3 zhukovsky

Critical points in zero-one laws for G(n, p)

Maksim Zhukovskii

MSU, MIPT, Yandex

Workshop on random graphs and their applications

26 October 2013

1/17 Maksim Zhukovskii Critical points in zero-one laws for G(n, p)

First-order properties.

First-order formulae:relational symbols ∼,=;logical connectivities ¬,⇒,⇔,∨,∧;variables x, y, x1, ...;quanti�ers ∀,∃.

• property of a graph to be complete

∀x∀y (¬(x = y)⇒ (x ∼ y)).

• property of a graph to contain a triangle

∃x ∃y ∃z ((x ∼ y) ∧ (y ∼ z) ∧ (x ∼ z)).

• property of a graph to have chromatic number k

∀x∀y (¬(x = y)⇒ (x ∼ y)).

∃x ∃y ∃z ((x ∼ y) ∧ (y ∼ z) ∧ (x ∼ z)).

∀x∀y (¬(x = y)⇒ (x ∼ y)).

∃x ∃y ∃z ((x ∼ y) ∧ (y ∼ z) ∧ (x ∼ z)).

∀x∀y (¬(x = y)⇒ (x ∼ y)).

∃x ∃y ∃z ((x ∼ y) ∧ (y ∼ z) ∧ (x ∼ z)).

Limit probabilities.

G(n, p): n � number of vertices, p � edge appearanceprobability

G � an arbitrary graph, LG � property of containing G.LG is the �rst order property.

maximal density ρmax(G) = maxH⊆G{ρ(H)}, ρ(H) = e(H)v(H)

p0 = n−1/ρmax(G);

p�p0 ⇒ limn→∞

Pn,p(LG) = 1,

Pn,p(LG) = 0,

p=p0 ⇒ limn→∞

Pn,p(LG)/∈ {0, 1}.

If G � set of graphs, L = {LG, LG, G ∈ G},p = n−α, α is irrational

then ∀L ∈ L limn→∞ Pn,p(L) ∈ {0, 1}.

G � an arbitrary graph, LG � property of containing G.

LG is the �rst order property.

Pn,p(LG) = 1,

Pn,p(LG) = 0,

p=p0 ⇒ limn→∞

Pn,p(LG)/∈ {0, 1}.

then ∀L ∈ L limn→∞ Pn,p(L) ∈ {0, 1}.

Pn,p(LG) = 1,

Pn,p(LG) = 0,

p=p0 ⇒ limn→∞

Pn,p(LG)/∈ {0, 1}.

then ∀L ∈ L limn→∞ Pn,p(L) ∈ {0, 1}.

Pn,p(LG) = 1,

Pn,p(LG) = 0,

p=p0 ⇒ limn→∞

Pn,p(LG)/∈ {0, 1}.

then ∀L ∈ L limn→∞ Pn,p(L) ∈ {0, 1}.

Pn,p(LG) = 1,

Pn,p(LG) = 0,

p=p0 ⇒ limn→∞

Pn,p(LG)/∈ {0, 1}.

then ∀L ∈ L limn→∞ Pn,p(L) ∈ {0, 1}.

Pn,p(LG) = 1,

Pn,p(LG) = 0,

p=p0 ⇒ limn→∞

Pn,p(LG)/∈ {0, 1}.

then ∀L ∈ L limn→∞ Pn,p(L) ∈ {0, 1}.

Pn,p(LG) = 1,

Pn,p(LG) = 0,

p=p0 ⇒ limn→∞

Pn,p(LG)/∈ {0, 1}.

then ∀L ∈ L limn→∞ Pn,p(L) ∈ {0, 1}.

Zero-one law

L � the set of all �rst-order properties.

De�nition

The random graph obeys zero-one law if ∀L ∈ L

limn→∞

Pn,p(n)(L) ∈ {0, 1}.

P is a set of functions p such that G(n, p) obeys zero-one law.

Theorem [J.H. Spencer, S. Shelah, 1988]

Let p = n−α, α ∈ (0, 1].

If α ∈ R \Q then p ∈ P.If α ∈ Q then p /∈ P.

Zero-one law

L � the set of all �rst-order properties.

De�nition

The random graph obeys zero-one law if ∀L ∈ L

limn→∞

Pn,p(n)(L) ∈ {0, 1}.

P is a set of functions p such that G(n, p) obeys zero-one law.

Theorem [J.H. Spencer, S. Shelah, 1988]

Let p = n−α, α ∈ (0, 1].

If α ∈ R \Q then p ∈ P.If α ∈ Q then p /∈ P.

Quanti�er depth.

Quanti�er depths of x = y, x ∼ y equal 0.

If quanti�er depth of φ equals k then quanti�er depth of ¬φequals k as well.

