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Class 3: Random Graphs

Class 3: Random Graphs. Prof. Albert-László Barabási Dr. Baruch Barzel, Dr. Mauro Martino. Network Science: Random Graphs 2012. RANDOM NETWORK MODEL. Network Science: Random Graphs 2012. RANDOM NETWORK MODEL. Alfréd Rényi (1921-1970). Pál Erdös (1913-1996). Erdös-Rényi model (1960)

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Class 3: Random Graphs

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  1. Class 3: Random Graphs Prof. Albert-László Barabási Dr. Baruch Barzel, Dr. Mauro Martino Network Science: Random Graphs 2012

  2. RANDOM NETWORK MODEL Network Science: Random Graphs 2012

  3. RANDOM NETWORK MODEL Alfréd Rényi (1921-1970) Pál Erdös (1913-1996) Erdös-Rényi model (1960) Connect with probability p p=1/6 N=10 k ~ 1.5 Network Science: Random Graphs 2012

  4. RANDOM NETWORK MODEL Definition: A random graph is a graph of N labeled nodes where each pair of nodes is connected by a preset probability p. We will call is G(N, p). Network Science: Random Graphs 2012

  5. RANDOM NETWORK MODEL p=1/6 N=12 Network Science: Random Graphs 2012

  6. RANDOM NETWORK MODEL p=0.03 N=100 Network Science: Random Graphs 2012

  7. RANDOM NETWORK MODEL Nand p do not uniquely define the network– we can have many different realizations of it. How many? N=10 p=1/6 The probability to form a particular graph (with some value of L)G(N,p) is That is, each graph G(N,p) appears with probability P(G(N,p)). Network Science: Random Graphs 2012

  8. RANDOM NETWORK MODEL P(L): the probability to have exactly L links in a network of N nodes and probability p: The maximum number of links in a network of N nodes. Binomial distribution... Number of different ways we can choose L links among all potential links. Network Science: Random Graphs 2012

  9. MATH TUTORIAL the mean of a binomial distribution There is a faster way using generating functions, see: http://planetmath.org/encyclopedia/BernoulliDistribution2.html Network Science: Random Graphs 2012

  10. MATH TUTORIAL the variance of a binomial distribution http://keral2008.blogspot.com/2008/10/derivation-of-mean-and-variance-of.html Network Science: Random Graphs 2012

  11. MATH TUTORIAL the variance of a binomial distribution http://keral2008.blogspot.com/2008/10/derivation-of-mean-and-variance-of.html Network Science: Random Graphs 2012

  12. MATH TUTORIAL Binomian Distribution: The bottom line http://keral2008.blogspot.com/2008/10/derivation-of-mean-and-variance-of.html Network Science: Random Graphs 2012

  13. RANDOM NETWORK MODEL P(L): the probability to have a network of exactly L links • The average number of links <L> in a random graph • The standard deviation for number of links Network Science: Random Graphs 2012

  14. DEGREE DISTRIBUTION OF A RANDOM GRAPH • probability of • missing N-1-k • edges Select k nodes from N-1 • probability of • having k edges As the network size increases, the distribution becomes increasingly narrow—we are increasingly confident that the degree of a node is in the vicinity of <k>. Network Science: Random Graphs 2012

  15. DEGREE DISTRIBUTION OF A RANDOM GRAPH For largeNand small k, we can use the following approximations: for Network Science: Random Graphs 2012

  16. DEGREE DISTRIBUTION OF A RANDOM GRAPH P(k) k Network Science: Random Graphs 2012

  17. DEGREE DISTRIBUTION OF A RANDOM NETWORK Exact Result -binomial distribution- Large N limit -Poisson distribution- Probability Distribution Function (PDF) Network Science: Random Graphs 2012

  18. NODES HAVE COMPARABLE DEGREES IN RANDOM NETWORKS What does it mean? Continuum formalism: If we consider a network with average degree <k> then the probability to have a node whose degree exceeds a degree k0 is: • For example, with <k>=10, • the probability to find a node whose degree is at least twice the average degree is 0.00158826. • the probability to find a node whose degree is at least ten times the average degree is 1.79967152 × 10-13 • the probability to find a node whose degree is less than a tenth of the average degree is 0.00049 • See http://www.stud.feec.vutbr.cz/~xvapen02/vypocty/po.php What does it mean? Discrete formalism: • The probability of seeing a node with very high of very low degree is exponentially small. • Most nodes have comparable degrees. • The larger the size of a random network, the more similar are the node degrees

