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1. Class 7: Evolving Network Models Prof. Albert-László Barabási Dr. Baruch Barzel, Dr. Mauro Martino Network Science: Evolving Network Models February 2012
2. P(k) ~ k- Empirical findings for real networks Clustered: clustering coefficient does not depend on network size. Scale-free: The degrees follow a power-laws distribution. Small World: distances scale logarithmically with the network size Network Science: Evolving Network Models February 2012
3. BENCHMARK 1: Regular Lattices Two-dimensional lattice: Average path-length: Degree distribution: P(k)=δ(k-6) Clustering coefficient: D-dimensional lattice: The average path-length varies as Constant degree P(k)=δ(k-kd) Constant clustering coefficient C=Cd Network Science: Evolving Network Models February 2012
4. BENCHMARK 2: Random Network Model Erdös-Rényi Model- Publ. Math. Debrecen 6, 290 (1959) fixed node number N connecting pairs of nodes with probability p Degree distribution: Path length: Clustering coefficient: Network Science: Evolving Network Models February 2012
5. BENCHMARK 3: Small World Model Watts-Strogatz algorithm – Nature 2008 fixed node number N connecting pairs of nodes with probability p Degree distribution: Exponential Path length: Clustering coefficient: Network Science: Evolving Network Models February 2012
6. P(k) ~ k- EMPIRICAL DATA FOR REAL NETWORKS Regular network Pathlenght Degree Distr. Clustering P(k)=δ(k-kd) Erdos- Renyi Watts- Strogatz Exponential Network Science: Evolving Network Models February 2012
7. SCALE-FREE MODEL(BA model) Network Science: Evolving Network Models February 2012
8. BA MODEL: Growth BA model: Growth ER, WS models: the number of nodes, N, is fixed (static models) Real networks continuously expand by the addition of new nodes Barabási & Albert, Science286, 509 (1999) Network Science: Evolving Network Models February 2012
9. BA MODEL: Growth (Actors/Internet) BA model: Growth Actor network Internet Growth of the Internet routing table Number of movies in IMDB http://www.trainsignaltraining.com/ccna-ipv6 Herr II, Bruce W., Ke, Weimao, Hardy, Elisha, and Börner, Katy. (2007) Movies and Actors: Mapping the Internet Movie Database. In Conference Proceedings of 11th Annual Information Visualization International Conference (IV 2007), Zurich, Switzerland, July 4-6, pp. 465-469. Network Science: Evolving Network Models February 2012
10. BA MODEL: Growth (www/Pubs) BA model: Growth WWW Scientific Publications http://website101.com/define-ecommerce-web-terms-definitions/ http://www.kk.org/thetechnium/archives/2008/10/the_expansion_o.php Barabási & Albert, Science286, 509 (1999) Network Science: Evolving Network Models February 2012
11. BA MODEL: Growth BA model: Growth (1) Networks continuously expand by the addition of new nodes Add a new node with m links Barabási & Albert, Science286, 509 (1999) Network Science: Evolving Network Models February 2012
12. BA MODEL: Preferential Attachment Where will the new node link to? ER, WS models: choose randomly. PREFERENTIAL ATTACHMENT: the probability that a node connects to a node with k links is proportional to k. New nodes prefer to link to highly connected nodes (www, citations, IMDB). Barabási & Albert, Science286, 509 (1999) Network Science: Evolving Network Models February 2012
13. (1) Networks continuously expand by the addition of new nodes WWW : addition of new documents (2) New nodes prefer to link to highly connected nodes. WWW : linking to well known sites P(k) ~k-3 Origin of SF networks: Growth and preferential attachment 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. Barabási & Albert, Science286, 509 (1999) Network Science: Evolving Network Models February 2012
14. All nodes follow the same growth law During a unit time (time step): Δk=m  A=m Use: β: dynamical exponent A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999) Network Science: Evolving Network Models February 2012
15. Fitness Model: Can Latecomers Make It? SF model: k(t)~t½(first mover advantage) Degree (k) time Network Science: Evolving Network Models February 2012
16. Degree distribution A node i can come with equal probability any time between ti=m0and t, hence: γ = 3 A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999) Network Science: Evolving Network Models February 2012
