WLU Mathematics Department Seminar March 12, The Web Graph. Anthony Bonato. Wilfrid Laurier University. Waterloo, Canada

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1 WLU Mathematics Department Seminar March 12, 2007 The Web Graph Anthony Bonato Wilfrid Laurier University Waterloo, Canada

2 Graph theory the last century 4/27/2007 The web graph - Anthony Bonato 2

3 Today 4/27/2007 The web graph - Anthony Bonato 3

4 a graph G=(V,E) consists of a nonempty set of vertices or nodes V, and a set of edges E, which is a reflexive, symmetric binary relation on V in directed graphs E need not be symmetric nodes edges 4/27/2007 The web graph - Anthony Bonato 4

5 The web graph/network nodes: web pages edges: links 4/27/2007 The web graph - Anthony Bonato 5

6 Google uses graph theory! 4/27/2007 The web graph - Anthony Bonato 6

7 How big is the web? The web is really infinite calendars, online organizers random strings: google raingod random strings Total web 54 billion static pages (Hirate, Kato, Yamana, 07) 4/27/2007 The web graph - Anthony Bonato 7

8 Collaboration networks nodes: mathematicians edges: co-authoring 4/27/2007 The web graph - Anthony Bonato 8

9 4/27/2007 The web graph - Anthony Bonato 9

10 Social networks nodes: people edges: social interaction (eg friendship, co-authorship, sexual relations) 4/27/2007 The web graph - Anthony Bonato 10

11 Biological networks nodes: proteins edges: biochemical interactions 4/27/2007 The web graph - Anthony Bonato 11

12 Hollywood network nodes: actors edges: star in same movie 4/27/2007 The web graph - Anthony Bonato 12

13 The web graph and related networks above are often called: self-organizing scale-free massive complex heterogeneous 4/27/2007 The web graph - Anthony Bonato 13

14 Important graph parameters distance degree diameter of between of a G, vertex diam(g) nodes, x in G, deg G (x,y) G (x) G deg diam(g)=3 G (1,6)=3 G (1)=2 deg G (6)=1 4/27/2007 The web graph - Anthony Bonato 14

15 Power laws in web graph P( k ) k b for some b>1 ratio of # nodes of degree k, to # nodes Broder et al, 01 4/27/2007 The web graph - Anthony Bonato 15

16 Interpreting a power law Many lowdegree nodes Few highdegree nodes 4/27/2007 The web graph - Anthony Bonato 16

17 Binomial Power law Highway network Air traffic network 4/27/2007 The web graph - Anthony Bonato 17

18 Other properties of self-organizing networks small world topology (Watts, Strogatz, 98): low diameter/average distance diameter of the web: 19 globally sparse, locally dense rich bipartite structures (Kumar et al, 00) 4/27/2007 The web graph - Anthony Bonato 18

19 Random graphs Paul Erdős Alfred Rényi 4/27/2007 The web graph - Anthony Bonato 19

20 4/27/2007 The web graph - Anthony Bonato 20

21 G(n,p) random graph model (Erdős, Rényi, 63) p a real number in (0,1), n a positive integer G(n,p): probability space on graphs with nodes {1,,n}, two nodes joined independently and with probability p 4/27/2007 The web graph - Anthony Bonato 21

22 Properties of G(n,p) Almost regular: node degrees concentrated around np Degree distribution is binomial Low diameter, low clustering, rich but uniform substructures 4/27/2007 The web graph - Anthony Bonato 22

23 G(n,p) as a model for the web graph Pros: Cons Corpus of existing literature:large arsenal of tools Independence Binomial degree distribution Independence Off-line 4/27/2007 The web graph - Anthony Bonato 23

24 (Bonato, 04) A model for selforganizing networks should have the following properties: 1. On-line property. The number of nodes and edges changes with time. 2. Power law degree distribution. 3. Small world property. 4. Many bipartite substructures. 4/27/2007 The web graph - Anthony Bonato 24

25 Preferential attachment (PA) model (Barabási, Albert, 99), (Bollobás et al, 01) Parameter: m a positive integer At time 0, add a single edge At time t+1, add m edges from a new node v t+1 to existing nodes the edge v t+1 v s is added with probability deg Gt 2t ( ) v s 4/27/2007 The web graph - Anthony Bonato 25

26 Simulation of the model: m=1 Wilensky, U. (2005). NetLogo Preferential Attachment model. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL. 4/27/2007 The web graph - Anthony Bonato 26