  19. NO OUTLIERS IN A RANDOM SOCIETY • According to sociological research, for a typical individual k ~1,000 • The probability to find an individual with degree k>2,000 is 10-27. • Given that N ~109, the chance of finding an individual with 2,000 acquaintances is so tiny that such nodes are virtually inexistent in a random society. • a random society would consist of mainly average individuals, with everyone with roughly the same number of friends. • It would lack outliers, individuals that are either highly popular or recluse. Network Science: Random Graphs 2012

  20. SIX DEGREES small worlds Sarah Jane Ralph Peter Frigyes Karinthy, 1929 Stanley Milgram, 1967 Network Science: Random Graphs 2012

  21. SIX DEGREES 1929: Frigyes Kartinthy 1929: Minden másképpen van (Everything is Different) Láncszemek (Chains) “Look, Selma Lagerlöf just won the Nobel Prize for Literature, thus she is bound to know King Gustav of Sweden, after all he is the one who handed her the Prize, as required by tradition. King Gustav, to be sure, is a passionate tennis player, who always participates in international tournaments. He is known to have played Mr. Kehrling, whom he must therefore know for sure, and as it happens I myself know Mr. Kehrling quite well.” "The worker knows the manager in the shop, who knows Ford; Ford is on friendly terms with the general director of Hearst Publications, who last year became good friends with Arpad Pasztor, someone I not only know, but to the best of my knowledge a good friend of mine. So I could easily ask him to send a telegram via the general director telling Ford that he should talk to the manager and have the worker in the shop quickly hammer together a car for me, as I happen to need one." Frigyes Karinthy (1887-1938) Hungarian Writer Network Science: Random Graphs 2012

  22. SIX DEGREES 1967: Stanley Milgram HOW TO TAKE PART IN THIS STUDY 1. ADD YOUR NAME TO THE ROSTER AT THE BOTTOM OF THIS SHEET, so that the next person who receives this letter will know who it came from. 2. DETACH ONE POSTCARD. FILL IT AND RETURN IT TO HARVARD UNIVERSITY. No stamp is needed. The postcard is very important. It allows us to keep track of the progress of the folder as it moves toward the target person. 3. IF YOU KNOW THE TARGET PERSON ON A PERSONAL BASIS, MAIL THIS FOLDER DIRECTLY TO HIM (HER). Do this only if you have previously met the target person and know each other on a first name basis. 4. IF YOU DO NOT KNOW THE TARGET PERSON ON A PERSONAL BASIS, DO NOT TRY TO CONTACT HIM DIRECTLY. INSTEAD, MAIL THIS FOLDER (POST CARDS AND ALL) TO A PERSONAL ACQUAINTANCE WHO IS MORE LIKELY THAN YOU TO KNOW THE TARGET PERSON. You may send the folder to a friend, relative or acquaintance, but it must be someone you know on a first name basis. Network Science: Random Graphs 2012

  23. SIX DEGREES 1991: John Guare "Everybody on this planet is separated by only six other people. Six degrees of separation. Between us and everybody else on this planet. The president of the United States. A gondolier in Venice…. It's not just the big names. It's anyone. A native in a rain forest. A Tierra del Fuegan. An Eskimo. I am bound to everyone on this planet by a trail of six people. It's a profound thought. How every person is a new door, opening up into other worlds." Network Science: Random Graphs 2012

  24. WWW: 19 DEGREES OF SEPARATION Image by Matthew Hurst Blogosphere Network Science: Random Graphs 2012

  25. DISTANCES IN RANDOM GRAPHS Random graphs tend to have a tree-like topology with almost constant node degrees. • nr. of first neighbors: • nr. of second neighbors: • nr. of neighbours at distance d: • estimate maximum distance: (dmax where N(dmax) = N) logN << N (small world) Network Science: Random Graphs 2012

  26. DISTANCES IN RANDOM GRAPHS compare with real data ????Given the huge differences in scope, size, and average degree, the agreement is excellent. See table 3.2 Network Science: Random Graphs 2012

  27. EVOLUTION OF A RANDOM GRAPH Network Science: Random Graphs 2012

  28. EVOLUTION OF A RANDOM NETWORK Until now we focused on the static properties of a random graph with fixes p value. What happens when vary the parameter p? GOTO http://cs.gmu.edu/~astavrou/random.html Choose Nodes=100. Note that the p goes up in increments of 0.001, which, for N=100, L=pN(N-1)/2~p*50,000, i.e. each increment is really about 50 new lines. Network Science: Random Graphs 2012

  29. EVOLUTION OF A RANDOM NETWORK disconnected nodesNETWORK. <k> How does this transition happen? Network Science: Random Graphs 2012

  30. I: Subcritical <k> < 1 II: Critical <k> = 1 III: Supercritical <k> > 1 IV: Connected <k> > ln N <k> N=100 <k>=1 <k>=3 <k>=5 <k>=0.5

  31. I: Subcritical <k> < 1 p < pc=1/N <k> No giant component. N-L isolated clusters, cluster size distribution is exponential The largest cluster is a tree, its size ~ lnN; N(giant comp)/N = ln N/N tends to 0 As N tends to infinity.