17. Degree distribution γ = 3 (i) The degree exponent is independent ofm. (ii) As the power-law describes systems of rather different ages and sizes, it is expected that a correct model should provide a time-independent degree distribution. Indeed, asymptotically the degree distribution of the BA model is independent of time (and of the system size N) the network reaches a stationary scale-free state. (iii) The coefficient of the power-law distribution is proportional to m2. A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999) Network Science: Evolving Network Models February 2012
18. NUMERICAL SIMULATION OF THE BA MODEL m=1,3,5,7 N=100,000;150,000;200,000 m-dependence Stationarity: P(k) independent of N Insert: degree dynamics Network Science: Evolving Network Models February 2012
19. The mean field theory offers the correct scaling, BUT it provides the wrong coefficient of the degree distribution. So assymptotically it is correct (k ∞), but not correct in details (particularly for small k). To fix it, we need to calculate P(k) exactly, which we will do next using a rate equation based approach. Network Science: Evolving Network Models February 2012
20. MFT - Degree Distribution: Rate Equation Number of nodes with degree k at time t. Since at each timestep we add one node, we have N=t (total number of nodes =number of timesteps) 2m: each node adds m links, but each link contributed to the degree of 2 nodes Total number of k-nodes Number of links added to degree k nodes after the arrival of a new node: Nr. of degree k-1 nodes that acquire a new link, becoming degree k New node adds m new links to other nodes Preferential attachment Nr. of degree k nodes that acquire a new link, becoming degree k+1 Loss of k-nodes via k k+1 Gain of k-nodes via k-1 k # k-nodes at time t # k-nodes at time t+1 A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999)
21. MFT - Degree Distribution: Rate Equation Loss of k-nodes via k k+1 Gain of k-nodes via k-1 k # k-nodes at time t # k-nodes at time t+1 We do not have k=0,1,...,m-1 nodes in the network (each node arrives with degree m)  We need a separate equation for degree m modes # m-nodes at time t # m-nodes at time t+1 Add one m-degeree node Loss of an m-node via m m+1 Network Science: Evolving Network Models February 2012 A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999)
22. MFT - Degree Distribution: Rate Equation k>m We assume that there is a stationary state in the N=t∞ limit, when P(k,∞)=P(k) k>m Network Science: Evolving Network Models February 2012 A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999)
23. MFT - Degree Distribution: Rate Equation  m+3  k ... for large k Krapivsky, Redner, Leyvraz, PRL 2000 Dorogovtsev, Mendes, Samukhin, PRL 2000 Bollobas et al, Random Struc. Alg. 2001 Network Science: Evolving Network Models February 2012 A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999)
24. MFT - Degree Distribution: A Pretty Caveat Start from eq. Its solution is: Dorogovtsev and Mendes, 2003 Network Science: Evolving Network Models February 2012 A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999)
25. Do we need both growth and preferential attachment? Network Science: Evolving Network Models February 2012
26. growth preferential attachment MODEL A Π(ki) : uniform Network Science: Evolving Network Models February 2012
27. MODEL B growth preferential attachment P(k) : power law (initially)  Gaussian Fully Connected Network Science: Evolving Network Models February 2012
28. Do we need both growth and preferential attachment?YEP. Network Science: Evolving Network Models February 2012
29. P(k) ~ k- EMPIRICAL DATA FOR REAL NETWORKS Regular network Pathlenght Degree Distr. Clustering P(k)=δ(k-kd) Erdos- Renyi Watts- Strogatz Exponential Barabasi-Albert P(k) ~ k- Network Science: Evolving Network Models February 2012
