The New World of Infinite Random Geometric Graphs - PowerPoint Presentation

Slide1

The New World of Infinite Random Geometric Graphs

Anthony BonatoRyerson University

East Coast Combinatorics ConferenceSlide2

co-author

talk

post-docSlide3

Into the infiniteSlide4

RInfinite random geometric graphs

111

110

101

011

100

010

001

000Slide5
Some properties

limit graph is countably infiniteevery finite graph gets added eventually

infinitely ofteneven holds for countable graphsadd an exponential number of vertices at each time-stepalso an on-line construction

Infinite random geometric graphsSlide6
Existentially closed (

e.c.)Infinite random geometric graphs

example of an

adjacency property

a.a.s

. true in

G(

n,p

)

solutionSlide7
Categoricity

e.c. captures R in a strong sense

Theorem (Fraïssé,53) Any two countable e.c. graphs are isomorphic.Proof:

back-and-forth argument.

Infinite random geometric graphsSlide8
Explicit construction

V = primes congruent to 1 (mod 4)

E: pq an edge if

=1

undirected by quadratic reciprocity

solutions to adjacency problems exist by:

Chinese remainder theorem

Dirichlet’s

theorem on primes in arithmetic progression

Infinite random geometric graphsSlide9
Infinite random graphs

G(N,1/2):

V = NE: sample independently with probability ½Theorem

(Erdős,Rényi,63)

With probability

1

, two graphs sampled from

G(N,1/2)

are e.c., and so isomorphic to R.

Infinite random geometric graphsSlide10
Proof sketch

with probability 1, any given adjacency problem has a solution

given A and B, a solution doesn’t exist with probability

countable union of measure

0

sets is measure

0

Infinite random geometric graphsSlide11
Properties of

Rdiameter 2universal

indestructibleindivisiblepigeonhole propertyaxiomatizes almost sure theory of graphs…Infinite random geometric graphsSlide12
More on

RA. Bonato, A Course on the Web Graph, AMS, 2008.

P.J. Cameron, The random graph, In: Algorithms and Combinatorics 14 (R.L. Graham and J. Nešetřil, eds.), Springer Verlag, New York (1997) 333-351

.

P.J. Cameron, The random graph revisited, In:

European Congress of Mathematics

Vol. I (C.

Casacuberta

, R. M. Miró-Roig, J. Verdera and S. Xambó-Descamps, eds.),

Birkhauser, Basel (2001) 267-274.Infinite random geometric graphsSlide13

And now for something completely differentSlide14
Graphs in normed spaces

fix a normed space: S

eg: 1 ≤ p ≤ ∞; ℓpd

:

R

d

with

L

p-normp < ∞:

=

p =

∞

:

=

V

: set of points in

S

E

: adjacency determined by relative distance

Infinite random geometric graphsSlide15
Aside: unit balls in

ℓp spaces

Infinite random geometric graphsballs converge to square as p → ∞Slide16
Random geometric graphs

Infinite random geometric graphsSlide17

Local Area Random Graph (LARG) modelparameters: p in

(0,1)a normed space SV: a countable set in SE: if || u – v || < 1

, then

uv

is an edge with probability

p

Infinite random geometric graphsSlide18

Geometric existentially closed (g.e.c.)

Infinite random geometric graphs

1

<1

 Slide19
Properties following from

g.e.clocally Rvertex sets are dense

Infinite random geometric graphsSlide20
LARG is almost surely

g.e.c.1-geometric graph: g.e.c. and

1-threshold: adjacency only may occur if distance < 1Theorem (BJ,11)

With probability

1

, and for any fixed p, LARG generates

1

-geometric graphs.

proof analogous to Erdős-Rényi result for R1

-geometric graphs “look like” R in their unit balls, but can have diameter > 2

Infinite random geometric graphsSlide21

Geometrization lemmain some settings, graph distance approximates the space’s metric geometry

