1. Erzsébet Ravasz, Zoltán Dezsö
The architecture of complexity
From the diameter of the www to the structure of the cell
Albert László Barabási
(Univ. of Notre Dame)
Zoltán Néda, Hawoong Jeong, Réka Albert, Ginestra Bianconi, Soonhyung Yook,
Erzsébet Ravasz, Zoltán Dezsö

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

3. Complexity What is Complexity?
 non-linear systems  chaos  fractals
A popular paradigm: Simple systems display complex behavior
3 Body Problem
Earth( ) Jupiter ( ) Sun ( )
Main Entry: 1com·plex Function: noun Etymology: Late Latin complexus totality, from Latin, embrace, from complecti Date: : a whole made up of complicated or interrelated parts

4. Six Degrees of Separation
Society
Nodes: individuals
Links: social relationship (family/work/friendship/etc.)
S. Milgram (1967)
Six Degrees of Separation
John Guare
Social networks: Many individuals with diverse social interactions between them.

5. Communication networks
The Earth is developing an electronic nervous system, a network with diverse nodes and links are
-computers
-routers
-satellites
-phone lines
-TV cables
-EM waves
Communication networks: Many non-identical components with diverse connections between them.

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

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

8. ARE COMPLEX NETWORKS REALLY RANDOM?
NO!
ARE COMPLEX NETWORKS REALLY RANDOM?

9. Cluster Coefficient
Clustering: My friends will likely know each other!
Probability to be connected C » p
# of links between 1,2,…n neighbors
C =
n(n-1)/2
Networks are clustered [large C(p)] but have a small characteristic path length [small L(p)].

10. (Watts and Strogatz, Nature 393, 440 (1998))
Watts-Strogatz Model
C(p) : clustering coeff L(p) : average path length
(Watts and Strogatz, Nature 393, 440 (1998))

11. WWW
World Wide Web
Nodes: WWW documents Links: URL links
800 million documents (S. Lawrence, 1999)
ROBOT: collects all URL’s found in a document and follows them recursively
R. Albert, H. Jeong, A-L Barabasi, Nature, (1999)

12. WWW-power What did we expect? We find: k ~ 6 P(k=500) ~ 10-99
NWWW ~ 109
 N(k=500)~10-90
We find:
out= 2.45
 in = 2.1
P(k=500) ~ 10-6
NWWW ~ 109
 N(k=500) ~ 103
Pout(k) ~ k-out
Pin(k) ~ k- in
J. Kleinberg, et. al, Proceedings of the ICCC (1999)

13. 19 degrees 19 degrees of separation l15=2 [125] l17=4 [1346  7]
 Finite size scaling: create a network with N nodes with Pin(k) and Pout(k)
< l > = log(N)
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

14. Power-law distribution
Airlines
What does it mean?
Poisson distribution
Exponential Network
Power-law distribution
Scale-free Network

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

16. Internet-Map

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

18. SCIENCE CITATION INDEX
2212 25
Nodes: papers Links: citations
Witten-Sander
PRL 1981
1736 PRL papers (1988)
P(k) ~k-
( = 3)
(S. Redner, 1998)

19. Coauthorship SCIENCE COAUTHORSHIP
Nodes: scientist (authors) Links: write paper together
(Newman, 2000, H. Jeong et al 2001)

20. Food Web Nodes: trophic species Links: trophic interactions
R.J. Williams, N.D. Martinez Nature (2000)
R. Sole (cond-mat/ )

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

22. Most real world networks have the same internal structure:
Scale-free networks
Why?
What does it mean?

23. Origins SF SCALE-FREE NETWORKS
(1) The number of nodes (N) is NOT fixed.
Networks continuously expand by the addition of new nodes
Examples: WWW : addition of new documents Citation : publication of new papers
(2) The attachment is NOT uniform.
A node is linked with higher probability to a node that already has a large number of links.
Examples : WWW : new documents link to well known sites (CNN, YAHOO, NewYork Times, etc) Citation : well cited papers are more likely to be cited again

24. BA model Scale-free model P(k) ~k-3
(1) GROWTH : At every timestep we add a new node with m edges (connected to the nodes already present in the system).
(2) PREFERENTIAL ATTACHMENT : The probability Π that a new node will be connected to node i depends on the connectivity ki of that node
P(k) ~k-3
A.-L.Barabási, R. Albert, Science 286, 509 (1999)

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

26. Model A
growth preferential attachment
Π(ki) : uniform

27. P(k) : power law (initially)
Model B
growth preferential attachment
P(k) : power law (initially)
 Gaussian

28. Preferential Attachment
For given t,k  (k)
k vs. k : increase in the No. of links in a unit time
Citation network
Internet
(cond-mat/ )

29. Universality? Extended Model prob. p : internal links
prob. q : link deletion
prob. 1-p-q : add node
P(k) ~ (k+(p,q,m))-(p,q,m)
  [1,)
• Predict the network topology from microscopic processes with parameters (p,q,m)
• Scaling but no universality
Actor network
p=0.937
m=1
 = 31.68
 = 3.07

30. More models Other Models
Non-linear preferential attachment : (k) ~ k  P(k) ~ no scaling for 1
 <1 : stretch-exponential
 >1 : no-scaling (>2 : “gelation”)
(Krapivsky et al (2000).)
Initial attractiveness : (k) ~ A+k
P(k) ~ k- where =2 + A/m
(Dorogovtsev et al (2000).)
Aging : each node has a lifetime  node cannot get links after retirement. (actor)
P(k) : power-law with exponential cutoff
(Amaral et al (2000).)

