1. Class 9: Barabasi-Albert Model Network Science: Evolving Network Models February 2015 Prof. Boleslaw Szymanski Prof. Albert-László Barabási Dr. Baruch Barzel, Dr. Mauro Martino

2. Empirical findings for real networks P(k) ~ k -  Small World: distances scale logarithmically with the network size Clustered: clustering coefficient does not depend on network size. Scale-free: The degrees follow a power-laws distribution. Network Science: Evolving Network Models February 2015

3. The average path-length varies as Constant degreeP(k)=δ(k-k d ) Constant clustering coefficientC=C d Two-dimensional lattice: D-dimensional lattice: Average path-length: Degree distribution: P(k)=δ(k-6) Clustering coefficient: BENCHMARK 1: Regular Lattices Network Science: Evolving Network Models February 2015

4. Erdös-Rényi Model- Publ. Math. Debrecen 6, 290 (1959) fixed node number N connecting pairs of nodes with probability p Clustering coefficient: Path length: Degree distribution: BENCHMARK 2: Random Network Model Network Science: Evolving Network Models February 2015

5. Watts-Strogatz algorithm – Nature 2008 For fixed node number N, first connect them into even number, k, degree ring in which k/2 nearest neighbors on each side of each node are connected to it Then, with probability p re-wire ring edges of each node to nodes not currently connected to and different from it Clustering coefficient: Path length: Degree distribution: Exponential BENCHMARK 3: Small World Model Network Science: Evolving Network Models February 2015

6. Missing Hubs Hubs represent the most striking difference between a random and a scale-free network. Their emergence in many real systems raises several fundamental questions: Why does the random network model of Erdős and Rényi fail to reproduce the hubs and the power laws observed in many real networks? Why do so different systems as the WWW or the cell converge to a similar scale-free architecture?

7. P(k) ~ k -  Regular network Erdos- Renyi Watts- Strogatz Pathlenght Clustering Degree Distr. P(k)=δ(k-k d ) Exponential EMPIRICAL DATA FOR REAL NETWORKS Network Science: Evolving Network Models February 2015

8. SCALE-FREE MODEL (BA model) Network Science: Evolving Network Models February 2015

9. Real networks continuously expand by the addition of new nodes Barabási & Albert, Science 286, 509 (1999) BA MODEL: Growth BA model: Growth ER, WS models: the number of nodes, N, is fixed (static models) Network Science: Evolving Network Models February 2015

10. networks expand through the addition of new nodes Barabási & Albert, Science 286, 509 (1999) BA MODEL: Growth BA model: Growth ER model: the number of nodes, N, is fixed (static models)

11. WWW Barabási & Albert, Science 286, 509 (1999) BA MODEL: Growth (www/Pubs) Scientific Publications BA model: Growth http://website101.com/define-ecommerce-web-terms-definitions/http://www.kk.org/thetechnium/archives/2008/10/the_expansion_o.php Network Science: Evolving Network Models February 2015

12. (1) Networks continuously expand by the addition of new nodes Add a new node with m links Barabási & Albert, Science 286, 509 (1999) BA MODEL: Growth BA model: Growth Network Science: Evolving Network Models February 2015

13. Barabási & Albert, Science 286, 509 (1999) 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). BA MODEL: Preferential Attachment Where will the new node link to? ER, WS models: choose randomly. Network Science: Evolving Network Models February 2015

14. Barabási & Albert, Science 286, 509 (1999) Network Science: Evolving Network Models Growth and Preferential Sttachment BA model: Growth The random network model differs from real networks in two important characteristics: Growth: While the random network model assumes that the number of nodes is fixed (time invariant), real networks are the result of a growth process that continuously increases. Preferential Attachment: While nodes in random networks randomly choose their interaction partner, in real networks new nodes prefer to link to the more connected nodes.

15. Barabási & Albert, Science 286, 509 (1999) P(k) ~k -3 (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 Origin of SF networks: Growth and preferential attachment Network Science: Evolving Network Models February 2015

16. A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999) All nodes follow the same growth law Use: During a unit time (time step): Δk=m  A=m β: dynamical exponent Network Science: Evolving Network Models February 2015

17. SF model: k(t)~t ½ (first mover advantage) Fitness Model: Can Latecomers Make It? time Degree (k) Network Science: Evolving Network Models February 2015

18. γ = 3 A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999) Degree distribution A node i can come with equal probability any time between t i =m 0 and t, hence: Network Science: Evolving Network Models February 2015

19. γ = 3 A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999) Degree distribution (i) The degree exponent is independent of m. (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 m 2. Network Science: Evolving Network Models February 2015

20. Stationarity: P(k) independent of N m=1,3,5,7 N=100,000;150,000;200,000 Insert: degree dynamics m-dependence NUMERICAL SIMULATION OF THE BA MODEL Network Science: Evolving Network Models February 2015

21. 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 2015

22. A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999) Number of nodes with degree k at time t. Nr. of degree k-1 nodes that acquire a new link, becoming degree k Preferential attachment 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 Number of links added to degree k nodes after the arrival of a new node: Total number of k-nodes New node adds m new links to other nodes Nr. of degree k nodes that acquire a new link, becoming degree k+1 # k-nodes at time t+1 # k-nodes at time t Gain of k- nodes via k-1  k Loss of k- nodes via k  k+1 MFT - Degree Distribution: Rate Equation

