The New World of Infinite Random Geometric Graphs - PowerPoint Presentation

Slide1

The New World of Infinite Random Geometric Graphs

Anthony BonatoRyerson University

CRM-ISM Colloquium

Université

Laval

Slide2

Complex networks in the era of Big Dataweb graph, social networks, biological networks, internet networks, …

Infinite random geometric graphs - Anthony Bonato

Slide3
Hidden geometry

Infinite random geometric graphs - Anthony Bonatovs

Slide4

Blau spaceOSNs live in social space or Blau space: each user identified with a point in a multi-dimensional space

coordinates correspond to socio-demographic variables/attributeshomophily principle: the flow of information between users is a declining function of distance in Blau space

Infinite random geometric graphs - Anthony Bonato

Slide5

Random geometric graphsn nodes are randomly placed in the unit square

each node has a constant sphere of influence, radius rnodes are joined if their Euclidean distance is at most

r

Infinite random geometric graphs - Anthony Bonato

Slide6

Spatially Preferred Attachment (SPA) model(Aiello, Bonato, Cooper, Janssen, Prałat,08)

volume of sphere of influence proportional to in-degree

nodes are added and spheres of influence shrink over time

a.a.s

.

leads to power laws graphs, low directed diameter, and small separators

Infinite random geometric graphs - Anthony Bonato

Slide7

Into the infinite

Slide8

RInfinite random geometric graphs - Anthony Bonato

111

110

101

011

100

010

001

000

Slide9
Properties of

Rlimit graph is countably infinite

every finite graph gets added eventuallyinfinitely oftenholds also for countable graphsadd an exponential number of vertices at each time-step

Infinite random geometric graphs - Anthony Bonato

Slide10
Existentially closed (

e.c.)Infinite random geometric graphs - Anthony Bonato

example of an

adjacency property

finite

 

 

solution

Slide11
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 graphs - Anthony Bonato

Slide12
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 graphs - Anthony Bonato

Slide13
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 graphs - Anthony Bonato

Slide14
Proof sketch

show that 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.

NB: proof works for

p

(0,1

)

.

 

Infinite random geometric graphs - Anthony Bonato

Slide15
Properties of

Rdiameter 2universalindestructibleindivisiblepigeonhole property

axiomatizes almost sure theory of graphs…Infinite random geometric graphs - Anthony Bonato

Slide16
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 graphs - Anthony Bonato

Slide17

And now for something completely different

Slide18
Graphs in normed spaces

fix a normed space: Seg

: 1 ≤ p ≤ ∞; ℓpd :

R

d

with

L

p

-norm

p < ∞:

=

p = ∞:

=

V

: set of points in

S

E

: adjacency determined by relative distance

 

Infinite random geometric graphs - Anthony Bonato

Slide19
Aside: unit balls in

ℓp spacesInfinite random geometric graphs - Anthony Bonato

balls converge to square as p → ∞

Slide20
Random geometric graphs

Infinite random geometric graphs - Anthony Bonato

Slide21

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 graphs - Anthony Bonato

Slide22

Geometric existentially closed (g.e.c.)

Infinite random geometric graphs - Anthony Bonato

 

 

1

 

<1

 

 

Slide23
Properties following from

g.e.clocally Rvertex sets are dense

Infinite random geometric graphs - Anthony Bonato

Slide24
LARG

graphs almost surely g.e.c.

1-geometric graph: g.e.c. and 1-threshold: adjacency only may occur if distance < 1

Theorem

(BJ,11)

With probability

1

, and for any fixed

p

, LARG generates 1-geometric graphs.proof analogous to Erdős-Rényi

result for

R

1

-geometric graphs “look like”

R

in their unit balls, but can have diameter

> 2

Infinite random geometric graphs - Anthony Bonato

Slide25
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 graphs - Anthony Bonato

Slide26
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

in

R

equivalent

to:

int

(x) =

int

(f(x))

frac

(x) <

frac

(y)

iff

frac

(f(x)) <

frac

((y))

 

Infinite random geometric graphs - Anthony Bonato

Slide27
Example:

ℓ∞V: dense countable set in

RE: LARG modelinteger distance free (IDF) set

pairwise

ℓ

∞

distance non-integer

dense sets contain

idf dense sets

“random” countable dense sets are idfInfinite random geometric graphs - Anthony Bonato

Slide28
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

GRd

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

Infinite random geometric graphs - Anthony Bonato

Slide29
Sketch of proof for

d = 1back-and-forthbuild

partial isomorphism from V = V(t) and W = W(t) to be a step-isomorphism via inductionadd v not in V, and

go-forth

(

back

similar)

a = max{

frac

(f(u)): u V,

frac

(u) <

frac

(v)}

,

b = min{

frac

(f(u)): u

V,

frac

(u) >

frac

(v)}

a < b

, as

fractional parts

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 graphs - Anthony Bonato

Slide30

The new world

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Properties of

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

automorphismsindestructiblelocally R, but infinite diameterInfinite random geometric graphs - Anthony Bonato

Slide32

Dimensionalityequilateral dimension

D of normed space:maximum number of points equal distancep = ∞: D = 2dpoints of hypercubep = 1

:

Kusner’s

conjecture

:

