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Computation and Logic on Dynamic Random Graphs Wesley Calvert - - PowerPoint PPT Presentation

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Computation and Logic on Dynamic Random Graphs Wesley Calvert Southern Illinois University ASL North American Annual Meeting Berkeley, California March 26, 2011 Wesley Calvert (Southern Illinois University) Dynamic Random Graphs ASL 2011 1

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Computation and Logic on Dynamic Random Graphs

Wesley Calvert

Southern Illinois University

ASL North American Annual Meeting Berkeley, California March 26, 2011

Wesley Calvert (Southern Illinois University) Dynamic Random Graphs ASL 2011 1 / 26

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Theorem (0-1 Law) Every sentence in the language of graphs is true for either almost all finite graphs or almost none of them.

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Theorem (0-1 Law) Every sentence in the language of graphs is true for either almost all finite graphs or almost none of them. Definition The theory of the random graph is the set of all sentences in the language

  • f graphs which are true for almost all finite graphs.

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Theorem The theory of the random graph is decidable.

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Theorem The theory of the random graph is decidable. Theorem The theory of the random graph is ℵ0-categorical.

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Theorem The theory of the random graph is decidable. Theorem The theory of the random graph is ℵ0-categorical. Proof. Back-and-forth

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Theorem The theory of the random graph is decidable. Theorem The theory of the random graph is ℵ0-categorical. Proof. Back-and-forth Theorem The theory of the random graph is computably categorical.

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Theorem The theory of the random graph is decidable. Theorem The theory of the random graph is ℵ0-categorical. Proof. Back-and-forth Theorem The theory of the random graph is computably categorical. Theorem The theory of the random graph is properly simple.

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When I showed this to some colleagues in my department, they didn’t like it much.

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When I showed this to some colleagues in my department, they didn’t like it much. “There’s nothing random in it!”

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When I showed this to some colleagues in my department, they didn’t like it much. “There’s nothing random in it!” “A model of that theory isn’t a random graph!”

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When I showed this to some colleagues in my department, they didn’t like it much. “There’s nothing random in it!” “A model of that theory isn’t a random graph!” “You mean your random graphs can be infinite?”

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When I showed this to some colleagues in my department, they didn’t like it much. “There’s nothing random in it!” “A model of that theory isn’t a random graph!” “You mean your random graphs can be infinite?” “Maybe a better name would be ’random theory of graphs.’ ”

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Here’s what they’re used to as a random graph: Definition The Erd˝

  • s-Renyi random graph Gp(n) is constructed by taking n vertices

and connecting each pair independently with probability p.

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Here’s what they’re used to as a random graph: Definition The Erd˝

  • s-Renyi random graph Gp(n) is constructed by taking n vertices

and connecting each pair independently with probability p. Proposition The theory of the random graph is the almost-sure theory of Gp(ω) if p ∈ (0, 1).

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Definition Continuous first-order logic is a logic taking truth values on [0, 1], and having all continuous functions on [0, 1] for its Boolean connectives and sup and inf for its quantifiers. We typically also include a metric d in the signature.

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Definition Continuous first-order logic is a logic taking truth values on [0, 1], and having all continuous functions on [0, 1] for its Boolean connectives and sup and inf for its quantifiers. We typically also include a metric d in the signature. Proposition (Ben Yaacov-Berenstein-Henson-Usvyatsov) Any continuous function [0, 1]n → [0, 1] can be approximated by ( . −, ¬ = x → 1 − x, 1

2).

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Definition A continuous structure M is a metric space (M, d)

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Definition A continuous structure M is a metric space (M, d) with Some distinguished uniformly continuous functions f : Mn → M

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Definition A continuous structure M is a metric space (M, d) with Some distinguished uniformly continuous functions f : Mn → M , and Some distinguished uniformly continuous predicates R : Mn → [0, 1].

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Example Consider the set {0, 1, . . . , n}, with the discrete metric.

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Example Consider the set {0, 1, . . . , n}, with the discrete metric. Define a predicate R such that R(x, y) = 1 − p if x = y

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Example Consider the set {0, 1, . . . , n}, with the discrete metric. Define a predicate R such that R(x, y) = 1 − p if x = y R(x, x) = 1.