27 Theorem (Bollobás et al, 01) Fix m a positive integer and fix ε > 0. For k a non-negative integer, define α m,k = 2m( m+1) ( k + m) ( k + m +1) ( k + m + 2) Then with probability tending to 1 as t, for all k satisfying 0 k t 1/5 ( 1 ε ) α m, k P tm ( k) ( 1+ ε ) α m, k G 4/27/2007 The web graph - Anthony Bonato 27

28 Theorem (Bollobás, Riordan, 04) Fix an integer m 1 and a positive real number ε. With probability 1 as t, G m (t) is connected and log t log log ( ) ( t 1 ε diam G ) ( 1 + ε ) t m log log t log t 4/27/2007 The web graph - Anthony Bonato 28

29 Other web graph models (Aiello, Chung Lu, 01) ACL. (Chung, Lu, 03) CL. (Kumar et al, 99) Copying model. (Chung, Lu, 04) CL-del growth-deletion model. (Cooper, Frieze, Vera, 04) CFV growth-deletion model. 4/27/2007 The web graph - Anthony Bonato 29

30 Properties of the models Model Online Power law Small world Bipartite Subgraphs PA Y Y Y? 3 ACL Y Y? N (2, ) CL N Y Y? (2, ) Copying Y Y? Y (2, ) CL-del Y Y Y? (2, ) CFV Y Y?? (2, ) b 4/27/2007 The web graph - Anthony Bonato 30

31 Open problem Design a model that provably has all of the four properties. 4/27/2007 The web graph - Anthony Bonato 31

32 Google and graph theory major component of Google s rankings: PageRank PageRank is a measure of the popularity of a page as a function of its indegree 4/27/2007 The web graph - Anthony Bonato 32

33 Pagerank PageRank is models the probability web surfing a random via a surfer random visits walka page surfer usually moves via out-links on occasion, the surfer teleports to a random page 4/27/2007 The web graph - Anthony Bonato 33

34 The Google Matrix given a digraph G with vertices {1,,n}, define the matrix P 1 form P 2 by replacing entries in zero rows of P 1 by 1/n define the Google matrix P as - c in (0,1) is the teleportation constant 4/27/2007 The web graph - Anthony Bonato 34

35 1 2 G 3 4 P(G) 4/27/2007 The web graph - Anthony Bonato 35

36 it can be shown that PageRank is the dominant eigenvector s of P, normalized so the sum of entries is 1 reduces to computing an eigenvector of an order 54 billion matrix numerical short-cut: power method applied to sparse matrices 4/27/2007 The web graph - Anthony Bonato 36

37 /27/2007 The web graph - Anthony Bonato 37

38 Infinite graphs A new research direction: As time tends to in on-line models, number of nodes grows Consider infinite limit: union of chain of nodes and edges Limit behaviour studied in other disciplines: Economics (continuum of agents) Physics(lattice structures) 4/27/2007 The web graph - Anthony Bonato 38

39 Infinite limit graph loses some properties of finite system: -nodes have infinite degree Certain structural properties are magnified: -self-similarity 4/27/2007 The web graph - Anthony Bonato 39

40 Theorem (Erdős,Rényi, 63): The random process generates a unique isomorphism type of graph with probability 1. paradoxical: random process with deterministic conclusion 4/27/2007 The web graph - Anthony Bonato 40

41 Infinite random graph unique isomorphism type, R Infinite random graph, Rado graph, universal homogeneous graph, R focus of intensive research for over 50 years Graph theorists Logicians Probabilists Group and semigroup theorists Topologists 4/27/2007 The web graph - Anthony Bonato 41

42 Homogeneity A graph G is homogeneous if each isomorphism between finite induced subgraphs extends to an automorphism of G. strongest symmetry a graph can possess for example C 5 is homogeneous, but C 6 is not R is homogeneous (by a back-and-forth argument) Countable homogeneous graphs were classified by: Gardiner, Sheehan, Gol fand & Klin: finite case, 1970 s Lachlan & Woodrow: infinite case, 1980 s there are countably many countable homogeneous graphs 4/27/2007 The web graph - Anthony Bonato 42

43 Isotropic graphs A graph G which is the limit of a sequence (G : t t 0) is isotropic if for all t 0 >0 there is an isomorphic copy of G contained in limg t t 0 t G 4/27/2007 The web graph - Anthony Bonato 43