23. A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999) # m-nodes at time t+1 # m- nodes at time t Add one m-degeree node Loss of an m-node via m  m+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 # k-nodes at time t+1 # k-nodes at time t Gain of k- nodes via k-1  k Loss of k- nodes via k  k+1 MFT - Degree Distribution: Rate Equation Network Science: Evolving Network Models February 2015

24. A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999) k>m We assume that there is a stationary state in the N=t  ∞ limit, when P(k,∞)=P(k) k>m MFT - Degree Distribution: Rate Equation Network Science: Evolving Network Models February 2015

25. A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999) ... m+3  k Krapivsky, Redner, Leyvraz, PRL 2000 Dorogovtsev, Mendes, Samukhin, PRL 2000 Bollobas et al, Random Struc. Alg. 2001 for large k MFT - Degree Distribution: Rate Equation Network Science: Evolving Network Models February 2015

26. A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999) Its solution is: Start from eq. Dorogovtsev and Mendes, 2003 MFT - Degree Distribution: A Pretty Caveat Network Science: Evolving Network Models February 2015

27. Do we need both growth and preferential attachment? Network Science: Evolving Network Models February 2015

28. growth preferential attachment Π(k i ) : uniform MODEL A Network Science: Evolving Network Models February 2015

29. growth preferential attachment P(k) : power law (initially)  Gaussian  Fully Connected MODEL B Network Science: Evolving Network Models February 2015

30. Do we need both growth and preferential attachment? YEP. Network Science: Evolving Network Models February 2015

31. P(k) ~ k -  Regular network Erdos- Renyi Watts- Strogatz P(k)=δ(k-k d ) Exponential Barabasi- Albert P(k) ~ k -  EMPIRICAL DATA FOR REAL NETWORKS Pathlenght Clustering Degree Distr. Network Science: Evolving Network Models February 2015

32. 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. 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 Small World DISTANCES IN SCALE-FREE NETWORKS

33. Bollobas, Riordan, 2002 PATH LENGTHS IN THE BA MODEL Network Science: Evolving Network Models February 2015

34. P(k) ~ k -  P(k)=δ(k-k d ) Exponential P(k) ~ k -  EMPIRICAL DATA FOR REAL NETWORKS Pathlenght Clustering Degree Distr. Regular network Erdos- Renyi Watts- Strogatz Barabasi- Albert Network Science: Evolving Network Models February 2015

35. The numerical results indicate a slightly slower decay. What is the functional form of C(N)? CLUSTERING COEFFICIENT OF THE BA MODEL Reminder: for a random graph we have: 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 2015

36. 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 k l participates is thus: We need to calculate P(i,j). CLUSTERING COEFFICIENT OF THE BA MODEL Network Science: Evolving Network Models February 2015

37. Calculate P(i,j). Node j arrives at time t j =j and the probability that it will link to node i with degree k i already in the network is determined by preferential attachment: Where we used that the arrival time of node j is t j =j and the arrival time of node is t i =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? CLUSTERING COEFFICIENT OF THE BA MODEL Network Science: Evolving Network Models February 2015

38. 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 2015

39. P(k) ~ k -  EMPIRICAL DATA FOR REAL NETWORKS Pathlenght Clustering Degree Distr. P(k)=δ(k-k d ) Exponential P(k) ~ k -  Regular network Erdos- Renyi Watts- Strogatz Barabasi- Albert Network Science: Evolving Network Models February 2015

40. The origins of preferential attachment. Network Science: Evolving Network Models February 2015

42. Plot the change in the degree  k during a fixed time  t for nodes with degree k, and you get   (k) (Jeong, Neda, A.-L. B, Europhys Letter 2003; cond-mat/0104131) No pref. attach: κ~k Linear pref. attach: κ~k 2 To reduce noise, plot the integral of Π(k) over k: CAN WE MEASURE PREFERENTIAL ATTACHMENT? Network Science: Evolving Network Models February 2015

43. neurosci collab actor collab. citation network Plots shows the integral of Π(k) over k: Internet CAN WE MEASURE PREFERENTIAL ATTACHMENT? No pref. attach: κ~k Linear pref. attach: κ~k 2 Network Science: Evolving Network Models February 2015

44. 1.Copying mechanism directed network select a node and an edge of this node attach to the endpoint of this edge 2.Walking on a network directed network the new node connects to a node, then to every first, second, … neighbor of this node 3.Attaching to edges select an edge attach to both endpoints of this edge 4.Node duplication duplicate a node with all its edges randomly prune edges of new node MECHANISMS RESPONSIBLE FOR PREFERENTIAL ATTACHMENT Network Science: Evolving Network Models February 2015

45. Copying Mechanism Network Science: Evolving Network Models February 2015

46. 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. ORIGIN OF THE SCALE-FREE TOPOLOGY IN THE CELL:Gene Duplication Network Science: Evolving Network Models February 2015

