1. Miniconference on the Mathematics of Computation
CRM-ISM Colloquium
Université Laval
The New World of Infinite Random Geometric Graphs
Anthony Bonato
Ryerson University

2. Complex networks in the era of Big Data
web graph, social networks, biological networks, internet networks, …
Infinite random geometric graphs - Anthony Bonato

3. Infinite random geometric graphs - Anthony Bonato
Hidden geometry vs
Infinite random geometric graphs - Anthony Bonato

4. Infinite random geometric graphs - Anthony Bonato
Blau space
OSNs live in social space or Blau space:
each user identified with a point in a multi-dimensional space
coordinates correspond to socio-demographic variables/attributes
homophily principle: the flow of information between users is a declining function of distance in Blau space
Infinite random geometric graphs - Anthony Bonato

5. Random geometric graphs
n nodes are randomly placed in the unit square
each node has a constant sphere of influence, radius r
nodes are joined if their Euclidean distance is at most r
Infinite random geometric graphs - Anthony Bonato

6. Infinite random geometric graphs - Anthony Bonato
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

7. Miniconference on the Mathematics of Computation
Into the infinite

8. Miniconference on the Mathematics of Computation
111
110
101
011
100
010
001
000
Infinite random geometric graphs - Anthony Bonato

9. Miniconference on the Mathematics of Computation
Properties of R
limit graph is countably infinite
every finite graph gets added eventually
infinitely often
holds also for countable graphs
add an exponential number of vertices at each time-step
Infinite random geometric graphs - Anthony Bonato

10. Existentially closed (e.c.)
Miniconference on the Mathematics of Computation
Existentially closed (e.c.)
∀ A, B finite
example of an adjacency property ∃z
solution
Infinite random geometric graphs - Anthony Bonato

11. Miniconference on the Mathematics of Computation
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

12. Explicit construction
Miniconference on the Mathematics of Computation
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 - Anthony Bonato

13. Infinite random graphs
Miniconference on the Mathematics of Computation
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.
Infinite random geometric graphs - Anthony Bonato

14. Miniconference on the Mathematics of Computation
Proof sketch
show that with probability 1, any given adjacency problem has a solution
given A and B, a solution doesn’t exist with probability
(1− 𝑎+𝑏 ) 𝑁 =o 1
countable union of measure 0 sets is measure 0.
NB: proof works for p ∈ (0,1).
Infinite random geometric graphs - Anthony Bonato

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

16. Infinite random geometric graphs - Anthony Bonato
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 - Anthony Bonato

17. And now for something completely different

18. 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

19. Aside: unit balls in ℓp spaces
balls converge to square as p → ∞
Infinite random geometric graphs - Anthony Bonato

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

21. 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

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

23. Properties following from g.e.c
locally R
vertex sets are dense
Infinite random geometric graphs - Anthony Bonato

24. 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

25. 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 convex, then
𝑑 𝐺 𝑥,𝑦 = 𝑑 𝑥,𝑦 +1.
graph distance integrally-approximates metric distance
Infinite random geometric graphs - Anthony Bonato

26. 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

27. 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

28. 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

29. 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

30. The new world

31. 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

32. 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

33. 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

34. 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

35. Enter functional analysis
Miniconference on the Mathematics of Computation
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 spaces
Infinite random geometric graphs - Anthony Bonato

36. 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

37. 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

38. Infinitely many parallel universes

39. 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

40. 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

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

42. 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

43. Infinite random geometric graphs - Anthony Bonato
Rado sets in c
Lemma (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

44. Infinite random geometric graphs - Anthony Bonato
Rado sets in c0
Theorem (BJ,17+): There exist countable dense in c0 that are Rado.
idea: consider the subspace of sequences which are eventually 0
almost all countable sets in this subspace are dense and idf
Infinite random geometric graphs - Anthony Bonato

45. The curious geometry of sequence spaces

46. 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

47. Infinite random geometric graphs - Anthony Bonato
Graph structure: c vs c0
Theorem (BJ,17+)
The graphs G(c) and G(c0) are not isomorphic to any GRd.
G(c) and G(c0) are non-isomorphic.
in G(c0), N≤3(x) contains: 3 3 6
Infinite random geometric graphs - Anthony Bonato

48. 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

49. Continuous functions

50. 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

51. Infinite random geometric graphs - Anthony Bonato
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

52. Infinite random geometric graphs - Anthony Bonato
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

53. 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
which Banach spaces have Rado sets?
program: interplay of graph structure and the geometry of Banach spaces
Infinite random geometric graphs - Anthony Bonato

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

55. Merci!