31. Other Models (continued)
Saturation : each node has maximum link number.
node cannot get links after finite # of links
P(k) : power-law with exponential cutoff
(Amaral et al (2000).)

32. Can Latecomers Make It? Fitness Model SF model: k(t)~t ½ (first mover advantage) Real systems: nodes compete for links -- fitness Fitness Model: fitness (h ) k(h,t)~tb(h) where b(h) =h/C G. Bianconi and A.-L. Barabási, Europhyics Letters. 54, 436 (2001).

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

34. Next Lecture: (9 am tomorrow morning)
-The Web of Life: Networks in biological systems
(metabolic and protein interaction networks)
-Modeling the Internet
-Achilles’ Heel: robustness, error and attack tolerance
-Why Bacon?

35. http://www.nd.edu/~networks Note:
Note:
There are two postdoctoral positions open in my research group.
For more details see

36. Erzsébet Ravasz, Zoltán Dezsö
The architecture of complexity
From the diameter of the www to the structure of the cell
Albert László Barabási
(Univ. of Notre Dame)
Zoltán Néda, Hawoong Jeong, Réka Albert, Ginestra Bianconi, Soonhyung Yook,
Erzsébet Ravasz, Zoltán Dezsö

37. Next Lecture: (9 am tomorrow morning)
-The Web of Life: Networks in biological systems
(metabolic and protein interaction networks)
-Modeling the Internet
-Achilles’ Heel: robustness, error and attack tolerance
-Why Bacon?

38. Internet-Map

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

40. BA model Scale-free model P(k) ~k-3
(1) GROWTH : At every timestep we add a new node with m edges (connected to the nodes already present in the system).
(2) PREFERENTIAL ATTACHMENT : The probability Π that a new node will be connected to node i depends on the connectivity ki of that node
P(k) ~k-3
A.-L.Barabási, R. Albert, Science 286, 509 (1999)

41. Spatial Distributions
Router density
Population density

42. Spatial Distribution of Routers
Fractal set
Box counting: N(l)  No. of boxes of size l that contain routers
N(l) ~ l -Df Df=1.5

43. Preferential Attachment Preferential Attachment:
Compare maps taken at different times (Dt = 6 months)
Measure Dk(k), increase in No. of links for a node with k links
Preferential Attachment:
Dk(k) ~ k

44. INTERNET
N(l) ~ l-Df Df=1.5
Dk(k) ~ ka a=1
P(d) ~ d-s s=1

45. Parameter (,,Df) dependence of P(k) and P(d)

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

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

48. Bio-chemical reactions
METABOLISM
Bio-chemical reactions
Citrate Cycle

49. Boehring-Mennheim

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

51. Organisms from all three domains of life are scale-free networks!
Meta-P(k)
Metabolic network
Archaea
Bacteria
Eukaryotes
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, (2000)

52. Larger organisms are expected to have a larger diameter!
Meta-diameter
Node-node distance in metabolic networks 3
D15=2 [125]
D17=4 [134  67]
… D = ?? 6 1 4 7 5 2
Scale-free networks:
D~log(N)
Larger organisms are expected
to have a larger diameter!

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

54. protein-protein interactions
PROTEOME
protein-protein interactions

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

56. Topology of the protein network
Prot P(k)
Topology of the protein network
H. Jeong, S.P. Mason, A.-L. Barabasi, Z.N. Oltvai, Nature 411, (2001)

57. P53
Nature (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’.”

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

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

60. Robustness of scale-free networks
Robust-SF
Robustness of scale-free networks
  3 : fc=1
(R. Cohen et al PRL, 2000)
Failures
Topological error tolerance 1 fc
Attacks S f 1

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

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

63. SUMMARY Network Complexity Scale-free network Science collaboration
WWW
Food Web
Scale-free network
Citation pattern
Cell
Internet
UNCOVERING ORDER HIDDEN WITHIN COMPLEX SYSTEMS

64. 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
identify the processes that contribute to the network topology
develop dynamical models that capture these processes
METHOD : 
BONUS: get the topology correctly.

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

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

67. http://www.nd.edu/~networks Note:
Note:
There are two postdoctoral positions open in my research group.
For more details see

68. References URL: http://www.nd.edu/~networks
R. Albert, H. Jeong, A.L. Barabasi, Nature (1999).
R. Albert, A.L. Barabasi, Science (1999).
A.L. Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999)
R. Albert, H. Jeong, A.L. Barabasi, Nature (2000).
H.Jeong, B.Tombor, R.Albert, Z.N.Oltvai, A.L.Barabasi, Nature (2000).
H. Jeong, S.P. Mason, A.L. Barabasi, Z.N. Oltvai, Nature (in press).
URL:

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

70. Properties of the protein network
Highly connected proteins are more essential (lethal) than less connected proteins.

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

72. Metabolic network Archaea Bacteria Eukaryotes
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, (2000)

73. Whole cellular network

74. Properties of metabolic networks
Average distances are independent of organisms!  by making more links between nodes  based on “design principles” of the cell through evolution.
cf. Other scale-free network: D~log(N)

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

76. Achilles’ Heel of complex network
failure
attack
Internet
Protein network
R. Albert, H. Jeong, A.L. Barabasi, Nature (2000)