Maksim Zhukovskii – Zero-one k-laws for G(n,n−α)

Zero-one k-laws and extended zero-one

k-laws for random distance graphs

Popova Svetlana

Lomonosov Moscow State University

Workshop on Extremal Graph Theory

Moscow, June 6, 2014

Denitions: Erdos-Renyi random graph G(n, p) and random graph G(Gn, p)

Denition. Erdos-Renyi random graph G(n, p) is a random element

with values in Ωn and distribution Pn,p on Fn, where

Ωn = (V = 1, ..., n, E), Fn = 2Ωn ,

Pn,p(G) = p|E|(1− p)C2n−|E|.

Denition. Let Gn be a sequence of graphs Gn = (Vn, En).

Random graph G(Gn, p) is a random element with values in ΩGnand distribution PGn,p on FGn , where

ΩGn = G = (V,E) : V = Vn, E ⊆ En,

FGn = 2ΩGn , PGn,p(G) = p|E|(1− p)|En|−|E|.

Denitions: Erdos-Renyi random graph G(n, p) and random graph G(Gn, p)

Denition. Erdos-Renyi random graph G(n, p) is a random element

with values in Ωn and distribution Pn,p on Fn, where

Ωn = (V = 1, ..., n, E), Fn = 2Ωn ,

Pn,p(G) = p|E|(1− p)C2n−|E|.

Denition. Let Gn be a sequence of graphs Gn = (Vn, En).

Random graph G(Gn, p) is a random element with values in ΩGnand distribution PGn,p on FGn , where

ΩGn = G = (V,E) : V = Vn, E ⊆ En,

FGn = 2ΩGn , PGn,p(G) = p|E|(1− p)|En|−|E|.

Denitions: rst-order properties and zero-one law

Denition. First-order properties of graphs are dened by

rst-order formulae, which are built of

predicate symbols ∼,=logical connectivities ¬,⇒,⇔,∨,∧variables x, y, . . .

quantiers ∀,∃

Denition. The random graph G(n, p) is said to follow zero-one law

if for any rst-order property L either

limn→∞

Pn,p(L) = 0

orlimn→∞

Pn,p(L) = 1.

Denitions: rst-order properties and zero-one law

Denition. First-order properties of graphs are dened by

rst-order formulae, which are built of

predicate symbols ∼,=logical connectivities ¬,⇒,⇔,∨,∧variables x, y, . . .

quantiers ∀,∃

Denition. The random graph G(n, p) is said to follow zero-one law

if for any rst-order property L either

limn→∞

Pn,p(L) = 0

orlimn→∞

Pn,p(L) = 1.

Denitions: zero-one k-law

Denition. The random graph G(n, p) is said to follow zero-one

k-law if for any property L dened by a rst-order formula with

quantier depth at most k either

limn→∞

Pn,p(L) = 0

orlimn→∞

Pn,p(L) = 1.

Zero-one law for Erdos-Renyi random graph G(n, p)

Theorem(Glebski et al., 1969; Fagin, 1976)

Let a function p = p(n) satisfy the property

∀β > 0 min(p, 1− p)nβ →∞ when n→∞.

Then the random graph G(n, p) follows the zero-one law.

Theorem(Shelah, Spencer, 1988)

Let p(n) = n−β and β be an irrational number, 0 < β < 1.

Zero-one law for Erdos-Renyi random graph G(n, p)

Theorem(Glebski et al., 1969; Fagin, 1976)

∀β > 0 min(p, 1− p)nβ →∞ when n→∞.

Theorem(Shelah, Spencer, 1988)

Let p(n) = n−β and β be an irrational number, 0 < β < 1.

Random distance graph

Random distance graph G(Gdistn , p)

Gdistn = (V distn , Edistn )

a = a(n), c = c(n)

V distn =

v = (v1, . . . , vn) : vi ∈ 0, 1,

n∑i=1

vi = a

Edistn = u,v ∈ V distn × V dist

n : (u,v) = c

Zero-one law for random distance graph

∀β > 0 min(p, 1− p)|V distn |β →∞ when n→∞.

Theorem

Let a(n) = αn, c(n) = α2n, α ∈ Q, 0 < α < 1. Then

the random graph G(Gdistn , p) doesn't follow the zero-one law, but

there exists a subsequence G(Gdistni, p) following the zero-one law.

Zero-one law for random distance graph

∀β > 0 min(p, 1− p)|V distn |β →∞ when n→∞.

Theorem

Let a(n) = αn, c(n) = α2n, α ∈ Q, 0 < α < 1. Then

the random graph G(Gdistn , p) doesn't follow the zero-one law, but

there exists a subsequence G(Gdistni, p) following the zero-one law.

Questions

When does a given subsequence G(Gdistni, p) follow zero-one

Does there exist a rst-order property L and a subsequenceG(Gdistni

, p) such that

limi→∞

PGdistni,p(L) ∈ (0, 1)

What limiting probabilities PGdistni,p(L) can we get?