If quanti�er depth of φ1 equals k1, quanti�er depth of φ2 equalsk2 then quanti�er depths of φ1 ⇒ φ2, φ1 ⇔ φ2, φ1 ∨ φ2,φ1 ∧ φ2 equal max{k1, k2}.

If quanti�er depth of φ(x) equals k, then quanti�er depths of∃xφ(x), ∀xφ(x) equal k + 1.

Quanti�er depth.

Bounded quanti�er depth.

Lk � the class of �rst-order properties de�ned by formulae withquanti�er depth bounded by k.

Random graph G(n, p) obeys zero-one k-law if ∀L ∈ LklimN→∞

Pn,p(n)(L) ∈ {0, 1}.

Pk � the class of all functions p = p(n) such that random graphG(n, p) obeys zero-one k-law.

Example: (∀x ∃y (x ∼ y)) ∧ (∀x ∃y ¬(x ∼ y)) k = 2∀x∃y ∃z ((x ∼ y) ∧ (¬(x ∼ z))) k = 3

Bounded quanti�er depth.

Lk � the class of �rst-order properties de�ned by formulae withquanti�er depth bounded by k.

Random graph G(n, p) obeys zero-one k-law if ∀L ∈ LklimN→∞

Pn,p(n)(L) ∈ {0, 1}.

Pk � the class of all functions p = p(n) such that random graphG(n, p) obeys zero-one k-law.

Example: (∀x ∃y (x ∼ y)) ∧ (∀x ∃y ¬(x ∼ y)) k = 2∀x ∃y ∃z ((x ∼ y) ∧ (¬(x ∼ z))) k = 3

Zero-one k-laws

Zhukovskii; 2010 (zero-one k-law)

Let p = n−α, k ∈ N, k ≥ 3.

If 0 < α < 1k−2 then p ∈ Pk.

Zhukovskii; 2012+ (extension of zero-one k-law)

Let p = n−α, k ∈ N, k ≥ 4, Q = {ab , a, b ∈ N, a ≤ 2k−1}

If α = 1− 12k−1+β

, β ∈ (0,∞) \ Q then p ∈ Pk.

)⋃(1−

2k − 1, 1−

)⋃. . .⋃

2k−1 + 2k−2, 1−

2k−1 + 2k−2 + 1

)⋃(1−

2k−1 + 2k−1−12

, 1−1

2k−1 + 2k−2

⋃. . .⋃1−

2k−1 +2k−1−

[2k−1

, 1−1

2k−1 + 2k−1

⋃ . . .? . . .⋃[

k − 2

Zero-one k-laws

Let p = n−α, k ∈ N, k ≥ 3.

If 0 < α < 1k−2 then p ∈ Pk.

If α = 1− 12k−1+β

, β ∈ (0,∞) \ Q then p ∈ Pk.

)⋃(1−

2k − 1, 1−

)⋃. . .⋃

2k−1 + 2k−2, 1−

2k−1 + 2k−2 + 1

)⋃(1−

2k−1 + 2k−1−12

, 1−1

2k−1 + 2k−2

⋃. . .⋃1−

2k−1 +2k−1−

[2k−1

, 1−1

2k−1 + 2k−1

⋃ . . .? . . .⋃[

k − 2

Zero-one k-laws

Let p = n−α, k ∈ N, k ≥ 3.

If 0 < α < 1k−2 then p ∈ Pk.

If α = 1− 12k−1+β

, β ∈ (0,∞) \ Q then p ∈ Pk.

)⋃(1−

2k − 1, 1−

)⋃. . .⋃

2k−1 + 2k−2, 1−

2k−1 + 2k−2 + 1

)⋃(1−

2k−1 + 2k−1−12

, 1−1

2k−1 + 2k−2

⋃. . .⋃1−

2k−1 +2k−1−

[2k−1

, 1−1

2k−1 + 2k−1

⋃ . . .? . . .⋃[

k − 2

Zero-one k-laws

Zhukovskii; 2013++ (no k-law)

Let p = n−α, k ∈ N.

If k ≥ 3, α = 1k−2 then p /∈ Pk.

If k ≥ 4,Q̃ =

2k−1 − 2 · 1, . . . , 1;2k−1−2·1−1·2

2, . . . , 1

2; 2k−1−2·1−2·2

3, . . . , 1

3; 2k−1−2·1−3·2

4, . . . , 1

2k−1−2·1−3·2−1·35

, . . . ; . . . ; 2k−1−2·1−3·2−4·38

, . . . ;. . . ; . . . ; . . .

α = 1− 1

2k−1+β, β ∈ Q̃ then p /∈ Pk.

Critical points

1− α : 0→ 1

Critical points

1− α : 0→ 1

Critical points

1− α : 0→ 1

Critical points

1− α : 0→ 1

12k−1

Critical points

1− α : 0→ 1

12k−1

12k−2

Critical points

1− α : 0→ 1

12k−1

12k−2

12k−3

Critical points

1− α : 0→ 1

12k−1

12k−2

12k−3

. . . 12k−2k−2

Critical points

1− α : 0→ 1

12k−1

12k−2

12k−3

. . . 12k−2k−2 . . .

