Isomorphism results for - PowerPoint Presentation

Slide1

Isomorphism results for infinite random geometric graphs

Anthony BonatoRyerson University

Random Geometric Graphs

and

Their Applications to Complex

Networks

BIRSSlide2

RInfinite random geometric graphs

111

110

101

011

100

010

001

000Slide3
Some properties

limit graph is countably infiniteevery finite graph gets added eventuallyinfinitely ofteneven holds for countable graphsadd an exponential number of vertices at each time-step

also an on-line constructionInfinite random geometric graphsSlide4
Existentially closed (

e.c.)Infinite random geometric graphs

example of an

adjacency property

a.a.s

. true in

G(

n,p

)

solutionSlide5
Categoricity

e.c. captures R in a strong senseTheorem (Fraïssé,53) Any two countable

e.c. graphs are isomorphic.Proof: back-and-forth argument.

Infinite random geometric graphsSlide6
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 graphsSlide7
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.holds also for any fixed

p

Infinite random geometric graphsSlide8
Proof sketch

with probability 1, any given adjacency problem has a solutiongiven A and B, a solution doesn’t exist with probability

countable union of measure

0

sets is measure

0

Infinite random geometric graphsSlide9
Properties of

Rdiameter 2universalindestructibleindivisiblepigeonhole propertyaxiomatizes almost sure theory of graphs

…Infinite random geometric graphsSlide10
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 graphsSlide11
Graphs in normed spaces

fix a normed space: Seg: 1 ≤ p ≤ ∞;

ℓpd : Rd

with

L

p

-norm

p < ∞:

=

p =

∞

:

=

V

: set of points in

S

E

: adjacency determined by relative distance

Infinite random geometric graphsSlide12

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 graphsSlide13

Geometric existentially closed (g.e.c.)Infinite random geometric graphs

1

<1

 Slide14
Properties following from

g.e.clocally Rvertex sets are dense Infinite random geometric graphsSlide15
LARG is almost surely

g.e.c.geometric 1-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 geometric

1

-graphs.

proof analogous to Erdős-

Rényi result for Rgeometric 1

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

Infinite random geometric graphsSlide16

Geometrization lemmain some settings, graph distance approximates the space’s metric geometryLemma (BJ,11) If G = (V,E) is a geometric

1-graph and is convex, then

graph distance

integrally-approximates

metric distance

Infinite random geometric graphsSlide17
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 graphsSlide18
Example:

ℓ∞V: dense countable set in RE: LARG model

integer distance free (IDF) setno element is integerpairwise ℓ

∞

distance non-integer

dense sets contain

idf

dense sets“random” countable dense sets are

idfInfinite random geometric graphsSlide19
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 graphsSlide20
Sketch of proof for

d = 1back-and-forthbuild isomorphism from V = V(t) and W = W(t) to be a step-isomorphism

add v not in V, and go-forth (back similar)a = max{frac

(f(u)):

frac

(u) <

frac

(v)}

,

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

frac(u) > frac(v)}a < b

, as remainders distinct by idfwant 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 graphsSlide21
Properties of

GRdsymmetry:step-isometric isomorphisms of finite induced subgraphs extend to automorphismsindestructible

locally R, but infinite diameterInfinite random geometric graphsSlide22

Dimensionalityequilateral dimension D of normed space:maximum number of points equal distance

p = ∞: D = 2dpoints of hypercube

p = 1

:

Kusner’s

conjecture

:

D = 2d

proven only for d ≤ 4equilateral clique number of a graph

, ω3: max |A| so that

A has all vertices of distance 3 apartTheorem

(BJ,15) ω3(GR

d) = 2d.if d ≠ d’,

then GRd

GRd’

Infinite random geometric graphsSlide23
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

isomorphic

Corollary (BJ,11) All countable dense sets in ℓ2

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

Infinite random geometric graphsSlide24
Honeycomb metric

Theoerem (BJ,12) Almost all countable dense sets R2 with the honeycomb metric are

strongly non-Rado.Infinite random geometric graphsSlide25
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

∞

-decomposition

Mazur-

Ulam

theorem

properties of extreme points in normed spacesInfinite random geometric graphsSlide26

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 graphsSlide27
Questions

classify which countable dense sets are Rado in ℓ∞d same question, but for finite-dimensional normed spaceswhat about infinite dimensional

spaces?Infinite random geometric graphsSlide28
Classical

Banach spacesC(X): continuous function on a compact Hausdorff space X eg:

C[0,1]ℓ∞: bounded sequencesc: convergent sequences

c

0

: sequences convergent to

0

Infinite random geometric graphsSlide29

Separabilitya normed space is separable if it contains a countable dense setC[0,1], c, and c

0 are separable ℓ∞ and ω

1

are not separable

Infinite random geometric graphsSlide30

HeirarchyInfinite random geometric graphs

c

c

0

C(X)

Banach

-MazurSlide31
Graphs on sequence spaces

fix V a countable dense set in cLARG model defined analogously to the finite dimensional caseNB: countably infinite graph defined over infinite-dimensional space

Infinite random geometric graphsSlide32
Rado sets in

cLemma (BJ,Quas,16+): Almost all countable sets in

c are dense and idf.Theorem (BJQ,16+): Almost all countable sets in c

are Rado.

Ideas of proof

:

Lemma: construct

fully supported, non-aligned

measures

proof of Theorem somewhat analogous to ℓ∞

d more machinery to deal with the fractional parts of limits of images in back-and-forth argumentInfinite random geometric graphsSlide33
Rado sets in

c0Lemma (BJQ,16+): Almost all countable sets in

c0 are dense, i.d.f., and satisfy the i.o.p

.

Theorem

(BJQ,16+)

: Almost all countable

dense

in

c0 that are Rado.

Ideas of proof: work in ca

Lemma follows by existence of measuresback-and-forth; i.o.p. acts to “extend collection of dimensions”

Infinite random geometric graphsSlide34
Geometric structure:

c vs c0c vs c0 are isomorphic as vector spaces

not isometrically isomorphic:c contains extreme pointseg: (1,1,1,1, …)unit ball of

c

0

contains no extreme points

Infinite random geometric graphsSlide35
Graph structure:

c vs c0Theorem (BJQ,16+)The graphs

G(c) and G(c0) are not isomorphic to any GR

n

.

G(c)

and

G(c

0

) are non-isomorphic.follows by result of

(Dilworth,99):δ-surjective ε-isometries of Banach

spaces are uniformly approximated by genuine isometriesIf geometric 1-graphs on dense subsets in

Banach spaces X and Y give rise to isomorphic graphs, then there is a surjective isometry from

X to Y.Infinite random geometric graphsSlide36

Questionsalmost all countable sets in C[0,1] are Rado?if yes, then non-isomorphic to those in c, c0

?which normed spaces have Rado sets?program: interplay of graph structure and the geometry of Banach spaces?

Infinite random geometric graphsSlide37
Contact

Web: http://www.math.ryerson.ca/~abonato/Blog: https://anthonybonato.com/

@Anthony_Bonato https://www.facebook.com/anthony.bonato.5

Zombies and SurvivorsSlide38
New book

Graph Searching Games and Probabilistic Methods (B,Pralat,17+)Discrete Mathematics and its Applications Series, CRC Press

Infinite random geometric graphsSlide39

CanaDAM 2017Ryerson University Toronto