1. 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

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

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

4. 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

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
(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. 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

8. 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

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. 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

11. 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

12. 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

13. Random Semi-Directed Graphs- Anthony Bonato1 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. 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

16. 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

17. 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

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. 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

22. 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

23. 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

24. 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

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

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