2 Networks, Complexity and Economic Development Class 2: Scale-Free NetworksCesar A. Hidalgo PhD
3 WATTS & STROGATZLatticeErdös-Rényi model (1960)Poisson distribution
4 High school friendship James Moody, American Journal of Sociology 107, (2001)
5 High school dating network Data: Peter S. Bearman, James Moody, and Katherine Stovel. American Journal of Sociology 110, (2004)Image: M. Newman
6 Previous Lecture Take Home Messages NETWORKS-Networks can be used to represent a wide set of systems-The properties of random networks emerge suddenly as a function of connectivity.-The distance between nodes in random networks is small compared to network size L~ log(N)Networks can exhibit simultaneously: short average path length and high clustering(SMALL WORLD PROPERTY)The coexistence of these last two properties cannot be explained by random networksThe small world property of networks is not exclusive of “social” networks.BONUSDeterministic Systems are not necessarily predictable.But you shouldn’t always blame the butterfly.
13 WWW World Wide Web Nodes: WWW documents Links: URL links ExpectedOver 3 billion documentsExponential NetworkROBOT: collects all URL’s found in a document and follows them recursivelyP(k) ~ k-FoundScale-free NetworkR. Albert, H. Jeong, A-L Barabasi, Nature, (1999).
19 Meta-P(k) Metabolic network Archaea Bacteria Eukaryotes Organisms from all three domains of life have scale-free metabolic networks!H. Jeong, B. Tombor, R. Albert, Z.N. Oltvai, and A.L. Barabasi, Nature, (2000)
23 BA model Scale-free model (1) Networks continuously expand by the addition of new nodesWWW : addition of new documents Citation : publication of new papersGROWTH:add a new node with m linksPREFERENTIAL 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 Citation : citing again highly cited papersWeb application:Barabási & Albert, Science 286, 509 (1999)
24 MFT Mean Field Theory γ = 3 , with initial conditionγ = 3A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999)
25 Model Agrowth preferential attachmentΠ(ki) : uniform
26 Model B P(k) : power law (initially) Gaussian growth preferential attachmentP(k) : power law (initially) Gaussian
30 Movie Actors WWW Lada A Adamic, Bernardo A Huberman Technical Comments Power-Law Distribution of the World Wide WebScience 24 March 2000: Vol no. 5461, p DOI: /science aA-L Barabasi, R Albert, H Jeong, G BianconiTechnical CommentsPower-Law Distribution of the World Wide WebScience 24 March 2000: Vol no. 5461, p DOI: /science aMovie ActorsWWW
31 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 Local Rules Random Walk Model qe qv 1-qe A Vazquez Growing network with local rules: Preferential attachment, clustering hierarchy, and degree correlationsPhysical Review E 67, (2003)Random Walk Modelqeqv1-qe
34 The easiest way to find a hub? Ask for a friend!!!Pick a random person and ask that person to name a friend.
35 Pick a link! Distribution of degrees on the edge of a link is = kP(k) P(k)=1/kPicking a link and looking for a node at the edge of it gives you a uniform distribution of degrees!
36 More models Other Models R. Albert, A.-L. Barabasi, Rev. Mod. Phys 2002
37 Why scale-free? What functions satisfy this functional relationship? F(ax)=bF(x)F(x)=xP(ax)P=aPxP=bxp
39 Tokyo~30 million in metro area Santiago ~ 6 million metro areaCurico~100k peopleNew York~18 million in metro area
40 P~1/x Number of Cities Size of Cities 16 x 4 millioncitiesTokyo ~30 million4 x 8 millioncitiesNew York,Mexico City~15 millionNumber of CitiesP~1/xSize of CitiesThere is an equivalent number of people living in cities of all sizes!
41 After Bill enters the arena the average income of the public ~ 1,000,000 ~ $50 billion
42 Power laws everywherePower-law distributions in empirical data, Aaron Clauset, Cosma Rohilla Shalizi, and M. E. J. Newman, submitted to SIAM Review.
43 Power laws everywherePower-law distributions in empirical data, Aaron Clauset, Cosma Rohilla Shalizi, and M. E. J. Newman, submitted to SIAM Review.
56 Robustness Robustness node failure Complex systems maintain their basic functions even under errors and failures (cell mutations; Internet router breakdowns)fc1Fraction of removed nodes, fSnode failure
57 Robust-SF Robustness of scale-free networks S f Attacks Failures 1SfAttacksFailures 3 : fc=1(R. Cohen et al PRL, 2000)CfcAlbert, Jeong, Barabasi, Nature (2000)
58 Achilles Heel Achilles’ Heel of complex networks failureattackInternetR. Albert, H. Jeong, A.L. Barabasi, Nature (2000)
62 dik/dt =-ik+rk(1-ik)S ik’P(k,k’) dik/dt =-ik+rk(1-ik)q We now have many compartmentsSk , Ikdik/dt =-ik+rk(1-ik)S ik’P(k,k’) dik/dt =-ik+rk(1-ik)qik=rkq/(1+rkq) (1)q=<k>-1S ikkP(k) (2)(1)->(2)q=<k>-1S kP(k) rkq/(1+rkq)R. Pastor-Satorras and A. Vespignani. Epidemic spreading in scale-free networks. Physical Review Letters 86, (2001).
63 df/dq |q=0≥1 -> r<k2>/<k> ≥ 1 r ≥ <k>/<k2> q=<k>-1S kP(k) rkq/(1+rkq)=f(q)q=qf(q)df/dq |q=0≥1 -> r<k2>/<k> ≥ 1r ≥ <k>/<k2>R. Pastor-Satorras and A. Vespignani. Epidemic spreading in scale-free networks. Physical Review Letters 86, (2001).
64 There is no epidemic threshold!!! InfectedrcR. Pastor-Satorras and A. Vespignani. Epidemic spreading in scale-free networks. Physical Review Letters 86, (2001).
65 Take home messages -Networks might look messy, but are not random. -Many networks in nature are Scale-Free (SF), meaning that just a few nodes have a disproportionately large number of connections.-Power-law distributions are ubiquitous in nature.-While power-laws are associated with critical points in nature, systems can self-organize to this critical state.- There are important dynamical implications of the Scale-Free topology.-SF Networks are more robust to failures, yet are more vulnerable to targeted attacks.-SF Networks have a vanishing epidemic threshold.