Infinite Limits of Other Random Graphs

Transcription

1 Infinite Limits of Other Random Graphs Richard Elwes, University of Leeds 29 July 2016

2 Erdős - Rényi Graphs Definition (Erdős & Rényi, 1960) Fix p (0, 1) and n N {ω}. Form a random graph G n (p) with n vertices by connecting each pair with an edge, independently, with probability p.

3 Erdős - Rényi Graphs Definition (Erdős & Rényi, 1960) Fix p (0, 1) and n N {ω}. Form a random graph G n (p) with n vertices by connecting each pair with an edge, independently, with probability p. Theorem (Erdős & Rényi, 1960) There exists a unique graph G such that, for any p (0, 1) with probability 1. G ω (p) = G

4 The Rado Graph Definition (The Rado Graph) You can get anything you want at Alice s restaurant - Arlo Guthrie

5 The Rado Graph Definition (The Rado Graph) You can get anything you want at Alice s restaurant - Arlo Guthrie Axioms for G: Given finite disjoint sets of vertices U and V, there exists a vertex joined to everything in U and nothing in V.

6 Facts about the Rado graph G is universal. (It contains every finite and countable graph as an induced subgraph.) G is ℵ 0 -categorical. (Countable models of the axioms are isomorphic.) G is 1-transitive. (Any vertex can be taken to any other via an automorphism.) G is ultrahomogeneous. (Local isomorphisms of finite substructures extend to automorphisms.) Supersimple, unstable, MS-measurable, 1-based, the model companion to the theory of graphs,...

7 Background: Erdo s - Re nyi Graphs Opte Project, Creative Commons Licence c 2014 by LyonLabs, LLC and Barrett Lyon Erdo s & Re nyi versus Real World

8 Erdős & Rényi versus Real World Degree Distributions In G n (p) degrees are distributed, in the large n limit, via the Poisson distribution Pois(np).

9 Erdős & Rényi versus Real World Degree Distributions In G n (p) degrees are distributed, in the large n limit, via the Poisson distribution Pois(np). Real-world networks often satisfy power laws: the number of vertices of degree k is approximately proportional to k γ for some fixed γ > 0 and all k.

10 Erdős & Rényi versus Real World Degree Distributions In G n (p) degrees are distributed, in the large n limit, via the Poisson distribution Pois(np). Real-world networks often satisfy power laws: the number of vertices of degree k is approximately proportional to k γ for some fixed γ > 0 and all k. Typically 2 < γ < 3, E.g. the WWW has γ 2.5, the Kevin Bacon graph has γ 2.3.

11 Erős-Rényi, V = 200 & p = 0.01:

12 Preferential attachment, f = 1 & V = 200:

13 Barabási & Albert, we assume that the probability that a new vertex will be connected to vertex i depends on the connectivity of that vertex.

14 A simple model Definition Start with a finite graph. At time t + 1 create G t+1 by introducing a new vertex t + 1 to G t. Independently connect t + 1 to each previous vertex u t with probability d t(u) where d t (u) is the t degree of u in G t.

15 A simple model Definition Start with a finite graph. At time t + 1 create G t+1 by introducing a new vertex t + 1 to G t. Independently connect t + 1 to each previous vertex u t with probability d t(u) where d t (u) is the t degree of u in G t. Theorem (E., 2016) So long as the initial graph is neither edgeless nor complete, with probability 1, G ω is isomorphic to the Rado graph, augmented with a finite number of either isolated vertices or universal vertices.

16 A multi-graph model Definition Fix a function f : N N such that f (n) 1 for all s.

17 A multi-graph model Definition Fix a function f : N N such that f (n) 1 for all s. M 0 (f ) consists of a single vertex v 0.

18 A multi-graph model Definition Fix a function f : N N such that f (n) 1 for all s. M 0 (f ) consists of a single vertex v 0. M 1 (f ) has two vertices: v 1 joined to v 0 by f (0) parallel edges.

19 A multi-graph model Definition Fix a function f : N N such that f (n) 1 for all s. M 0 (f ) consists of a single vertex v 0. M 1 (f ) has two vertices: v 1 joined to v 0 by f (0) parallel edges. Thereafter, form M n+1 (f ) by adding v n+1 to M n (f ) with f (n) outgoing edges

20 A multi-graph model Definition Fix a function f : N N such that f (n) 1 for all s. M 0 (f ) consists of a single vertex v 0. M 1 (f ) has two vertices: v 1 joined to v 0 by f (0) parallel edges. Thereafter, form M n+1 (f ) by adding v n+1 to M n (f ) with f (n) outgoing edges with each endpoint v chosen from {v 0,..., v n } independently with probability deg n (v) n i=0 deg n(v i ).

