1. Miniconference on the Mathematics of Computation
Random Geometric Graphs
and Their Applications to Complex Networks
BIRS
Isomorphism results for infinite random geometric graphs
Anthony Bonato
Ryerson University

2. Infinite random geometric graphs
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Infinite random geometric graphs

3. Infinite random geometric graphs
Some properties
limit graph is countably infinite
every finite graph gets added eventually
infinitely often
even holds for countable graphs
add an exponential number of vertices at each time-step
also an on-line construction
Infinite random geometric graphs

4. Existentially closed (e.c.)
∀ A, B
example of an adjacency property a.a.s. true in G(n,p) ∃z
solution
Infinite random geometric graphs

5. Infinite random geometric graphs
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

6. Explicit construction
V = primes congruent to 1 (mod 4)
E: pq an edge if 𝑝 𝑞 =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

7. Infinite random graphs
G(N,1/2):
V = N
E: sample independently with probability ½
Theorem (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 𝜖 (0,1)
Infinite random geometric graphs

8. Infinite random geometric graphs
Proof sketch
with probability 1, any given adjacency problem has a solution
given A and B, a solution doesn’t exist with probability
(1− 𝑚+𝑛 ) 𝑁 =o 1
countable union of measure 0 sets is measure 0
Infinite random geometric graphs

9. Infinite random geometric graphs
Properties of R
diameter 2
universal
indestructible
indivisible
pigeonhole property
axiomatizes almost sure theory of graphs …
Infinite random geometric graphs

10. Infinite random geometric graphs
More on R
A. 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)
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)
Infinite random geometric graphs

11. 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

12. 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

13. Geometric existentially closed (g.e.c.)
∀ A, B
∀𝛿<1 ∃z 𝛿 1
∀ x
Infinite random geometric graphs

14. Properties following from g.e.c
locally R
vertex sets are dense
Infinite random geometric graphs

15. LARG is almost surely g.e.c.
geometric 1-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 geometric 1-graphs.
proof analogous to Erdős-Rényi result for R
geometric 1-graphs “look like” R in their unit balls, but can have diameter > 2
Infinite random geometric graphs

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

17. Infinite random geometric graphs
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
equivalent to:
int(x) = int(f(x))
frac(x) < frac(y) iff frac(f(x)) < frac((y))
Infinite random geometric graphs

18. Infinite random geometric graphs
Example: ℓ∞
V: dense countable set in R
E: LARG model
integer distance free (IDF) set
no element is integer
pairwise ℓ∞ distance non-integer
dense sets contain idf dense sets
“random” countable dense sets are idf
Infinite random geometric graphs

19. Infinite random geometric graphs
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

20. Infinite random geometric graphs
Sketch of proof for d = 1
back-and-forth
build 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 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

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

22. Infinite random geometric graphs
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 distance 3 apart
Theorem (BJ,15) ω3(GRd) = 2d.
if d ≠ d’, then GRd ≄ GRd’
Infinite random geometric graphs

23. Infinite random geometric graphs
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

24. Infinite random geometric graphs
Honeycomb metric
Theoerem (BJ,12) Almost all countable dense sets R2 with the honeycomb metric are strongly non-Rado.
Infinite random geometric graphs

25. Enter functional analysis
Miniconference on the Mathematics of Computation
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 spaces
Infinite random geometric graphs

26. Infinite random geometric graphs
ℓ∞d are special spaces
ℓ∞d are the only finite-dimensional normed spaces with Rado sets
interpretation:
ℓ∞d is the only space whose geometry is approximated by graph structure
Infinite random geometric graphs

27. Infinite random geometric graphs
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

28. 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

29. Infinite random geometric graphs
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

30. Infinite random geometric graphs
Heirarchy c c0
Banach-Mazur
C(X)
Infinite random geometric graphs

31. 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

32. Infinite random geometric graphs
Rado sets in c
Lemma (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 argument
Infinite random geometric graphs

33. Infinite random geometric graphs
Rado sets in c0
Lemma (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 measures
back-and-forth; i.o.p. acts to “extend collection of dimensions”
Infinite random geometric graphs

34. 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

35. Infinite random geometric graphs
Graph structure: c vs c0
Theorem (BJQ,16+)
The graphs G(c) and G(c0) are not isomorphic to any GRn.
G(c) and G(c0) are non-isomorphic.
follows by result of (Dilworth,99):
δ-surjective ε-isometries of Banach spaces are uniformly approximated by genuine isometries
If 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 graphs

36. Infinite random geometric graphs
Questions
almost 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 graphs

37. Contact Web: http://www.math.ryerson.ca/~abonato/
Blog:
@Anthony_Bonato
Zombies and Survivors

38. Infinite random geometric graphs
New book
Graph Searching Games and Probabilistic Methods (B,Pralat,17+)
Discrete Mathematics and its Applications Series, CRC Press
Infinite random geometric graphs

39. Ryerson University Toronto
CanaDAM 2017
Ryerson University Toronto