Questions

, p) such that

limi→∞

Questions

, p) such that

limi→∞

Extended zero-one k-law

Denition. The random graph G(Gn, p) is said to follow extended

zero-one k-law if for every property L dened by a rst-order

formula with quantier depth at most k any partial limit of the

sequence PGn,p(L) equals either 0 or 1.

Goal. Find conditions on the sequence G(Gdistni, p) under which one

of the following takes place:

zero-one k-law holds

zero-one k-law doesn't hold, but extended zero-one k-law holds

extended zero-one k-law doesn't hold

Extended zero-one k-law

Denition. The random graph G(Gn, p) is said to follow extended

zero-one k-law if for every property L dened by a rst-order

formula with quantier depth at most k any partial limit of the

sequence PGn,p(L) equals either 0 or 1.

Goal. Find conditions on the sequence G(Gdistni, p) under which one

of the following takes place:

zero-one k-law holds

zero-one k-law doesn't hold, but extended zero-one k-law holds

extended zero-one k-law doesn't hold

Ehrenfeucht game EHR(G,H, k)

EHR(G,H, k)

Graphs G,H, number of rounds kTwo players Spoiler and Duplicator

i-th round:

Spoiler chooses a vertex either from G or from H

Duplicator chooses a vertex of the other graph

Let x1, . . . , xk, y1, . . . , yk be vertices chosen from graphs G andH respectively.Duplicator wins if and only if G|x1,...,xk ∼= H|y1,...,yk.

Theorem

The random graph G(Gn, p) follows zero-one k-law if and only if

P(Duplicator wins the game EHR(G(Gn, p), G(Gm, p), k))→ 1

as n,m→∞.

Ehrenfeucht game EHR(G,H, k)

EHR(G,H, k)

Graphs G,H, number of rounds kTwo players Spoiler and Duplicator

i-th round:

Spoiler chooses a vertex either from G or from H

Duplicator chooses a vertex of the other graph

Let x1, . . . , xk, y1, . . . , yk be vertices chosen from graphs G andH respectively.Duplicator wins if and only if G|x1,...,xk ∼= H|y1,...,yk.

Theorem

The random graph G(Gn, p) follows zero-one k-law if and only if

P(Duplicator wins the game EHR(G(Gn, p), G(Gm, p), k))→ 1

as n,m→∞.

Full level extension property

Denition. The graph G = (V,E) is said to satisfy full level t

extension property if for any vertices v1, . . . ,vl, u1, . . . ,ur(l + r ≤ t) there exists a vertex v adjacent to v1, . . . ,vl and

non-adjacent to u1, . . . ,ur.

Proposition

Let G(Gn, p) satisfy full level (k − 1) extension property

asymptotically almost surely. Then the random graph G(Gn, p)follows zero-one k-law.

Corollary

Let G(Gn, p) satisfy full level t extension property a.a.s for every

t ∈ N. Then the random graph G(Gn, p) follows the zero-one law.

Proposition

Corollary

Proposition

Corollary

Full level extension property for random distance graph

Proposition

Let a(n) = αn, α ∈ Q, 0 < α < 1. Then G(Gdistni, p) satises

full level t extension property a.a.s for every t ∈ N if and only if

c = α2n and ∀m ∈ N m|ni for suciently large i.

Proposition

Let a(n) = αn, c = α2n, α ∈ Q, 0 < α < 1, t ≤ 5. Then

G(Gdistni, p) satises full level t extension property a.a.s if and only if

Dt|a(ni)− c(ni) for suciently large i, where

D2 = 1, D3 = 2, D4 = 6, D5 = 60.

Full level extension property for random distance graph

Proposition

Let a(n) = αn, α ∈ Q, 0 < α < 1. Then G(Gdistni, p) satises

full level t extension property a.a.s for every t ∈ N if and only if

c = α2n and ∀m ∈ N m|ni for suciently large i.

Proposition

Let a(n) = αn, c = α2n, α ∈ Q, 0 < α < 1, t ≤ 5. Then

G(Gdistni, p) satises full level t extension property a.a.s if and only if

Dt|a(ni)− c(ni) for suciently large i, where

D2 = 1, D3 = 2, D4 = 6, D5 = 60.

Zero-one k-laws for random distance graph

Notation. a = αn, c = α2n, α = s/q, (s, q) = 1.

Theorem (zero-one 4-law)

The random graph G(Gdistn , p) follows extended zero-one 4-law.

The sequence G(Gdistni, p) follows zero-one 4-law if and only if

∃i0 such that all the numbers a(ni)− c(ni) for i > i0 have thesame parity.

Let a sequence ni be such that a(ni)− c(ni) are even for

suciently large i. Then

G(Gdistni, p) follows extended zero-one 5-law,

G(Gdistni, p) follows zero-one 5-law if and only if ∃i0 such that

either ∀i > i0 3|a(ni)− c(ni) or ∀i > i0 3 - a(ni)− c(ni).

