1. Miniconference on the Mathematics of Computation
2018 CMS Winter Meeting
The New World of Infinite Random Geometric Graphs
Anthony Bonato
Ryerson University

2. Happy Birthday, Robert!

3. Graphs in normed spaces
fix a normed space: S
eg: 1 ≤ p ≤ ∞; ℓpd : Rd with Lp-norm
p < ∞: 𝑥 𝑝 = 𝑛 𝑥 𝑛 𝑝 1/𝑝
p = ∞: 𝑥 𝑝 = max 𝑛 𝑥 𝑛
V: set of points in S
E: adjacency determined by relative distance
Infinite random geometric graphs - Anthony Bonato

4. Random geometric graphs
Infinite random geometric graphs - Anthony Bonato

5. Local Area Random Graph (LARG) model
parameters:
p in (0,1)
a normed space S
V: a countable set in S
E: if || u – v || < 1, then uv is an edge with probability p
Infinite random geometric graphs - Anthony Bonato

6. Geometric existentially closed (g.e.c.)
∀ A, B finite
∀𝛿<1 ∃z 𝛿 1
∀ x
Infinite random geometric graphs - Anthony Bonato

7. 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 the Rado graph R
1-geometric graphs “look like” R in their unit balls, but can have diameter > 2
Infinite random geometric graphs - Anthony Bonato

8. Infinite random geometric graphs - Anthony Bonato
Geometrization lemma
in some settings, graph distance approximates the space’s metric geometry
Lemma (BJ,11) If G = (V,E) is a 1-geometric graph and V is dense, then
𝑑 𝐺 𝑥,𝑦 = 𝑑 𝑥,𝑦 +1.
graph distance integrally-approximates metric distance
Infinite random geometric graphs - Anthony Bonato

9. Infinite random geometric graphs - Anthony Bonato
Step-isometries
S and T normed spaces, f: S → T is a step-isometry if
𝑑 𝑆 𝑥,𝑦 = 𝑑 𝑇 𝑓(𝑥),𝑓(𝑦)
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

10. Infinite random geometric graphs - Anthony Bonato
Example: ℓ∞
V: dense countable set in R
E: LARG model
integer distance free (IDF) set
pairwise ℓ∞ distance non-integer
dense sets contain idf dense sets
“random” countable dense sets are idf
Infinite random geometric graphs - Anthony Bonato

11. Infinite random geometric graphs - Anthony Bonato
Categoricity
countable V is Rado if the LARG graphs on it are isomorphic with probability 1
Theorem (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

12. Infinite random geometric graphs - Anthony Bonato
Sketch of proof for d = 1
back-and-forth
build partial isomorphism from V = V(t) and W = W(t) to be a step-isomorphism via induction
add 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 𝐼∩𝑊 (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

13. The new world

14. Infinite random geometric graphs - Anthony Bonato
Properties of GRd
symmetry:
step-isometric isomorphisms of finite induced subgraphs extend to automorphisms
indestructible
locally R, but infinite diameter
Infinite random geometric graphs - Anthony Bonato

15. Infinite random geometric graphs - Anthony Bonato
Dimensionality
equilateral dimension D of normed space:
maximum number of points equal distance
p = ∞: D = 2d
points of hypercube
p = 1: Kusner’s conjecture: D = 2d
proven only for d ≤ 4
equilateral clique number of a graph, ω3:
max |A| so that A has all vertices of graph distance 3 apart
Theorem (BJ,15) ω3(GRd) = 2d.
if d ≠ d’, then GRd ≄ GRd’
Infinite random geometric graphs - Anthony Bonato

16. Infinite random geometric graphs - Anthony Bonato
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 ℓ22 are strongly non-Rado.
non-trivial proof, but ad hoc
Infinite random geometric graphs - Anthony Bonato

17. Infinite random geometric graphs - Anthony Bonato
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

18. Enter functional analysis
Miniconference on the Mathematics of Computation
Enter functional analysis
(Balister,Bollobás,Gunderson,Leader,Walters,18)
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 spaces
Infinite random geometric graphs - Anthony Bonato

19. Infinite random geometric graphs - Anthony Bonato
ℓ∞d are special spaces
ℓ∞d are the only finite-dimensional normed spaces where almost all countable sets are Rado
interpretation:
ℓ∞d is the only space whose geometry is approximated by graph structure
Infinite random geometric graphs - Anthony Bonato

20. Infinite random geometric graphs - Anthony Bonato
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

21. Infinitely many parallel universes

22. Classical Banach spaces
C(X): continuous function on a compact Hausdorff space X
eg: C[0,1]
ℓ∞ bounded sequences
c: convergent sequences
c0: sequences convergent to 0
Infinite random geometric graphs - Anthony Bonato

23. Infinite random geometric graphs - Anthony Bonato
Separability
a normed space is separable if it contains a countable dense set
C[0,1], c, and c0 are separable
 ℓ∞ and ω1 are not separable
Infinite random geometric graphs - Anthony Bonato

24. Infinite random geometric graphs - Anthony Bonato
Hierarchy c c0
Banach-Mazur
C(X)
Infinite random geometric graphs - Anthony Bonato

25. Graphs on sequence spaces
fix V a countable dense set in c
LARG model defined analogously to the finite dimensional case
NB: countably infinite graph defined over infinite-dimensional space
Infinite random geometric graphs - Anthony Bonato

26. Infinite random geometric graphs - Anthony Bonato
Rado sets in c and c0
Theorem (BJ,Quas,18+): Almost all countable sets in c are Rado.
Theorem (BJQ,18+): Almost all countable sets in c are Rado.
Ideas of proof:
almost all sets are dense and idf
like ℓ∞d, but more machinery to deal with the fractional parts of limits of images in back-and-forth argument
Infinite random geometric graphs - Anthony Bonato

27. The curious geometry of sequence spaces

28. Geometric structure: c vs c0
c vs c0 are isomorphic as vector spaces
not isometrically isomorphic:
c contains extreme points
eg: (1,1,1,1, …)
unit ball of c0 contains no extreme points
Infinite random geometric graphs - Anthony Bonato

29. Interpolating the space from the graph
Theorem (BJQ,18+) 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

30. Infinite random geometric graphs - Anthony Bonato
Graph structure: c vs c0
Theorem (BJQ,18+)
The graphs G(c) and G(c0) are not isomorphic to any GRd.
G(c) and G(c0) are non-isomorphic.
Infinite random geometric graphs - Anthony Bonato

31. Continuous functions

32. Dense sets in C[0,1] (BJQ,18+)
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

33. Infinite random geometric graphs - Anthony Bonato
Isomorphism in C[0,1]
Theorem (BJQ,18+)
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

34. Infinite random geometric graphs - Anthony Bonato
Non-isomorphism
Theorem (BJQ,18+) 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

35. Infinite random geometric graphs - Anthony Bonato
Questions
“almost all” countable sets in C[0,1] are Rado?
need a suitable measure of random continuous function
explicit representations? automorphism groups?
which Banach spaces have Rado sets?
program: interplay of graph structure and the geometry of Banach spaces
Infinite random geometric graphs - Anthony Bonato

36. Contact Web: http://www.math.ryerson.ca/~abonato/
Blog:
@Anthony_Bonato