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Example Consider the set {0, 1, . . . , n}, with the discrete metric. Define a predicate R such that R(x, y) = 1 − p if x = y R(x, x) = 1. Note that this is a continuous structure, and “is” the Erd˝

  • s-Renyi random

graph Gp(n).

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Definition Let 2ω be the set of infinite binary sequences, with the usual Lebesgue probability measure µ.

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Definition Let 2ω be the set of infinite binary sequences, with the usual Lebesgue probability measure µ.

1 A randomized Turing machine is a Turing machine equipped with an

  • racle for an element of 2ω, called the random bits, with output in

{0, 1}.

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Definition Let 2ω be the set of infinite binary sequences, with the usual Lebesgue probability measure µ.

1 A randomized Turing machine is a Turing machine equipped with an

  • racle for an element of 2ω, called the random bits, with output in

{0, 1}.

2 We say that a randomized Turing machine M accepts n with

probability p if and only if µ{x ∈ 2ω : Mx(n) ↓= 0} = p.

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Definition Let 2ω be the set of infinite binary sequences, with the usual Lebesgue probability measure µ.

1 A randomized Turing machine is a Turing machine equipped with an

  • racle for an element of 2ω, called the random bits, with output in

{0, 1}.

2 We say that a randomized Turing machine M accepts n with

probability p if and only if µ{x ∈ 2ω : Mx(n) ↓= 0} = p.

3 We say that a randomized Turing machine M rejects n with

probability p if and only if µ{x ∈ 2ω : Mx(n) ↓= 1} = p.

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Definition The continuous atomic diagram D(M) of a continuous structure M is the set of pairs (ϕ, p), where ϕ is an atomic CFO formula (in M with unary distance) and the truth value of ϕ in M is equal to p.

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Definition The continuous atomic diagram D(M) of a continuous structure M is the set of pairs (ϕ, p), where ϕ is an atomic CFO formula (in M with unary distance) and the truth value of ϕ in M is equal to p. Definition A probabilistically computable structure M is a continuous structure equipped with a randomized Turing machine which, for any pair (ϕ, p) ∈ D(M), accepts ϕ with probability equal to p.

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Proposition Every classically computable structure, with the discrete metric, is a probabilistically computable structure.

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Proposition Every classically computable structure, with the discrete metric, is a probabilistically computable structure. Theorem Classically computable structures are to RCA0 as probabilistically computable structures are to ACA0.

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Note that the Erd˝

  • s-Renyi random graph we described earlier is

probabilistically computable.

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But, the Erd˝

  • s-Renyi graphs are too homogeneous.

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But, the Erd˝

  • s-Renyi graphs are too homogeneous.

Example The FaceBook friend graph.

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But, the Erd˝

  • s-Renyi graphs are too homogeneous.

Example The FaceBook friend graph. Example The web graph.

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But, the Erd˝

  • s-Renyi graphs are too homogeneous.

Example The FaceBook friend graph. Example The web graph. Example The collaboration graph.

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But, the Erd˝

  • s-Renyi graphs are too homogeneous.

Example The FaceBook friend graph. Example The web graph. Example The collaboration graph. Example Consider the reactions catalyzed by E. Coli. Make a node for each collection of reactants and products, and a connection for any two that share a metabolite (intermediate compound).

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Major features we want to capture:

1 Development over time Wesley Calvert (Southern Illinois University) Dynamic Random Graphs ASL 2011 14 / 26

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Major features we want to capture:

1 Development over time 2 Non-homogeneity [number of vertices of degree k proportional to

1 kβ

for fixed β]

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The language: Infinitely many predicates (Vi)i∈ω, which will be interpreted as the vertex set at stage i.

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The language: Infinitely many predicates (Vi)i∈ω, which will be interpreted as the vertex set at stage i. Infinitely many predicates (Ei)i∈ω, interpreted as the edge set at stage i.

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Definition Let G0 be a finite graph with vertex set V∗ and edge set E∗. The preferential attachment graph G(p, G0) is the random graph process with V0 = V∗, with E0 = E∗, and with the following property:

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Definition Let G0 be a finite graph with vertex set V∗ and edge set E∗. The preferential attachment graph G(p, G0) is the random graph process with V0 = V∗, with E0 = E∗, and with the following property: For each t > 0, we form Gt by independently

With probability p let Vt consist of Vt−1 plus a single new element v, and independently choose a vertex u in proportion to its degree in Gt−1, setting Et = Et+1 ∪ {(u, v)}.