30. DISTANCES IN SCALE-FREE NETWORKS Distances in scale-free networks Size of the biggest hub is of order O(N). Most nodes can be connected within two layers of it, thus the average path length will be independent of the system size. The average path length increases slower than logarithmically. In a random network all nodes have comparable degree, thus most paths will have comparable length. In a scale-free network the vast majority of the path go through the few high degree hubs, reducing the distances between nodes. Some key models produce γ=3, so the result is of particular importance for them. This was first derived by Bollobas and collaborators for the network diameter in the context of a dynamical model, but it holds for the average path length as well. The second moment of the distribution is finite, thus in many ways the network behaves as a random network. Hence the average path length follows the result that we derived for the random network model earlier. Small World Cohen, Havlin Phys. Rev. Lett. 90, 58701(2003); Cohen, Havlin and ben-Avraham, in Handbook of Graphs and Networks, Eds. Bornholdt and Shuster (Willy-VCH, NY, 2002) Chap. 4; Confirmed also by: Dorogovtsev et al (2002), Chung and Lu (2002); (Bollobas, Riordan, 2002; Bollobas, 1985; Newman, 2001
31. PATH LENGTHS IN THE BA MODEL Bollobas, Riordan, 2002 Network Science: Evolving Network Models February 2012
32. P(k) ~ k- EMPIRICAL DATA FOR REAL NETWORKS Pathlenght Degree Distr. Clustering Regular network P(k)=δ(k-kd) Erdos- Renyi Watts- Strogatz Exponential Barabasi-Albert P(k) ~ k- Network Science: Evolving Network Models February 2012
33. CLUSTERING COEFFICIENT OF THE BA MODEL Reminder: for a random graph we have: The numerical results indicate a slightly slower decay. What is the functional form of C(N)? Konstantin Klemm, Victor M. Eguiluz, Growing scale-free networks with small-world behavior, Phys. Rev. E 65, 057102 (2002), cond-mat/0107607 Network Science: Evolving Network Models February 2012
34. CLUSTERING COEFFICIENT OF THE BA MODEL 1 2 Denote the probability to have a link between node i and j with P(i,j) The probability that three nodes i,j,l form a triangle is P(i,j)P(i,l)P(j,l) The expected number of triangles in which a node l with degree kl participates is thus: We need to calculate P(i,j). Network Science: Evolving Network Models February 2012
35. CLUSTERING COEFFICIENT OF THE BA MODEL Calculate P(i,j). Node j arrives at time tj=j and the probability that it will link to node i with degree ki already in the network is determined by preferential attachment: Where we used that the arrival time of node j is tj=j and the arrival time of node is ti=i Let us approximate: Which is the degree of node l at current time, at time t=N There is a factor of two difference... Where does it come from? Network Science: Evolving Network Models February 2012
36. Clustering Coefficient of the BA model CLUSTERING COEFFICIENT OF THE BA MODEL Konstantin Klemm, Victor M. Eguiluz, Phys. Rev. E 65, 057102 (2002) Network Science: Evolving Network Models February 2012
37. P(k) ~ k- EMPIRICAL DATA FOR REAL NETWORKS Pathlenght Degree Distr. Clustering Regular network P(k)=δ(k-kd) Erdos- Renyi Watts- Strogatz Exponential Barabasi-Albert P(k) ~ k- Network Science: Evolving Network Models February 2012
38. The origins of preferential attachment. Network Science: Evolving Network Models February 2012
39. Preferential Attachment: a brief history of collective amnesia GyorgyPolya (1887-1985) 1923: Polya process in the mathematics literature George Udmy Yule (1871-1951) in 1925: the number of species per genus of flowering plants; Yule process in statistics Robert Gibrat (1904-1980), 1931: rule of proportional growth (the size of the growth and rate of a firm are independent). Gibrat process in economics George Kinsley Zipf (1902-1950), 1949: the distribution of the wealth on the society. Herbert Alexander Simon (1916-2001), 1955, the distribution of city sizes and other phenomena Derek de Solla Price (1922-1983), 1976, used it to explain the citation statistics of scientific publications, "cumulative advantage” Robert Merton (1910-2003), 1968: Matthew effect, Gospel of Matthew: "For everyone who has will be given more, and he will have an abundance. Whoever does not have, even what he has will be taken from him." "Not the first, but the last" Network Science: Evolving Network Models February 2012