Let q 6= 5 and a sequence ni be such that a(ni)− c(ni) are

divisible by 12 for suciently large i. Then

Disproof of extended zero-one laws for random distance graph

∀β > 0 min(p, 1− p)|V distn |β →∞ as n→∞. (∗)

Theorem (disproof of extended zero-one 6-law)

Let one of the following two cases take place:

q = 5 and a sequence ni is such that a(ni)− c(ni) are not

divisible by 5 for suciently large i,

α = 12 and a sequence ni is such that a(ni)− c(ni) are not

divisible by 4 for suciently large i.

Then there exists a function p(n) satisfying (∗) such thatG(Gdistni

, p) doesn't follow extended zero-one 6-law.

Disproof of extended zero-one laws for random distance graph

Theorem (disproof of extended zero-one law)

Let q be even, α ∈ (14 ,

34) and a sequence ni be such that

a(ni)− c(ni) are not divisible by 4 for suciently large i. Then

there exists a function p(n) satisfying (∗) such that G(Gdistni, p)

doesn't follow extended zero-one law.

Special sets of vertices

Denition. Vertices v1, . . . ,vt of a graph G = (V,E) are said to

form a special t-set if there doesn't exist a vertex v ∈ V adjacent

to all of the vertices v1, . . . ,vt.

Let Rt be a property of spanning subgraphs of Gn:for any vertices v1, . . . ,vt not forming a special t-set in Gn and

for any subset U ⊆ v1, . . . ,vt there exists a vertex v

adjacent to all vertices from U and non-adjacent to all vertices

from v1, . . . ,vt \ U .

Proposition

For every t ∈ N the random graph G(Gdistn , p) satisfyes Rt a.a.s.

from v1, . . . ,vt \ U .

Proposition

from v1, . . . ,vt \ U .

Proposition

Proof of zero-one k-laws: special sets of vertices without edges

Suppose

(1) Gn = (Vn, En) doesn't have special (t− 1)-sets

(2) G(Gn, p) satisfyes Rt a.a.s.

Proposition

Let a sequence Gn = (Vn, En) satisfy (1), (2) and the followingconditions:

Gn has special t-sets,

for every special t-set any two of its vertices are non-adjacent.

Then the random graph G(Gn, p) follows zero-one (t+ 1)-law.

Proof of zero-one k-laws: special sets of vertices without edges

Suppose

(1) Gn = (Vn, En) doesn't have special (t− 1)-sets

(2) G(Gn, p) satisfyes Rt a.a.s.

Proposition

Let a sequence Gn = (Vn, En) satisfy (1), (2) and the followingconditions:

Gn has special t-sets,

for every special t-set any two of its vertices are non-adjacent.

Then the random graph G(Gn, p) follows zero-one (t+ 1)-law.

Proof of zero-one k-laws: special sets of vertices with edges

Proposition

Suppose Gn = (Vn, En) satises (1), (2) and for any vertices

v1, . . . ,vi where i < t one of the following holds:

for any vertex vi+1 such that v1, . . . ,vi+1 can be extended toa special t-set there exist Ω(|Vn|β) dierent vertices each of

which can be mapped onto vi+1 by an automorphism of Gnxing v1, . . . ,vi (where β is a positive constant),

|(vi+1, . . . ,vt) : v1, . . . ,vt is a special t-set| = O(1).

Then the random graph G(Gn, p) follows extended zero-one(t+ 1)-law.

Disproof of extended zero-one k-laws

Let L be a property of subgraphs G ⊆ Gn:for any (v1, . . . ,vi) that can be extended to a special t-set with

edges in Gn there exist vi+1, . . . ,vt extending (v1, . . . ,vi) to a

special t-set with edges in G.

Let K(v1, . . . ,vi) be the number of (vi+1, . . . ,vt) extending

(v1, . . . ,vi) to a special t-set with edges in Gn.

If there exists (v1, . . . ,vi) with

K(v1, . . . ,vi)→∞, K(v1, . . . ,vi) = |Vn|o(1),

then PGn,p(L) can approach any number from (0, 1).

K(v1, . . . ,vi)→∞, K(v1, . . . ,vi) = |Vn|o(1),

Replace L by a rst-order property L:

L = ∀v1 . . . ∀vi ∃vi+1 . . . ∃vt Q(v1, . . . ,vt),

where Q approximately says that either (v1, . . . ,vi) can't be

extended to a special t-set with edges in Gn or (v1, . . . ,vt) forms a

special t-set with edges in G(Gn, p).

Thanks

Thank you for your attention!

Maksim Zhukovskii – Zero-one k-laws for G(n,n−α)

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Transcript of Maksim Zhukovskii – Zero-one k-laws for G(n,n−α)