Shifing the thresold of phase transition in 2-SAT and random graphs - - PowerPoint PPT Presentation

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results

Shifing the thresold of phase transition in 2-SAT and random graphs

Sergey Dovgal1,2,3,4 Vlady Ravelomanana2

1Université Paris-13, 2Université Paris-Diderot 3Moscow Institute of Physics and

Technology 4Institute for Information Transmission Problems, Moscow Acknowledgements: Élie de Panafieu, Fedor Petrov, ipython+sympy+cpp

May 19, 2017

D., Ravelomanana Shifing the phase transition 1 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results

Outline

1

2-SAT, phase transitions and degree constraints

2

Lower bound for 2-SAT

3

Saddle-point method and analytic lemma

4

Related results

D., Ravelomanana Shifing the phase transition 2 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Phase transition Shifing the phase transition Graphs with degree constraints Experimental results

Outline

1

2-SAT, phase transitions and degree constraints

2

Lower bound for 2-SAT

3

Saddle-point method and analytic lemma

4

Related results

D., Ravelomanana Shifing the phase transition 3 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Phase transition Shifing the phase transition Graphs with degree constraints Experimental results

Phase transition in Erdős–Rényi random graphs

n vertices, m edges, m = 1 2n(1 + µn−1/3)

1

“gas” µ → −∞ : planar graph, trees and unicycles, max component size O(log n).

2

“liquid” |µ| = O(1) : complex components appear, max component size O(n2/3).

3

“crystal” µ → +∞ : non-planar, complex compontnes, max component size linear O(n).

D., Ravelomanana Shifing the phase transition 4 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Phase transition Shifing the phase transition Graphs with degree constraints Experimental results

Phase transition in Erdős–Rényi random graphs

n vertices, m edges, m = 1 2n(1 + µn−1/3)

1

“gas” µ → −∞ : planar graph, trees and unicycles, max component size O(log n).

2

“liquid” |µ| = O(1) : complex components appear, max component size O(n2/3).

3

“crystal” µ → +∞ : non-planar, complex compontnes, max component size linear O(n).

D., Ravelomanana Shifing the phase transition 4 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Phase transition Shifing the phase transition Graphs with degree constraints Experimental results

Phase transition in Erdős–Rényi random graphs

n vertices, m edges, m = 1 2n(1 + µn−1/3)

1

“gas” µ → −∞ : planar graph, trees and unicycles, max component size O(log n).

2

“liquid” |µ| = O(1) : complex components appear, max component size O(n2/3).

3

“crystal” µ → +∞ : non-planar, complex compontnes, max component size linear O(n).

D., Ravelomanana Shifing the phase transition 4 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Phase transition Shifing the phase transition Graphs with degree constraints Experimental results

Phase transition :: largest component, n = 1000

D., Ravelomanana Shifing the phase transition 5 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Phase transition Shifing the phase transition Graphs with degree constraints Experimental results

Phase transition :: planarity, n = 1000

D., Ravelomanana Shifing the phase transition 6 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Phase transition Shifing the phase transition Graphs with degree constraints Experimental results

Phase transition :: diameter, n = 1000

D., Ravelomanana Shifing the phase transition 7 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Phase transition Shifing the phase transition Graphs with degree constraints Experimental results

Phase transition :: connected components, n = 1000

D., Ravelomanana Shifing the phase transition 8 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Phase transition Shifing the phase transition Graphs with degree constraints Experimental results

2SAT Transition

1 [Bollobás, Borgs, Chayes, Kim, and Wilson ’99]

2SAT Transition

2 [Coppersmith, Gamarnik, Hajaghayi, Sorkin ’03]

MAX 2-SAT Transition

3 [Cooper, Freize, Sorkin ’07]

2SAT with degree sequence constraints

D., Ravelomanana Shifing the phase transition 9 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Phase transition Shifing the phase transition Graphs with degree constraints Experimental results

Shifing the phase transition

m = 1 2n(1 + µn−1/3) ⇒ m = αn(1 + µn−1/3)

1 Achlioptas percolation process (α = 0.535?) 2 Degree sequence models (less detailed information) 3 Degree set constraint :: current talk

D., Ravelomanana Shifing the phase transition 10 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Phase transition Shifing the phase transition Graphs with degree constraints Experimental results

Shifing the phase transition

m = 1 2n(1 + µn−1/3) ⇒ m = αn(1 + µn−1/3)

1 Achlioptas percolation process (α = 0.535?) 2 Degree sequence models (less detailed information) 3 Degree set constraint :: current talk

D., Ravelomanana Shifing the phase transition 10 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Phase transition Shifing the phase transition Graphs with degree constraints Experimental results

Shifing the phase transition

m = 1 2n(1 + µn−1/3) ⇒ m = αn(1 + µn−1/3)

1 Achlioptas percolation process (α = 0.535?) 2 Degree sequence models (less detailed information) 3 Degree set constraint :: current talk

D., Ravelomanana Shifing the phase transition 10 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Phase transition Shifing the phase transition Graphs with degree constraints Experimental results

Example of graph with degree constraints

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Figure: Random labeled graph from G26,30,Ω with the set of degree constraints Ω = {1, 2, 3, 5, 7}.

