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Universal Random Semi-Directed Graphs

ROGICS’08 May 14, 2008. Universal Random Semi-Directed Graphs. Anthony Bonato Wilfrid Laurier University Ryerson University Canada. Joint work with Dejan Delić and Changping Wang. Web graph. nodes : web pages edges : links. The web graph. How big is the web?.

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Universal Random Semi-Directed Graphs

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  1. ROGICS’08 May 14, 2008 Universal Random Semi-Directed Graphs Anthony Bonato Wilfrid Laurier University Ryerson University Canada Joint work with Dejan Delić and Changping Wang

  2. Web graph Random Semi-Directed Graphs- Anthony Bonato

  3. nodes: web pages edges: links The web graph Random Semi-Directed Graphs- Anthony Bonato

  4. How big is the web? • the web is infinite… • calendars, online organizers • random strings: • google “raingod random strings” • total web ≈ 54 billionstatic pages (Hirate, Kato, Yamana, 07) Random Semi-Directed Graphs- Anthony Bonato

  5. Key property of the web graph: Power laws for some b > 1, where Ni,tis the number of nodes of (in- out-) degree i in a graph of order t (Broder et al, 01) Random Semi-Directed Graphs- Anthony Bonato

  6. Other properties of the web graph • small world property(Watts, Strogatz, 98): • in a graph of order t, diameter O(log t), average distance: O(loglogt) • globally sparse, locally dense • many bipartite subgraphs, sparse cuts, strong conductance, eigenvalue power law, … Random Semi-Directed Graphs- Anthony Bonato

  7. Complex networks • graphs with these properties (power law, small world,…) are now called complex networks • examples of complex networks arise also in the social and biological sciences Facebook graph Random Semi-Directed Graphs- Anthony Bonato

  8. Preferential attachment (PA) modelfor complex networks(Barabási, Albert, 99), (Bollobás,Riordan,Spencer,Tusnady,01) • parameter: m a positive integer • at time 0, add a single directed edge • at time t+1, add m directed edges from a new node vt+1to existing nodes • the edge vt+1 vsis added with probability Random Semi-Directed Graphs- Anthony Bonato

  9. Properties of the PA model • (BRST,01) For integers m > 0, a.a.s. (that is, with probability tending to 1 as t→∞) for all k satisfying 0 ≤ k ≤ t1/15 • (Bollobás, Riordan, 04) For integers m > 0, a.a.s. the diameter of the graph at time t is Random Semi-Directed Graphs- Anthony Bonato

  10. several web graph models introduced and rigorously analyzed • Bollobás, Chung, Frieze, Kleinberg, Luczak,… • in most models, nodes are born joined to an m-set of vertices satisfying some properties • high degree • in a neighbour set • older nodes Random Semi-Directed Graphs- Anthony Bonato

  11. Semi-directed graphs • the following assumptions are common to most models of the web graph and complex networks • on-line: nodes are added over a countable sequence of discrete time-steps • constant out-degree: new vertices point only to existing ones, and for a fixed integer m > 0, there are exactly m such directed edges • a digraph satisfying 1) and 2) is called semi-directed • name recently coined by Bollobás • emphasizes that orientation arises according to time: “new point to old” Random Semi-Directed Graphs- Anthony Bonato

  12. semi-directed graphs lead naturally to • countably infinite limits: • unions of chains of finite semi-directed graphs • are the limits unique? • do the limits naturally arise from a random • graph process? • what properties do the limits satisfy? Random Semi-Directed Graphs- Anthony Bonato

  13. 0 1 2 3 4 5 6 • toss a coin to generate edges on the • nonnegative integers: G(N,p) • Theorem (Erdős,Rényi, 63): With probability 1, any two graphs sampled from G(N,p) are isomorphic. Random Semi-Directed Graphs- Anthony Bonato

  14. The infinite random graph • unique isomorphism type, R • infinite random graph, Rado graph • existentially closed (e.c.): A B z • R is the unique countable e.c. graph • Fraïssé:R is the unique universal homogeneous graph Random Semi-Directed Graphs- Anthony Bonato

  15. Rm,H • fix R0 = H a finite digraph with m vertices • suppose Rtis defined • to form Rt+1, for each m-set S in Rt, add a vertex zs joined to each vertex of S and to no other vertices of Rt • the limit graph is Rm,H Rt S zs Random Semi-Directed Graphs- Anthony Bonato

  16. Properties of Rm,H • acyclic; constant out-degree m, sensitive to H • unlike R, Rm,H is not inexhaustible: • deleting vertices changes constant out-degree S zs Random Semi-Directed Graphs- Anthony Bonato

  17. m-e.c. • fix m > 0 an integer • A and B finite sets of vertices, |A| = m B A z Random Semi-Directed Graphs- Anthony Bonato

  18. Uniqueness and universality Theorem (Bonato, Delić, Wang, 08) A countable digraph G is isomorphic to Rm,H iff G is semi-directed with initial graph H, and satisfies the m-e.c. property. • proved by a back-and-forth argument • corollary: each countable semi-directed digraph embeds in Rm,H Random Semi-Directed Graphs- Anthony Bonato

  19. Age Dependent Process (ADP) Random Semi-Directed Graphs- Anthony Bonato

  20. Universal random semi-directed graphs Theorem (BDW, 08) With probability 1, a countable digraph generated by ADP with parameters m and H is isomorphic to Rm,H. Random Semi-Directed Graphs- Anthony Bonato

  21. Generalization • theory may be generalized so that the isotypes induced by out-neighbour sets are in a specified infinite hereditary class of finite digraphs: • all digraphs • tournaments; linear orders • digraphs with bounded in-degree… Random Semi-Directed Graphs- Anthony Bonato

  22. Group of R • R is homogeneous(eg vertex- and edge-transitive) • R has a rich automorphism group (see P.Cameron’s surveys) • cardinality and is simple • cyclic automorphisms • strong small index property • embeds all countable groups Random Semi-Directed Graphs- Anthony Bonato

  23. Group of Rm,H • Rm,H is not vertex-transitive Theorem (BDW, 08)Aut(Rm,H) embeds all countable groups. • implies that Aut(Rm,H): • generates the variety of all groups • has undecidable universal theory Random Semi-Directed Graphs- Anthony Bonato

  24. Future research • further investigate the automorphism group and endomorphism monoid of Rm,H • distinguishing number is 2 • consider limits of other recent models of complex networks • (Kleinberg, Kleinberg, 05): limits of PA model • (Bonato, Janssen, 04/08): limits of copying model… • geometric models? Chung, Frieze, Bonato et al. Random Semi-Directed Graphs- Anthony Bonato

  25. preprints, reprints, contact: Google: “Anthony Bonato” Random Semi-Directed Graphs- Anthony Bonato

  26. New book Random Semi-Directed Graphs- Anthony Bonato

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