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

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

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

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

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

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

Miniconference on the Mathematics of Computation Into the infinite

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

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

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

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

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

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

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− 1 2 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

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

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) 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

And now for something completely different

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

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

Random geometric graphs Infinite random geometric graphs - Anthony Bonato

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

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

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

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

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

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

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

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

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

The new world

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

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

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

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

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

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

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

Infinitely many parallel universes

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

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

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

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

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

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

The curious geometry of sequence spaces

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

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

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

Continuous functions

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

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

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

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

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Merci!