D., Ravelomanana Shifing the phase transition 11 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Phase transition Shifing the phase transition Graphs with degree constraints Experimental results

Constant of phase transition

Ω — the set of degree constraints

1 Random graphs

m = 1 2n(1 + µn−1/3)

?

⇒ m = αn(1 + µn−1/3)

2 Random 2-CNF

m = 1 · n(1 + µn−1/3)

?

⇒ m = 2αn(1 + µn−1/3)

3 How to compute α depending on Ω?

D., Ravelomanana Shifing the phase transition 12 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Phase transition Shifing the phase transition Graphs with degree constraints Experimental results

Constant of phase transition

Ω — the set of degree constraints

1 Random graphs

m = 1 2n(1 + µn−1/3)

?

⇒ m = αn(1 + µn−1/3)

2 Random 2-CNF

m = 1 · n(1 + µn−1/3)

?

⇒ m = 2αn(1 + µn−1/3)

3 How to compute α depending on Ω?

D., Ravelomanana Shifing the phase transition 12 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Phase transition Shifing the phase transition Graphs with degree constraints Experimental results

Constant of phase transition

Ω — the set of degree constraints

1 Random graphs

m = 1 2n(1 + µn−1/3)

?

⇒ m = αn(1 + µn−1/3)

2 Random 2-CNF

m = 1 · n(1 + µn−1/3)

?

⇒ m = 2αn(1 + µn−1/3)

3 How to compute α depending on Ω?

D., Ravelomanana Shifing the phase transition 12 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Phase transition Shifing the phase transition Graphs with degree constraints Experimental results

Exponential generating function

1 Set of degree constraints. Ω = {1, 2, 3, 5, 7}. Can be infinite. 2 Exponential generating function connected to Ω

ω(z) =

• d∈Ω

zd d! = z1 1! + z2 2! + z3 3! + z5 5! + z7 7! .

3 Definition of the point α(Ω):

      

• z ω′′(

z) ω′( z) = 1,

• z ω′(

z) ω( z) = 2α (1)

D., Ravelomanana Shifing the phase transition 13 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Phase transition Shifing the phase transition Graphs with degree constraints Experimental results

Exponential generating function

1 Set of degree constraints. Ω = {1, 2, 3, 5, 7}. Can be infinite. 2 Exponential generating function connected to Ω

ω(z) =

• d∈Ω

zd d! = z1 1! + z2 2! + z3 3! + z5 5! + z7 7! .

3 Definition of the point α(Ω):

      

• z ω′′(

z) ω′( z) = 1,

• z ω′(

z) ω( z) = 2α (1)

D., Ravelomanana Shifing the phase transition 13 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Phase transition Shifing the phase transition Graphs with degree constraints Experimental results

Exponential generating function

1 Set of degree constraints. Ω = {1, 2, 3, 5, 7}. Can be infinite. 2 Exponential generating function connected to Ω

ω(z) =

• d∈Ω

zd d! = z1 1! + z2 2! + z3 3! + z5 5! + z7 7! .

3 Definition of the point α(Ω):

      

• z ω′′(

z) ω′( z) = 1,

• z ω′(

z) ω( z) = 2α (1)

D., Ravelomanana Shifing the phase transition 13 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Phase transition Shifing the phase transition Graphs with degree constraints Experimental results

Experimental results

(1/3)

D., Ravelomanana Shifing the phase transition 14 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Phase transition Shifing the phase transition Graphs with degree constraints Experimental results

Experimental results

(2/3)

D., Ravelomanana Shifing the phase transition 14 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Phase transition Shifing the phase transition Graphs with degree constraints Experimental results

Experimental results

(3/3)

D., Ravelomanana Shifing the phase transition 14 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Phase transition Shifing the phase transition Graphs with degree constraints Experimental results

Ipython session :: let’s compute the threshold point!

D., Ravelomanana Shifing the phase transition 15 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results 2-CNF formula and digraph model

Outline

1

2-SAT, phase transitions and degree constraints

2

Lower bound for 2-SAT

3

Saddle-point method and analytic lemma

4

Related results

D., Ravelomanana Shifing the phase transition 16 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results 2-CNF formula and digraph model

Precise statement of the theorem

Theorem Let Fn,m,Ω be random 2-CNF with Ω-degree constraints. n – number of variables m – number of clauses m = αn(1 + µn−1/3)

1 P(Fn,m,Ω is sat) ≥ 1 − O(|µ|−3) as µ → −∞, 2 P(Fn,m,Ω is sat) ≥ Θ(1) as |µ| = O(1), 3 P(Fn,m,Ω is sat) ≥ exp(−Θ(µ3)) as µ → +∞.