40. CAN WE MEASURE PREFERENTIAL ATTACHMENT? Plot the change in the degree k during a fixed time t for nodes with degree k, and you get  (k) To reduce noise, plot the integral of Π(k) over k: No pref. attach: κ~k Linear pref. attach: κ~k2 (Jeong, Neda, A.-L. B, Europhys Letter 2003; cond-mat/0104131) Network Science: Evolving Network Models February 2012
41. CAN WE MEASURE PREFERENTIAL ATTACHMENT? citation network Plots shows the integral of Π(k) over k: Internet No pref. attach: κ~k actor collab. neurosci collab Linear pref. attach: κ~k2 Network Science: Evolving Network Models February 2012
42. MECHANISMS RESPONSIBLE FOR PREFERENTIAL ATTACHMENT Copying mechanism directed network select a node and an edge of this node attach to the endpoint of this edge Walking on a network directed network the new node connects to a node, then to every first, second, … neighbor of this node Attaching to edges select an edge attach to both endpoints of this edge Node duplication duplicate a node with all its edges randomly prune edges of new node Network Science: Evolving Network Models February 2012
43. Copying Mechanism Network Science: Evolving Network Models February 2012
44. ORIGIN OF THE SCALE-FREE TOPOLOGY IN THE CELL:Gene Duplication Proteins with more interactions are more likely to obtain new links: Π(k)~k (preferential attachment) Wagner 2001; Vazquez et al. 2003; Sole et al. 2001; Rzhetsky & Gomez 2001; Qian et al. 2001; Bhan et al. 2002. Network Science: Evolving Network Models February 2012
45. PREFERENTIAL ATTACHMENT IN PROTEIN INTERACTION NETWORKS k vs. k : increase in the No. of links in a unit time No PA:k is independent of k PA:k ~k Eisenberg E, Levanon EY, Phys. Rev. Lett. 2003 Jeong, Neda, A.-L.B, Europhys. Lett. 2003 Network Science: Evolving Network Models February 2012
46. SUMMARY: PROPERTIES OF THE BA MODEL Nr. of nodes: Nr. of links: Average degree: Degree dynamics Degree distribution: Average Path Length: Clustering Coefficient: β: dynamical exponent γ: degree exponent The network grows, but the degree distribution is stationary. Network Science: Evolving Network Models February 2012
47. DEGREE EXPONENTS γcollab γmetab γintern γsynonyms γwin γwout γactor γsex γcita γ=1 γ=2 γ=3 <k2> diverges <k2> finite BA model Can we change the degree exponent? Network Science: Evolving Network Models February 2012
48. Evolving network models Network Science: Evolving Network Models February 2012
49. EVOLVING NETWORK MODELS The BA model is only a minimal model. Makes the simplest assumptions: linear growth linear preferential attachment Does not capture variations in the shape of the degree distribution variations in the degree exponent the size-independent clustering coefficient Hypothesis: The BA model can be adapted to describe most features of real networks. We need to incorporate mechanisms that are known to take place in real networks: addition of links without new nodes, link rewiring, link removal; node removal, constraints or optimization Network Science: Evolving Network Models February 2012
50. BA ALGORITHM WITH DIRECTED EDGES (the simplest way to change the degree exponent) Undirected BA network: Directed BA network: β=1: dynamical exponent γin=2: degree exponent; P(kout)=δ(kout-m) Undirected BA: β=1/2; γ=3 Network Science: Evolving Network Models February 2012 γ = 3 A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999)
51. EXTENDED MODEL: Other ways to change the exponent 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,) Network Science: Evolving Network Models February 2012
52. EXTENDED MODEL: Small-k cutoff P(k) ~ (k+(p,q,m))-(p,q,m)   [1,) Extended Model prob. p : internal links prob. q : link deletion prob. 1-p-q : add node Predicts a small-k cutoff a correct model should predict all aspects of the degree distribution, not only the degree exponent. Degree exponent is a continuous function of p,q, m p=0.937 m=1  = 31.68  = 3.07 Actor network Network Science: Evolving Network Models February 2012
53. More models NONLINEAR PREFERENTIAL ATTACHMENT Non-linear preferential attachment:  P(k) does not follow a power law for 1  <1 : stretch-exponential  >1 : no-scaling (>2 : “gelation”) P. Krapivsky, S. Redner, F. Leyvraz, Phys. Rev. Lett. 85, 4629 (2000) Network Science: Evolving Network Models February 2012
