oltvai, A.-L. B. Nature, 2004. SUCC: Succinate uptake GLU : Glutamate uptake Central Metabolism, Emmerling et. al, J Bacteriol 184, 152 (2002)

35. Inhomogeneity in the local flux distribution ~ k -0.27 Mass flows along linear pathways Glutamate rich substrate Succinate rich substrate Mass flows along linear pathways

36. Pyramid Life’s Complexity Pyramid Z.N. Oltvai and A.-L. B. (2002).

37. Http://www.nd.edu/~networks http://www.nd.edu/~networks Zoltán N. Oltvai, Northwestern Med. School Hawoong Jeong, KAIST, Corea Réka Albert, Penn State Ginestra Bianconi, Friburg/Trieste Erzsébet Ravasz, Notre Dame Stefan Wuchty, Notre Dame Eivind Almaas, Notre Dame Baldvin Kovács, Budapest Tamás Vicsek, Budapest

38. Http://www.nd.edu/~networks http://www.nd.edu/~networks

39. Rod Steiger Martin Sheen Donald Pleasence #1 #2 #3 #876 Kevin Bacon Bacon-map

40. Bonus: Why Kevin Bacon? Measure the average distance between Kevin Bacon and all other actors. No. of movies : 46 No. of actors : 1811 Average separation: 2.79 Kevin Bacon Is Kevin Bacon the most connected actor? NO! 876 Kevin Bacon 2.786981 46 1811 Bacon-list

41. Inhomogeneity in the local flux distribution

42. Scale-free Science collaborationWWW Internet CellCitation pattern Language SUMMARY Scale-free P(k)~k -γ Hierarchical C(k)~k -β Modular C(N)=const. Hierarchical Networks

43. Traditional modeling: Network as a static graph Given a network with N nodes and L links Create a graph with statistically identical topology RESULT: model the static network topology PROBLEM: Real networks are dynamical systems!  Evolving networks OBJECTIVE: capture the network dynamics METHOD : identify the processes that contribute to the network topology develop dynamical models that capture these processes BONUS: get the topology correctly. 

44. Whole cellular network Meta+all P(k)

45. Protein network Nodes : proteins Links : physical interaction (binding) Proteomics : identify and determine the properties of the proteins. (related to structure of proteins)

46. Metabolic Network Nodes : chemicals (substrates) Links : chem. reaction

47. Whole cellular network

48. Achilles’ Heel of complex network InternetProtein network failure attack Achilles Heel R. Albert, H. Jeong, A.L. Barabasi, Nature 406 378 (2000)

49. Taxonomy using networks A: Archaea B: Bacteria E: Eukaryotes

50. Watts-Strogatz (Nature 393, 440 (1998)) N nodes forms a regular lattice. With probability p, each edge is rewired randomly. Clustering: My friends will know each other with high probability! Probability to be connected C » p C = # of links between 1,2,…n neighbors n(n-1)/2

51.  Finite size scaling: create a network with N nodes with P in (k) and P out (k) = 0.35 + 2.06 log(N) 19 degrees of separation l 15 =2 [1  2  5] l 17 =4 [1  3  4  6  7] … = ?? 1 2 3 4 5 6 7 nd.edu 19 degrees of separation R. Albert et al Nature (99) based on 800 million webpages [S. Lawrence et al Nature (99)] A. Broder et al WWW9 (00) IBM 19 degrees

52. SCIENCE CITATION INDEX (  = 3) Nodes: papers Links: citations (S. Redner, 1998) P(k) ~k -  3212 33 1736 PRL papers (1988) Citation Hopfield J.J., PNAS1982

53. Complexity Network Scale-free network Science collaborationWWW Internet CellCitation pattern UNCOVERING ORDER HIDDEN WITHIN COMPLEX SYSTEMS Food Web SUMMARY

54. Combining Modularity and the Scale-free Property Deterministic Scale-Free Networks Barabási, A.-L., Ravasz, E., & Vicsek, T. (2001) Physica A 299, 559. Dorogovtsev, S. N., Goltsev, A. V., & Mendes, J. F. F. (2001) cond-mat/0112143. (DGM)

55. Problems with the scale-free model C is independent of NC decreases with N C i =2n i /k i (k i -1) Watts, Strogatz, 1998

56. Exceptions: Geographically Organized Networks: Common feature: economic pressures towards shorter links Internet (router), Vazquez et al, ‘01 Power Grid

57. Is the hierarchical exponent β universal?   For most systems: Connect a p fraction of nodes to the central module using preferential attachment