  32. II: Critical <k> = 1 p=pc=1/N <k> • Unique giant component: NG~ N2/3 • contains a vanishing fraction of all nodes, NG/N ~ N-1/3 • Small components are trees, GC has loops. A jump in the cluster size: N=1,000  ln N~ 6.9; N2/3~95 N=7 109 ln N~ 22; N2/3~3,659,250

  33. III: Supercritical <k> > 1 p > pc=1/N <k>=3 <k> • Unique giant component: NG~ (p-pc)N (GC has a finite fraction of nodes) • GC has loops. • Cluster size distribution: exponential

  34. IV: Connected <k> > ln N p > (ln N)/N <k>=5 <k> • Only one cluster: NG=N • GC is dense. • Cluster size distribution: None Network Science: Random Graphs 2012

  35. I: Subcritical <k> < 1 II: Critical <k> = 1 III: Supercritical <k> > 1 IV: Connected <k> > ln N <k> N=100 <k>=1 <k>=3 <k>=5 <k>=0.5

  36. CLUSTERING COEFFICIENT • Since edges are independent and have the same probability p, This is valid for random networks only, with arbitrary degree distribution The clustering coefficient of random graphs is small. For fixed degree C decreases with the system size N. Network Science: Random Graphs 2012 13.47 from Newman 2010

  37. Erdös-Rényi MODEL (1960) • Degree distribution • Binomial, Poisson (exponential tails) • Clustering coefficient • Vanishing for large network sizes • Average distance among nodes • Logarithmically small Network Science: Random Graphs 2012

  38. HISTORICAL NOTE Network Science: Random Graphs 2012

  39. HISTORICAL NOTE 1951, Rapoport and Solomonoff:  first systematic study of a random graph. demonstrates the phase transition. natural systems: neural networks; the social networks of physical contacts (epidemics); genetics. 1959: G(N,p) Anatol Rapoport 1911- 2007 Edgar N. Gilbert (b.1923) Why do we call it the Erdos-Renyi random model? Network Science: Random Graphs 2012

  40. Erdös-Rényi MODEL (1960) Network Science: Random Graphs 2012

  41. Are real networks like random graphs? Network Science: Random Graphs 2012

  42. ARE REAL NETWORKS LIKE RANDOM GRAPHS? As quantitative data about real networks became available, we can compare their topology with the predictions of random graph theory. Note that once we have N and <k> for a random network, from it we can derive every measurable property. Indeed, we have: Average path length: Clustering Coefficient: Degree Distribution: Network Science: Random Graphs 2012

  43. PATH LENGTHS IN REAL NETWORKS Prediction: Data: Real networks have short distances like random graphs. Network Science: Random Graphs 2012

  44. CLUSTERING COEFFICIENT Prediction: Data: Crand underestimates with orders of magnitudes the clustering coefficient of real networks. Network Science: Random Graphs 2012

  45. THE DEGREE DISTRIBUTION Prediction: Data: • Internet; • Movie Actors; • Coauthorship, high energy physics; • (d) Coauthorship, neuroscience Network Science: Random Graphs 2012

  46. ARE REAL NETWORKS LIKE RANDOM GRAPHS? As quantitative data about real networks became available, we can compare their topology with the predictions of random graph theory. Note that once we have N and <k> for a random network, from it we can derive every measurable property. Indeed, we have: Average path length: Clustering Coefficient: Degree Distribution: Network Science: Random Graphs 2012

  47. IS THE RANDOM GRAPH MODEL RELEVANT TO REAL SYSTEMS? (A) Problems with the random network model: -the degree distribution differs from that of real networks -the giant component in most real network does NOT emerge through a phase transition -the clustering coefficient in most systems will now vanish, as predicted by the model. (B) Most important: we need to ask ourselves, are real networks random? The answer is simply: NO Hence it is IRRELEVANT. A WRONG B  IRRELEVANT There is no network in nature that we know of that would be described by the random network model. Network Science: Random Graphs 2012

  48. IF IT IS WRONG AND IRRELEVANT, WHY DID WE DEVOT TO IT A FULL CLASS? It is the reference model for the rest of the class. It will help us calculate many quantities, that can then be compared to the real data, understanding to what degree is a particular property the result of some random process. Organizing principles: patterns in real networks that are shared by a large number of real networks, yet which deviate from the predictions of the random network model. In order to identify these, we need to understand how would a particular property look like if it is driven entirely by random processes. While WRONG and IRRELEVANT, it will turn out to be extremly USEFUL! Network Science: Random Graphs 2012

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