54. INITIAL ATTRACTIVENESS BA model: k=0 nodes cannot aquire links, as Π(k=0)=0 (the probability that a new node will attach to it is zero) A- initial attractiveness Initial attractiveness shifts the degree exponent: Note: the parameter A can be measured from real data, being the rate at which k=0 nodes acquire links, i.e. Π(k=0)=A Dorogovtsev, Mendes, Samukhin, Phys. Rev. Lett. 85, 4633 (2000) Network Science: Evolving Network Models February 2012
55. GROWTH CONSTRAINTS AND AGING CAUSE CUTOFFS Finite lifetime to acquire new edges L. A. N. Amaral et al., PNAS 97, 11149 (2000) Gradual aging: S. N. Dorogovtsev and J. F. F. Mendes, Phys. Rev. E 62, 1842 (2000) Network Science: Evolving Network Models February 2012
56. P(k) ~ k- THE LAST PROBLEM: HIGH, SYSTEM-SIZE INDEPENDENT C(N) Pathlenght Degree Distr. Clustering Regular network P(k)=δ(k-kd) Erdos- Renyi Watts- Strogatz Exponential Barabasi-Albert P(k) ~ k- Network Science: Evolving Network Models February 2012
57. A MODEL WITH HIGH CLUSTERING COEFFICIENT Each node of the network can be either active or inactive. There arem active nodes in the network in any moment. Start with m active, completely connected nodes. Each timestep add a new node (active) that connects to m active nodes. Deactivate one active node with probability: C C* when N∞ K. Klemm and V. Eguiluz, Phys. Rev. E 65, 036123 (2002) Network Science: Evolving Network Models February 2012

58. Fitness Model

    Network Science: Evolving Network Models February 2012
59. Fitness Model: Can Latecomers Make It? SF model: k(t)~t ½(first mover advantage) Fitness model: fitness (h ) k(h,t)~tb(h) b(h) =h/C Degree (k) time Bianconi & Barabási, Physical Review Letters 2001; Europhys. Lett. 2001.
60. Network Bose gas MAPPING TO A QUANTUM GAS Fitnessη Energy level ε New node with fitness η New energy levelε Link pointing to node η  Particle at level ε Network  quantum gas G. Bianconi and A.-L. Barabási, Physical Review Letters 2001; cond-mat/0011029 Network Science: Evolving Network Models February 2012
61. BOSE-EINSTEIN CONDENSATION f(e)=e-b(e-m) . The dynamic exponent f(e) depends on m, determined by the self-consistent equation: Network Science: Evolving Network Models February 2012
62. Bose-Einstein Condensation Degree (k) time Bianconi & Barabási, Physical Review Letters 2001; Europhys. Lett. 2001. Network Science: Evolving Network Models February 2012
63. Bose-Einstein Condensation Bianconi & Barabási, Physical Review Letters 2001; Europhys. Lett. 2001. Network Science: Evolving Network Models February 14, 2011
64. Fit-gets-rich Bose-Einstein condensation FITNESS MODEL: Can Latecomers Make It?
65. LESSONS LEARNED: evolving network models There is no universal exponent characterizing all networks. Growth and preferential attachment are responsible for the emergence of the scale-free property. The origins of the preferential attachment is system-dependent. Modeling real networks: identify the microscopic processes that take place in the system measure their frequency from real data develop dynamical models that capture these processes. 5. If the model is correct, it should correctly predict not only the degree exponent, but both small and large k-cutoffs. Network Science: Evolving Network Models February 2012
66. LESSONS LEARNED: evolving network models Philosophical change in network modeling: ER, WS models are static models – the role of the network modeler it to cleverly place the links between a fixed number of nodes to that the network topology mimic the networks seen in real systems. BA and evolving network models are dynamical models: they aim to reproduce how the network was built and evolved. Thus their goal is to capture the network dynamics, not the structure.  as a byproduct, you get the topology correctly Network Science: Evolving Network Models February 2012
67. 5 slides Discuss: What are your nodes and links How will you collect the data Expected size of the network (Nr of nodes, links) What questions you plan to ask (they may change as we move along with the class). Why do we care about the network you plan to study. Network Science: Evolving Network Models February 2012
68. The end Network Science: Evolving Network Models February 2012