21 Theorem (Kleinberg & Kleinberg, 2005) Let i = 1 or i = 2 and suppose f (n) = i for all n. Then then there is a unique infinite di-multi-graph M(i), such that with probability 1 M ω (f ) = M(i) as di-multi-graphs.

22 Theorem (Kleinberg & Kleinberg, 2005) Let i = 1 or i = 2 and suppose f (n) = i for all n. Then then there is a unique infinite di-multi-graph M(i), such that with probability 1 M ω (f ) = M(i) as di-multi-graphs. Definition (Kleinberg & Kleinberg, 2005) Axioms for M(1): There exists a unique root with outdegree 0. Every other vertex has outdegree 1. Every vertex has infinite indegree. There is a path to the root from every vertex.

23 Definition (Kleinberg & Kleinberg, 2005) Axioms for M(2): There exists a unique vertex with outdegree 0. Every other vertex has outdegree 2. Given any (not necessarily distinct) v 1 & v 2, there are infinitely many vertices whose two outgoing edges arrive at v 1 & v 2. M(2) does not contain any infinite forward path.

24 Definition (Kleinberg & Kleinberg, 2005) Axioms for M(2): There exists a unique vertex with outdegree 0. Every other vertex has outdegree 2. Given any (not necessarily distinct) v 1 & v 2, there are infinitely many vertices whose two outgoing edges arrive at v 1 & v 2. M(2) does not contain any infinite forward path. Theorem (Kleinberg & Kleinberg, 2005) For f 1, 2, there is a positive probability that two instantiations of M ω (f ) will be non-isomorphic as di-multi-graphs.

25 Definition (Kleinberg & Kleinberg, 2005) Axioms for M(2): There exists a unique vertex with outdegree 0. Every other vertex has outdegree 2. Given any (not necessarily distinct) v 1 & v 2, there are infinitely many vertices whose two outgoing edges arrive at v 1 & v 2. M(2) does not contain any infinite forward path. Theorem (Kleinberg & Kleinberg, 2005) For f 1, 2, there is a positive probability that two instantiations of M ω (f ) will be non-isomorphic as di-multi-graphs. For f = C 1, 2, there is a positive probability that two instantiations of M ω (f ) will be non-isomorphic as multi-graphs.

26 Linear growth Theorem (E., 2015) There exists a multigraph M(L) such that if a n f (n) b n for some a, b R >0 and all large enough n, then with probability 1 as multigraphs. M ω (f ) = M(L)

27 Definition ( The Rado Multigraph ) Axioms for M(L) in a language E i i ω.

28 Definition ( The Rado Multigraph ) Axioms for M(L) in a language E i i ω. Each E j is a symmetric irreflexive binary relation.

29 Definition ( The Rado Multigraph ) Axioms for M(L) in a language E i i ω. Each E j is a symmetric irreflexive binary relation. E 0 (u, v) for all u, v

30 Definition ( The Rado Multigraph ) Axioms for M(L) in a language E i i ω. Each E j is a symmetric irreflexive binary relation. E 0 (u, v) for all u, v E j+1 (u, v) E j (u, v)

31 Definition ( The Rado Multigraph ) Axioms for M(L) in a language E i i ω. Each E j is a symmetric irreflexive binary relation. E 0 (u, v) for all u, v E j+1 (u, v) E j (u, v) [Finitariness] For all u, v there is some i such that E i (u, v).

32 Definition ( The Rado Multigraph ) Axioms for M(L) in a language E i i ω. Each E j is a symmetric irreflexive binary relation. E 0 (u, v) for all u, v E j+1 (u, v) E j (u, v) [Finitariness] For all u, v there is some i such that E i (u, v). [Alice s Restaurant Property] For any n N, any nodes u 1,..., u n and any m 1,..., m n N there exists v such that E mi (u i, v) E mi +1(u i, v) for each i.

33 Facts about the Rado multigraph M(L) is universal. (It contains every finite/countable finitary multigraph as an induced submultigraph.) M(L) is ℵ 0 -categorical. (Countable models of the axioms are isomorphic.) M(L) is 1-transitive. (Any vertex can be taken to any other via an automorphism.) M(L) is ultrahomogeneous. (Local isomorphisms of finite substructures extend to automorphisms.)...?

34 THANK YOU!