47.  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 PREFERENTIAL ATTACHMENT IN PROTEIN INTERACTION NETWORKS Network Science: Evolving Network Models February 2015

48. Nr. of nodes: Nr. of links: Average degree: Degree dynamics Degree distribution: Average Path Length: Clustering Coefficient: The network grows, but the degree distribution is stationary. β: dynamical exponent γ: degree exponent SUMMARY: PROPERTIES OF THE BA MODEL Network Science: Evolving Network Models February 2015

49. γ=1γ=2γ=3 diverges finite γ w in γ w out γ intern γ actor γ collab γ metab γ cita γ synonyms γ sex BA model Can we change the degree exponent? DEGREE EXPONENTS Network Science: Evolving Network Models February 2015

50. Evolving network models Network Science: Evolving Network Models February 2015

51. 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 EVOLVING NETWORK MODELS Network Science: Evolving Network Models February 2015

52. (the simplest way to change the degree exponent) γ = 3 A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999) Undirected BA network: Directed BA network: β=1: dynamical exponentγ in =2: degree exponent; P(k out )=δ(k out -m) Undirected BA: β=1/2; γ=3 BA ALGORITHM WITH DIRECTED EDGES Network Science: Evolving Network Models February 2015

53. Extended Model prob. p : internal links prob. q : link deletion prob. 1-p-q : add node EXTENDED MODEL: Other ways to change the exponent P(k) ~ (k+  (p,q,m)) -  (p,q,m)   [1,  ) Network Science: Evolving Network Models February 2015

54. P(k) ~ (k+  (p,q,m)) -  (p,q,m)   [1,  ) Extended Model p=0.937 m=1  = 31.68  = 3.07 Actor network 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 EXTENDED MODEL: Small-k cutoff Network Science: Evolving Network Models February 2015

55. More models 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) NONLINEAR PREFERENTIAL ATTACHMENT Network Science: Evolving Network Models February 2015

56. Initial attractiveness shifts the degree exponent: A - initial attractiveness Dorogovtsev, Mendes, Samukhin, Phys. Rev. Lett. 85, 4633 (2000) 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) 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 INITIAL ATTRACTIVENESS Network Science: Evolving Network Models February 2015

57. Finite lifetime to acquire new edges Gradual aging: S. N. Dorogovtsev and J. F. F. Mendes, Phys. Rev. E 62, 1842 (2000) L. A. N. Amaral et al., PNAS 97, 11149 (2000) GROWTH CONSTRAINTS AND AGING CAUSE CUTOFFS Network Science: Evolving Network Models February 2015

58. P(k) ~ k -  Pathlenght Clustering Degree Distr. P(k)=δ(k-k d ) Exponential P(k) ~ k -  THE LAST PROBLEM: HIGH, SYSTEM-SIZE INDEPENDENT C(N) Regular network Erdos- Renyi Watts- Strogatz Barabasi- Albert Network Science: Evolving Network Models February 2015

59. Each node of the network can be either active or inactive. There are m active nodes in the network in any moment. 1.Start with m active, completely connected nodes. 2.Each timestep add a new node (active) that connects to m active nodes. 3.Deactivate one active node with probability: K. Klemm and V. Eguiluz, Phys. Rev. E 65, 036123 (2002) C  C* when N  ∞ A MODEL WITH HIGH CLUSTERING COEFFICIENT Network Science: Evolving Network Models February 2015

61. Fitness Model Network Science: Evolving Network Models February 2015

62. SF model: k(t)~t ½ (first mover advantage) Fitness model: fitness (  k( ,t)~t   =  C Fitness Model: Can Latecomers Make It? time Degree (k) Bianconi & Barabási, Physical Review Letters 2001; Europhys. Lett. 2001.

63. G. Bianconi and A.-L. Barabási, Physical Review Letters 2001; cond-mat/0011029 NetworkBose gas Fitness η  Energy level ε New node with fitness η  New energy level ε Link pointing to node η  Particle at level ε Network  quantum gas MAPPING TO A QUANTUM GAS Network Science: Evolving Network Models February 2015

64. f(  )=e . The dynamic exponent f(e) depends on m, determined by the self-consistent equation: BOSE-EINSTEIN CONDENSATION Network Science: Evolving Network Models February 2015

65. time Degree (k) Bianconi & Barabási, Physical Review Letters 2001; Europhys. Lett. 2001. Bose-Einstein Condensation Network Science: Evolving Network Models February 2015

66. Bianconi & Barabási, Physical Review Letters 2001; Europhys. Lett. 2001. Network Science: Evolving Network Models February 14, 2011 Bose-Einstein Condensation

67. Bose-Einstein condensation Fit-gets-rich FITNESS MODEL: Can Latecomers Make It?

68. 1. There is no universal exponent characterizing all networks. 2.Growth and preferential attachment are responsible for the emergence of the scale-free property. 3.The origins of the preferential attachment is system-dependent. 4. 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. LESSONS LEARNED: evolving network models Network Science: Evolving Network Models February 2015

69. 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 LESSONS LEARNED: evolving network models Network Science: Evolving Network Models February 2015

70. The end Network Science: Evolving Network Models February 2015