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Definition Let G0 be a finite graph with vertex set V∗ and edge set E∗. The preferential attachment graph G(p, G0) is the random graph process with V0 = V∗, with E0 = E∗, and with the following property: For each t > 0, we form Gt by independently

With probability p let Vt consist of Vt−1 plus a single new element v, and independently choose a vertex u in proportion to its degree in Gt−1, setting Et = Et+1 ∪ {(u, v)}. Otherwise, independently choose two vertices u, v with probability proportional to their degrees in Gt−1, and set Vt = Vt−1 and Et = Et−1 ∪ {(u, v)}.

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Proposition There is an effective procedure which, given σ ∈ 2ω, will produce an index for the sample path corresponding to σ; that is, for the uniformly computable sequence of (classical) relations (V σ

i , E σ i )i∈ω.

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Lemma For any p ∈ [0, 1], there exists an infinite uniformly computable sequence

  • f subsets (Sp

i )i∈ω of 2ω which are independent, such that Sp i has

probability p.

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Lemma For any p ∈ [0, 1], there exists an infinite uniformly computable sequence

  • f subsets (Sp

i )i∈ω of 2ω which are independent, such that Sp i has

probability p. Lemma For any positive integer n, there exist infinite independent uniformly computable sequences of tuples (Qn

t )t∈ω and (Pn t )t∈ω, where each Qn t and

each Pn

t has the form (Qn t,k)k<n and the Qn t,k are disjoint subsets of 2ω,

each with measure 1

n.

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Proof of Proposition. Start with G0.

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Proof of Proposition. Start with G0. Check σ against Sp to decide whether to do a vertex step

  • r an edge step.

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Proof of Proposition. Start with G0. Check σ against Sp to decide whether to do a vertex step

  • r an edge step. If a vertex step, check with the appropriate Q to figure
  • ut where the new vertex should connect.

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Proof of Proposition. Start with G0. Check σ against Sp to decide whether to do a vertex step

  • r an edge step. If a vertex step, check with the appropriate Q to figure
  • ut where the new vertex should connect. If an edge step, check with the

appropriate P.

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Theorem For any p ∈ [0, 1] and any finite graph G0, there is a probabilistically computable random graph process of form G(p, G0).

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Theorem For any p ∈ [0, 1] and any finite graph G0, there is a probabilistically computable random graph process of form G(p, G0). Proof. Take the Turing machine we just built, and let it work over all oracles.

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Problem What is Th (G(p, G0))?

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Proposition The axioms of preferential attachment are neither complete nor categorical.

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Proposition The axioms of preferential attachment are neither complete nor categorical. Proof. Do the same construction as before, but arrange that individual numbers have small probability of coming in as vertices at stage t.

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The metabolic pathway graphs tend to look different.

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The metabolic pathway graphs tend to look different. Recall: nk ∝

1 kβ .

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The metabolic pathway graphs tend to look different. Recall: nk ∝

1 kβ .

Preferential attachment β ≥ 2 (generally) Metabolic Networks β ≈ 1.5.

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Definition Let G0 be a finite graph with vertex set V∗ and edge set E∗. The duplication graph B(p, G0) is the random graph process with V0 = V∗, with E0 = E∗, and with the following property: For each t > 0, we form Gt by Independently selecting a vertex vt from Vt−1 uniformly at random, add a new vertex yt, and For each neighbor u connected to vt at stage t − 1, we independently attach u to yt with probability p.

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Theorem For any p ∈ [0, 1] and any finite graph G0, there is a probabilistically computable random graph process of the form B(p, G0).

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Theorem For any p ∈ [0, 1] and any finite graph G0, there is a probabilistically computable random graph process of the form B(p, G0). And it has all the same issues with completeness.

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Computation and Logic on Dynamic Random Graphs

Wesley Calvert

Southern Illinois University

ASL North American Annual Meeting Berkeley, California March 26, 2011

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