Universal Random Semi-Directed Graphs Anthony Bonato Wilfrid Laurier University Ryerson University...

Universal Random Semi-Directed Graphs

Anthony BonatoWilfrid Laurier University

Ryerson UniversityCanada

ROGICS’08May 14, 2008

Joint work with Dejan Delić and Changping Wang

Random Semi-Directed Graphs- Anthony Bonato

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Web graph


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The web graph

• nodes: web pages

• edges: links


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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)


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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

tiN bti

,

(Broder et al, 01)


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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, …


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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


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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+1 to existing nodes

– the edge vt+1 vs is added with probability

t

vsGtdeg


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• (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

.))1(1( 3, kot

N tk

Properties of the PA model

.loglog

log)1(1

t

to


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• 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


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Semi-directed graphs• the following assumptions are common to most models

of the web graph and complex networks1. on-line: nodes are added over a countable

sequence of discrete time-steps2. 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”


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• 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?


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• 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.

0 1 2 3 4 5 6


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The infinite random graph

• unique isomorphism type, R– infinite random graph, Rado graph– existentially closed (e.c.):

• R is the unique countable e.c. graph• Fraïssé: R is the unique universal homogeneous graph

A B

z


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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


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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


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m-e.c.

A B

z

• fix m > 0 an integer

• A and B finite sets of vertices, |A| = m


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Uniqueness and universalityTheorem (Bonato, Delić, Wang, 08)

A countable digraph G is isomorphic to Rm,H iffG is semi-directed with initial graph H, and satisfiesthe m-e.c. property.

• proved by a back-and-forth argument• corollary: each countable semi-directed digraph

embeds in Rm,H


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Age Dependent Process (ADP)


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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.


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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…


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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


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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


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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.


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• preprints, reprints, contact:

Google: “Anthony Bonato”


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Universal Random Semi-Directed Graphs Anthony Bonato Wilfrid Laurier University Ryerson University...

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