Universal Random Semi-Directed Graphs Miniconference on the Mathematics of Computation 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

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

Miniconference on the Mathematics of Computation The web graph nodes: web pages edges: links Random Semi-Directed Graphs- Anthony Bonato

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

Key property of the web graph: Power laws for some b > 1, where Ni,t is 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

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(loglog t) globally sparse, locally dense many bipartite subgraphs, sparse cuts, strong conductance, eigenvalue power law, … Random Semi-Directed Graphs- Anthony Bonato

Random Semi-Directed Graphs- Anthony Bonato 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

Random Semi-Directed Graphs- Anthony Bonato Preferential attachment (PA) model for 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+1 to existing nodes the edge vt+1 vs is added with probability Random Semi-Directed Graphs- Anthony Bonato

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

Random Semi-Directed Graphs- Anthony Bonato 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

Random Semi-Directed Graphs- Anthony Bonato 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

Random Semi-Directed Graphs- Anthony Bonato 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

Random Semi-Directed Graphs- Anthony Bonato 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

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

Random Semi-Directed Graphs- Anthony Bonato Rm,H fix R0 = H a finite digraph with m vertices suppose Rt is 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

Random Semi-Directed Graphs- Anthony Bonato 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

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

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

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

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

Random Semi-Directed Graphs- Anthony Bonato 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

Random Semi-Directed Graphs- Anthony Bonato 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

Random Semi-Directed Graphs- Anthony Bonato 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

Random Semi-Directed Graphs- Anthony Bonato 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

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

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