58. What does it mean? Real Networks Have a Hierarchical Topology Many highly connected small clusters combine into few larger but less connected clusters combine into even larger and even less connected clusters  The degree of clustering follows:

59. Stochastic Hierarchical Model

60. Hierarchy in biological systems Metabolic networks Protein networks

61. Mean Field Theory γ = 3, with initial condition A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999) MFT

62. Nature 408 307 (2000) … “One way to understand the p53 network is to compare it to the Internet. The cell, like the Internet, appears to be a ‘scale-free network’.” P53

63. p53 network (mammals) P53 P(k)

64. Real Networks HollywoodLanguage Internet (AS) Vaquez et al,'01 WWW Eckmann & Moses, ‘02

65. Achilles’ Heel of complex networks Internet failure attack Achilles Heel R. Albert, H. Jeong, A.L. Barabasi, Nature 406 378 (2000)

66. What is the topology of cellular networks? Argument 2: Cellular networks are exponential! Reason: They have been streamlined by evolution... Argument 1: Cellular networks are scale-free! Reason: They formed one node at a time… Cells-SF or ER?

67. ACTOR CONNECTIVITIES Nodes: actors Links: cast jointly N = 212,250 actors  k  = 28.78 P(k) ~k -  Days of Thunder (1990) Far and Away (1992) Eyes Wide Shut (1999)  =2.3 Actors

69. Yeast protein network Nodes : proteins Links : physical interactions (binding) P. Uetz, et al. Nature 403, 623-7 (2000). Prot Interaction map

70. Http://www.nd.edu/~networks Interplay between network structure and evolution S. Wuchty, Z.N. Oltvai, A.-L.B., 2003. Removing the complexes

71. 2. Clustering coefficient independent of N Properties of hierarchical networks 1. Scale-free

72. Node-node distance in metabolic networks D 15 =2 [1  2  5] D 17 =4 [1  3  4  6  7] … D = ?? 1 2 3 4 5 6 7 Scale-free networks: D~log(N) Larger organisms are expected to have a larger diameter! Meta-diameter

73. Erdös-Rényi model (1960) - Democratic - Random Pál Erdös Pál Erdös (1913-1996) Connect with probability p p=1/6 N=10  k  ~ 1.5 Poisson distribution

74. World Wide Web Over 1 billion documents ROBOT: collects all URL’s found in a document and follows them recursively Nodes: WWW documents Links: URL links R. Albert, H. Jeong, A-L Barabasi, Nature, 401 130 (1999). WWW Expected P(k) ~ k -  Found

75. What does it mean? Poisson distribution Random Network Power-law distribution Scale-free Network Airlines

76. INTERNET BACKBONE (Faloutsos, Faloutsos and Faloutsos, 1999) Nodes: computers, routers Links: physical lines Internet

77. Complex systems Made of many non-identical elements connected by diverse interactions. NETWORK New York Times

78. Restriction of solution space by optimization for maximal growth

79. Random Networks Connect each pair of nodes with probability p p=1/6 N=10  k  ~ 1.5 Erdös-Rényi, 1960

80. Scale-free networks A.-L.Barabási, R. Albert, Science 286, 509 (1999) P(k) ~k -3 BA model Growth: Networks expand by the addition of new nodes Preferential attachment: New nodes prefer to link to highly connected nodes

81. Small World Features: distance in metabolic networks D 15 =2 [1  2  5] D 17 =4 [1  3  4  6  7] … D = ?? 1 2 3 4 5 6 7 Random Networks: D~log(N) (small world effect) Meta-diameter Scale-Free Networks: P(k)~k - γ log N γ>3 D = log log N 2<γ<3 const γ=2 (ultra small world) Cohen,Havlin, PRL’03

82. The New York Times

83. Modularity in the metabolism  Metabolic network (43 organisms)  Scale-free model Clustering Coefficient: C(k)= # links between k neighbors k(k-1)/2

84. A Few Good Man Robert Wagner Austin Powers: The spy who shagged me Wild Things Let’s make it legal Barry Norton What Price Glory Monsieur Verdoux Bacon 1

85. Can Latecomers Make It? Fitness Model SF model: k(t)~t ½ (f irst mover advantage) Real systems: nodes compete for links -- fitness Fitness Model: fitness (   k( ,t)~t  where  =  C G. Bianconi and A.-L. Barabási, Europhyics Letters. 54, 436 (2001).

86. Bose-Einstein Condensation in Evolving Networks G. Bianconi and A.-L. Barabási, Physical Review Letters 2001; cond-mat/0011029 NetworkBose gas Fit-gets-richBose-Einstein condensation