Lemma (BJ,11) If G = (V,E) is a 1-geometric graph and

is convex, then

graph distance

integrally-approximates

metric distance

Infinite random geometric graphsSlide22
Step-isometries

S and T

normed spaces, f: S → T is a step isometry if

restriction

of notion of

isometry

remove floors

captures integer distances only

equivalent to:

int

(x) =

int

(f(x))

frac

(x) <

frac

(y) iff frac(f(x))

< frac((y))

Infinite random geometric graphsSlide23
Example:

ℓ∞V: dense countable set in

RE: LARG modelinteger distance free (IDF) set

pairwise

ℓ

∞

distance non-integerdense sets contain

idf dense sets“random” countable dense sets are idf

Infinite random geometric graphsSlide24
Categoricity

countable V is Rado if the LARG graphs on it are isomorphic with probability

1Theorem (BJ,11) Dense idf sets in ℓ

∞

d

are Rado for all

d > 0.new class of infinite graphs GR

d which are unique limit objects of random graph processes in normed spaces

Infinite random geometric graphsSlide25
Sketch of proof for

d = 1back-and-forth

build isomorphism from V = V(t) and W = W(t) to be a step-isomorphismadd v not in V, and go-forth (

back

similar)

a = max{

frac

(f(u)): r(u) < r(v)}

, b = min{frac(f(u)):

r(u) > r(v)}a < b, as remainders distinct by idf

want f(v) to satisfy:int(f(v)) =

int(v)frac(f(v))

[

a,b)I = (int

(v) + a, int(v) + b)choose vertex in

(using density)will maintain step-isometry in (IS)use

g.e.c to find f(v) in co-domain correctly joined to W.

Infinite random geometric graphsSlide26

The new worldInfinite random geometric graphsSlide27
Properties of

GRdsymmetry:step-isometric isomorphisms

of finite induced subgraphs extend to automorphismsindestructiblelocally R, but infinite diameter

Infinite random geometric graphsSlide28

Dimensionalitye

quilateral dimension D of normed space:maximum number of points equal distancep = ∞: D = 2d

points of hypercube

p = 1

:

Kusner’s

conjecture

: D = 2dproven only for d ≤ 4

equilateral clique number of a graph, ω3:

max |A| so that A has all vertices of distance 3

apartTheorem (BJ,15) ω3

(GRd) = 2d.

if d ≠ d’, then GRd

GRd’

Infinite random geometric graphsSlide29
Euclidean distance

Lemma (BJ,11) In

ℓ22, every step-isometry is an isometry.countable dense V is

strongly

non-Rado

if any two

such LARG

graphs on

V are with probability 1 not isomorphicCorollary

(BJ,11) All countable dense sets in ℓ22

are strongly non-Rado.non-trivial proof, but ad hoc

Infinite random geometric graphsSlide30
Honeycomb metric

Theoerem (BJ,12) Almost all countable dense sets

R2 with the honeycomb metric are strongly non-Rado.Infinite random geometric graphsSlide31
Enter functional analysis

(Balister,Bollobás,Gunderson,Leader,Walters,16+) Let

S be finite-dimensional normed space not isometric to ℓ∞d . Then almost all countable dense sets in

S

are strongly non-Rado.

proof uses functional analytic tools:

ℓ

∞

-decompositionMazur-Ulam theoremproperties of extreme points in normed spaces

Infinite random geometric graphsSlide32

ℓ∞d are special spacesℓ

∞d are the only finite-dimensional normed spaces with Rado setsinterpretation:ℓ

∞

d

is the only space whose geometry is approximated by graph structure

Infinite random geometric graphsSlide33
Questions

classify which countable dense sets are Rado in ℓ∞d

same question, but for finite-dimensional normed spaces.what about infinite dimensional spaces?Infinite random geometric graphsSlide34

CanaDAM 2017Ryerson University TorontoSlide35
Math PhD program at Ryerson

starts Fall 2016emphasis on modelling, applicationsdiscrete math/graph theory is a fieldtaking applications in October for Fall 2017

http://math.ryerson.ca/graduate/overview-phd.htmlZombies and Survivors