44 isotropic anisotropic 4/27/2007 The web graph - Anthony Bonato 44

45 Limits of models of self-organizing networks are: isotropic and heterogeneous Self-organizing networks are: self-similar and heterogeneous 4/27/2007 The web graph - Anthony Bonato 45

46 Copying models New nodes copy some of the link structure of an existing node Motivation: 1. web page generation 2. mutation in biology 4/27/2007 The web graph - Anthony Bonato 46

47 Copying model for W: R. Kumar, P. Raghavan, S. Rajagopalan, D. Sivakumar, A. Tomkins, E. Upfal, Stochastic models for the web graph, In: Proceedings of the 41th IEEE Symp. on Foundations of Computer Science, (2000) Duplication model for biological networks: F.R.K. Chung, G. Dewey, D.J. Galas, L. Lu, Duplication models for biological networks, Journal of Computational Biology 10 (2003) Generalized copying model: generalizes and unifies the above A. Bonato, J. Janssen, in preparation. 4/27/2007 The web graph - Anthony Bonato 47

48 N(v) v N(u) u y x 4/27/2007 The web graph - Anthony Bonato 48

49 Limits of copying models (Bonato,Janssen, 04) start with a finite initial graph H at each time-step, choose a node u, and choose a subset S of N(u) add a node z joined to S do for all u and S resulting limit is R H 4/27/2007 The web graph - Anthony Bonato 49

50 Non-isomorphic R H The graph R H admits a homomorphism to H (using a greedy colouring) There are infinitely many non-isomorphic R H for example, R K3 is not isomorphic to R K4 4/27/2007 The web graph - Anthony Bonato 50

51 Properties of R H with probability 1, limits of graphs generated by duplication model (with initial graph H) isomorphic to R H inexhaustible: for all nodes x, R H -x isomorphic to R H isomorphism type related to dismantling orderings on finite graphs studied by (Nowakowski, Winkler, 83) rich endomorphisms (Bonato, P. Cameron, Delic, 06) heterogeneous, isotropic 4/27/2007 The web graph - Anthony Bonato 51

52 Limits for PA models (J. Kleinberg, R. Kleinberg,05) considered limits of preferential attachment models limits of PA models: m=1,2: unique limit m>2: limit not unique proofs use martingale techniques heterogeneous, isotropic 4/27/2007 The web graph - Anthony Bonato 52

53 A new geometric model for the web Idea: web pages exist in a topic-space (similar to word-document space, used in text-mining) a page is more likely to link to pages close to it in topic-space 4/27/2007 The web graph - Anthony Bonato 53

54 Random geometric graphs well-studied alternative to G(n,p) (Frieze et al.,04) proposes a geometric PA model for the web graph drawback: new nodes need global knowledge 4/27/2007 The web graph - Anthony Bonato 54

55 Locally Random Geometric (LRG) model (Bonato,Cooper,Janssen, 07) on-line model generating directed graphs parameter: p a real number in (0,1] let S be a sphere in R 3 with radius 1 at time 0, add a single node chosen u.a.r. at time t, each node v has a neighbourhood of influence B v with radius ( v) deg + G 1 t t at time t+1, node z is added with position u.a.r. on S if z is in B v, then add (z,v) independently with probability p 4/27/2007 The web graph - Anthony Bonato 55

56 As nodes are born, they are more likely to enter some B v with larger radius (indegree) Over time, a power law degree distribution results 4/27/2007 The web graph - Anthony Bonato 56

57 Theorem (Bonato, Cooper, Janssen, 07) With probability 1, the LRG model generates power law graphs with exponent 1 1+ [2, ) p other properties under investigation: sparse cuts, local structure, diameter 4/27/2007 The web graph - Anthony Bonato 57

58 Research Directions 1. Limits of graphs generated by models. graph searching and sweeping, graph homomorphisms, endo/automorphisms, adjacency properties. 2. Global theory for self-organizing networks. definitions, concentration results (eg via differential equations); node deletion models. 3. Dynamics in networks. modelling viruses and worms, network attacks, network congestion and routing. 4/27/2007 The web graph - Anthony Bonato 58

59 Preprints, reprints, contact: Google: Anthony Bonato 4/27/2007 The web graph - Anthony Bonato 59

60 new book by me: A course on the web graph due out later this year Fall 07 term: MA338 Graph Theory MWF 1:30 2:20 pm 4/27/2007 The web graph - Anthony Bonato 60

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