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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
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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
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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
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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.
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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.
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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
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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
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ARE COMPLEX NETWORKS REALLY RANDOM?
NO! ARE COMPLEX NETWORKS REALLY RANDOM?
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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)].
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(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))
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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)
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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)
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19 degrees 19 degrees of separation l15=2 [125] l17=4 [1346 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
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Power-law distribution
Airlines What does it mean? Poisson distribution Exponential Network Power-law distribution Scale-free Network
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(Faloutsos, Faloutsos and Faloutsos, 1999)
Internet INTERNET BACKBONE Nodes: computers, routers Links: physical lines (Faloutsos, Faloutsos and Faloutsos, 1999)
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Internet-Map
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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
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SCIENCE CITATION INDEX
2212 25 Nodes: papers Links: citations Witten-Sander PRL 1981 1736 PRL papers (1988) P(k) ~k- ( = 3) (S. Redner, 1998)
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Coauthorship SCIENCE COAUTHORSHIP
Nodes: scientist (authors) Links: write paper together (Newman, 2000, H. Jeong et al 2001)
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Food Web Nodes: trophic species Links: trophic interactions
R.J. Williams, N.D. Martinez Nature (2000) R. Sole (cond-mat/ )
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Sex-web Nodes: people (Females; Males) Links: sexual relationships
4781 Swedes; 18-74; 59% response rate. Liljeros et al. Nature 2001
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Most real world networks have the same internal structure:
Scale-free networks Why? What does it mean?
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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
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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)
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MFT Mean Field Theory γ = 3 , with initial condition
A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999)
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Model A growth preferential attachment Π(ki) : uniform
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P(k) : power law (initially)
Model B growth preferential attachment P(k) : power law (initially) Gaussian
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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/ )
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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
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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).)
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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).)
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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).
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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/
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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?
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http://www.nd.edu/~networks Note:
Note: There are two postdoctoral positions open in my research group. For more details see
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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
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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?
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Internet-Map
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(Faloutsos, Faloutsos and Faloutsos, 1999)
Internet INTERNET BACKBONE Nodes: computers, routers Links: physical lines (Faloutsos, Faloutsos and Faloutsos, 1999)
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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)
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Spatial Distributions
Router density Population density
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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
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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
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INTERNET N(l) ~ l-Df Df=1.5 Dk(k) ~ ka a=1 P(d) ~ d-s s=1
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Parameter (,,Df) dependence of P(k) and P(d)
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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...
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Bio-Map GENOME protein-gene interactions PROTEOME
protein-protein interactions Citrate Cycle METABOLISM Bio-chemical reactions
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Bio-chemical reactions
METABOLISM Bio-chemical reactions Citrate Cycle
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Boehring-Mennheim
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Metab-movie Metabolic Network Nodes: chemicals (substrates)
Links: bio-chemical reactions Metabolic Network
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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)
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Larger organisms are expected to have a larger diameter!
Meta-diameter Node-node distance in metabolic networks 3 D15=2 [125] D17=4 [134 67] … D = ?? 6 1 4 7 5 2 Scale-free networks: D~log(N) Larger organisms are expected to have a larger diameter!
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Bio-Map GENOME protein-gene interactions PROTEOME
protein-protein interactions Citrate Cycle METABOLISM Bio-chemical reactions
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protein-protein interactions
PROTEOME protein-protein interactions
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Prot Interaction map Yeast protein network Nodes: proteins
Links: physical interactions (binding) P. Uetz, et al. Nature 403, (2000).
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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)
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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’.”
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P53 P(k) p53 network (mammals)
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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
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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
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Achilles’ Heel of complex networks
failure attack Internet R. Albert, H. Jeong, A.L. Barabasi, Nature (2000)
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- 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)
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SUMMARY Network Complexity Scale-free network Science collaboration
WWW Food Web Scale-free network Citation pattern Cell Internet UNCOVERING ORDER HIDDEN WITHIN COMPLEX SYSTEMS
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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.
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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
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Bacon-map #1 Rod Steiger #876 Kevin Bacon Donald Pleasence #2 #3
Martin Sheen
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http://www.nd.edu/~networks Note:
Note: There are two postdoctoral positions open in my research group. For more details see
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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:
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Whole cellular network
Meta+all P(k) Whole cellular network
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Properties of the protein network
Highly connected proteins are more essential (lethal) than less connected proteins.
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Nodes: chemicals (substrates)
Links: chem. reaction Metabolic Network
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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)
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Whole cellular network
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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)
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Taxonomy using networks
A: Archaea B: Bacteria E: Eukaryotes
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Achilles’ Heel of complex network
failure attack Internet Protein network R. Albert, H. Jeong, A.L. Barabasi, Nature (2000)
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