D = 2d

proven only for

d ≤ 4equilateral clique number of a graph, ω

3

:

max

|A|

so that

A

has all vertices of graph distance

3

apart

Theorem

(BJ,15)

ω

3

(

GR

d

) = 2

d

.

if

d ≠ d’,

then

GR

d

GR

d’

 

Infinite random geometric graphs - Anthony Bonato

Slide33
Euclidean distance

Lemma (BJ,11) In ℓ

22, 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 ℓ22

are strongly non-Rado.

non-trivial proof, but ad hoc

Infinite random geometric graphs - Anthony Bonato

Slide34
Honeycomb metric

Theorem (BJ,12) Almost all countable dense sets R2 with the

honeycomb metric are strongly non-Rado.Infinite random geometric graphs - Anthony Bonato

Slide35
Enter functional analysis

(Balister,Bollobás,Gunderson,Leader,Walters,17+) 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 graphs - Anthony Bonato

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ℓ∞d are special spacesℓ

∞d are the only finite-dimensional normed spaces where almost all countable sets are Radointerpretation:ℓ

∞

d

is the only space whose geometry is approximated by graph structure

Infinite random geometric graphs - Anthony Bonato

Slide37
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 graphs - Anthony Bonato

Slide38

Infinitely many parallel universes

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Classical

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

X eg: C[0,1]ℓ∞ bounded sequences

c

: convergent sequences

c

0

: sequences convergent to

0

Infinite random geometric graphs - Anthony Bonato

Slide40
Separability

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

, c, and c0 are separable 

ℓ

∞

and

ω

1

are not separable

Infinite random geometric graphs - Anthony Bonato

Slide41

HeirarchyInfinite random geometric graphs - Anthony Bonato

c

c

0

C(X)

Banach

-Mazur

Slide42
Graphs on sequence spaces

fix V a countable dense set in cLARG model defined analogously to the finite dimensional case

NB: countably infinite graph defined over infinite-dimensional spaceInfinite random geometric graphs - Anthony Bonato

Slide43
Rado sets in

cLemma (BJ,Quas,17+): Almost all countable sets in

c are dense and idf.Theorem (BJQ,17+): Almost all countable sets in c are Rado.

Ideas of proof

:

Lemma: functional analysis

Proof of Theorem somewhat analogous to

ℓ

∞

d more machinery to deal with the fractional parts of limits of images in back-and-forth argument

Infinite random geometric graphs - Anthony Bonato

Slide44
Rado sets in

c0Theorem (BJ,17+): There exist countable dense in

c0 that are Rado.idea: consider the subspace of sequences which are eventually 0almost all countable sets in this subspace are dense and

idf

Infinite random geometric graphs - Anthony Bonato

Slide45

The curious geometry of sequence spaces

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Geometric structure:

c vs c0

c vs c0 are isomorphic as vector spacesnot isometrically isomorphic:c contains extreme points

eg

: (1,1,1,1, …)

unit ball of

c

0

contains no extreme points

Infinite random geometric graphs - Anthony Bonato

Slide47
Graph structure:

c vs c0Theorem

(BJ,17+)The graphs G(c) and G(c0) are not isomorphic to any GR

d

.

G(c)

and

G(c

0

) are non-isomorphic.in

G(c

0

)

,

N

≤3

(x)

contains:

Infinite random geometric graphs - Anthony Bonato

3

3

6

Slide48

Interpolating the space from the graph

Theorem (BJQ,17+) Suppose V and W are Banach spaces with dense sets

X

and

Y

. If

G

and H

are the 1-geometric graphs on X and

Y

(

resp

) and are isomorphic, then there is a surjective isometry from

V

to

W

.

hidden geometry: if we know LARG graphs almost surely, then we can recover the

Banach

space!

Idea

- use Dilworth’s theorem:

δ

-surjective

ε

-isometries

of

Banach

spaces are uniformly approximated by genuine isometries

Infinite random geometric graphs - Anthony Bonato

Slide49

Continuous

functions

Slide50

Dense sets in C[0,1](AJQ,17+)

piecewise linear functions and polynomials almost all sets are smoothly dense

Brownian motion path functions

almost all sets are

IC-dense

Infinite random geometric graphs - Anthony Bonato

Slide51
Isomorphism in C[0,1]

Theorem (AJQ,17+)

Smoothly dense sets give rise to a unique isotype of LARG graphs: GR(SD).Almost surely IC-sets give rise to a unique isotype of LARG graphs:

GR(ICD

)

.

Infinite random geometric graphs - Anthony Bonato

Slide52

Non-isomorphism

Theorem (AJQ,17+) The graphs GR(SD) and GR(ICD) are non-isomorphic.Idea

: Dilworth’s

theorem and

Banach

-Stone

theorem: isometries

on C[0,1]

induce homeomorphisms on [0,1]

Infinite random geometric graphs - Anthony Bonato

Slide53
Questions

“almost all” countable sets in C[0,1] are Rado?

need a suitable measure of random continuous functionwhich Banach spaces have Rado sets?program: interplay of graph structure and the geometry of Banach

spaces

Infinite random geometric graphs - Anthony Bonato

Slide54
Contact

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

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

Slide55

Merci!