Post on 15-Jan-2016
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
2
Web graph
Random Semi-Directed Graphs- Anthony Bonato
3
The web graph
• nodes: web pages
• edges: links
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 billion static 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,t is the number of nodes of (in- out-) degree i in a graph of order t
tiN bti
,
(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(loglog t)
– 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+1 to existing nodes
– the edge vt+1 vs is added with probability
t
vsGtdeg
Random Semi-Directed Graphs- Anthony Bonato
9
• (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
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 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”
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
• 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
Random Semi-Directed Graphs- Anthony Bonato
14
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
Random Semi-Directed Graphs- Anthony Bonato
15
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
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.
A B
z
• fix m > 0 an integer
• A and B finite sets of vertices, |A| = m
Random Semi-Directed Graphs- Anthony Bonato
18
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
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