Critical points

1− α : 0→ 1

12k−1

12k−2

12k−3

. . . 12k−2k−2 . . . ?

Large gap: [1/2k−1, 1− 1/(k − 2))

Critical points

1− α : 0→ 1

12k−1

12k−2

12k−3

. . . 12k−2k−2 . . . ? 1− 1

Large gap: [1/2k−1, 1− 1/(k − 2))

Critical points

1− α : 0→ 1

0 12k−2

12k−3

. . . 12k−2k−2 . . . ? 1− 1

Large gap: [1/2k−1, 1− 1/(k − 2))

Intervals

Ehrenfeucht game

EHR(G,H, k)

G,H � two graphs

k � number of rounds

Spoiler & Duplicator � two players

Ehrenfeucht game

EHR(G,H, k)

G,H � two graphs

Ehrenfeucht game

EHR(G,H, k)

G,H � two graphs

Ehrenfeucht game

EHR(G,H, k)

G,H � two graphs

Ehrenfeucht game

EHR(G,H, k)

G,H � two graphs

Ehrenfeucht game

EHR(G,H, k)

G,H � two graphs

Ehrenfeucht game

EHR(G,H, k)

G,H � two graphs

Ehrenfeucht game

EHR(G,H, k)

G,H � two graphs

Ehrenfeucht game

EHR(G,H, k)

G,H � two graphs

Ehrenfeucht game

EHR(G,H, k)

G,H � two graphs

Ehrenfeucht game

EHR(G,H, k)

G,H � two graphs

Ehrenfeucht theorem

Theorem [A. Ehrenfeucht, 1960]

Let G,H be two graphs. For any �rst-order property L expressedby formula with quanti�er depth bounded by a number kG ∈ L⇔ H ∈ L if and only if Duplicator has a winning strategy inthe game EHR(G,H, k).

Example: ∃x1 ∃x2 ∃x3 ((x1 ∼ x2) ∧ (x2 ∼ x3) ∧ (x1 ∼ x3)).

Ehrenfeucht theorem

Proofs of k-laws

Corollary

Zero-one law holds if and only if for any k ∈ N almost surelyDuplicator has a winning strategy in the Ehrenfeucht game onk rounds.

Random graph G(n, p) obeys zero-one k-law if and only ifalmost surely Duplicator has a winning strategy in theEhrenfeucht game on k rounds.

Case k = 4, α = 13/14

FISH GRAPH

e = 14, v = 13, ρ = 14/13

G(n, n−13/14) contains a copy of FISH GRAPH with positiveasymptotical probability;

G(n, n−13/14) does not contain a copy of any graph with at most13 vertices and maximal density at least 14/13 with positive

asymptotical probability.

EHR(G,H, 4):

G contains FISH GRAPH;H does not contain a copy of any graph with at most 13vertices and maximal density at least 14/13.

Case k = 4, α = 13/14

FISH GRAPH

e = 14, v = 13, ρ = 14/13

EHR(G,H, 4):

Case k = 4, α = 13/14

FISH GRAPH

e = 14, v = 13, ρ = 14/13

EHR(G,H, 4):

Case k = 4, α = 13/14

FISH GRAPH

e = 14, v = 13, ρ = 14/13

EHR(G,H, 4):

Case k = 4, α = 13/14

FISH GRAPH

e = 14, v = 13, ρ = 14/13

EHR(G,H, 4):

G contains FISH GRAPH;

H does not contain a copy of any graph with at most 13vertices and maximal density at least 14/13.

Case k = 4, α = 13/14

FISH GRAPH

e = 14, v = 13, ρ = 14/13

EHR(G,H, 4):

Winning strategy of Spoiler

Open questions

What happens when α ∈(

1k−2 , 1−

12k−1

Spencer and Shelah, 1988: There exists a �rst-order property andan in�nite set S of rational numbers from (0, 1) such that for anyα ∈ S random graph G(n, n−α) follows the property withprobability which doesn't tend to 0 or to 1.

For any �xed k and any ε > 0 in(1− 1

2k−1 + ε, 1)there is only

�nite number of critical points. Does this property holds for(1− 1

2k−1 , 1)? Are there any other such subintervals in(

1k−2 , 1−

12k−1

Open questions

1k−2 , 1−

12k−1

1k−2 , 1−

12k−1

Open questions

1k−2 , 1−

12k−1

1k−2 , 1−

12k−1

Thank you!

Thank you very much for your

attention!

3 zhukovsky

Technology

lg property

trianglex y z x y y

completexy x

variables x

arbitrary graph

p edge appearanceprobabilityg

g set of graphs

order formulae