D., Ravelomanana Shifing the phase transition 17 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results 2-CNF formula and digraph model

2-CNF formula and digraph model

Digraph representation and sum-representation of a 2-sat formula (x1 ∨ x2)(x2 ∨ x3)(x2 ∨ x1)(x4 ∨ x3)(x4 ∨ x2)(x4 ∨ x4)

1 2 3 1 4 3 2 4 + 1 2 3 1 4 3 2 4 = 1 2 3 4 1 2 3 4

D., Ravelomanana Shifing the phase transition 18 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results 2-CNF formula and digraph model

Tools from random graphs

n – number of vertices m – number of edges Framework: m = αn, linear dependence.

1 m = (1 − ε)αn

← − only trees and unicycles

2 m = αn

← − complex components with positive probability

3 m = (1 + ε)αn

← − probability of fixed excess is exponentially small

D., Ravelomanana Shifing the phase transition 19 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results 2-CNF formula and digraph model

Structural theorem for random graphs

Theorem ( Regime: m = αn(1 + µn−1/3) )

1 if µ → −∞, |µ| = O(n1/12), then

P(Gn,m,Ω has only trees and unicycles) = 1 − Θ(|µ|−3) ;

2 if |µ| = O(1), i.e. µ is fixed, then

P(Gn,m,Ω has only trees and unicycles) → constant ∈ (0, 1) , P(Gn,m,Ω has a complex part with total excess q) → constant ∈ (0, 1) ,

3 if µ → +∞, |µ| = O(n1/12), then

P(Gn,m,Ω has only trees and unicycles) = Θ(e−µ3/6µ−3/4) , P(Gn,m,Ω has a complex part with excess q) = Θ(e−µ3/6µ3q/2−3/4) .

D., Ravelomanana Shifing the phase transition 20 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results 2-CNF formula and digraph model

Structural theorem for random graphs

Theorem ( Regime: m = αn(1 + µn−1/3) )

1 if µ → −∞, |µ| = O(n1/12), then

P(Gn,m,Ω has only trees and unicycles) = 1 − Θ(|µ|−3) ;

2 if |µ| = O(1), i.e. µ is fixed, then

P(Gn,m,Ω has only trees and unicycles) → constant ∈ (0, 1) , P(Gn,m,Ω has a complex part with total excess q) → constant ∈ (0, 1) ,

3 if µ → +∞, |µ| = O(n1/12), then

P(Gn,m,Ω has only trees and unicycles) = Θ(e−µ3/6µ−3/4) , P(Gn,m,Ω has a complex part with excess q) = Θ(e−µ3/6µ3q/2−3/4) .

D., Ravelomanana Shifing the phase transition 20 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results 2-CNF formula and digraph model

Structural theorem for random graphs

Theorem ( Regime: m = αn(1 + µn−1/3) )

1 if µ → −∞, |µ| = O(n1/12), then

P(Gn,m,Ω has only trees and unicycles) = 1 − Θ(|µ|−3) ;

2 if |µ| = O(1), i.e. µ is fixed, then

P(Gn,m,Ω has only trees and unicycles) → constant ∈ (0, 1) , P(Gn,m,Ω has a complex part with total excess q) → constant ∈ (0, 1) ,

3 if µ → +∞, |µ| = O(n1/12), then

P(Gn,m,Ω has only trees and unicycles) = Θ(e−µ3/6µ−3/4) , P(Gn,m,Ω has a complex part with excess q) = Θ(e−µ3/6µ3q/2−3/4) .

D., Ravelomanana Shifing the phase transition 20 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results 2-CNF formula and digraph model

Structural theorem for random graphs

Theorem ( Regime: m = αn(1 + µn−1/3) )

1 if µ → −∞, |µ| = O(n1/12), then

P(Gn,m,Ω has only trees and unicycles) = 1 − Θ(|µ|−3) ;

2 if |µ| = O(1), i.e. µ is fixed, then

P(Gn,m,Ω has only trees and unicycles) → constant ∈ (0, 1) , P(Gn,m,Ω has a complex part with total excess q) → constant ∈ (0, 1) ,

3 if µ → +∞, |µ| = O(n1/12), then

P(Gn,m,Ω has only trees and unicycles) = Θ(e−µ3/6µ−3/4) , P(Gn,m,Ω has a complex part with excess q) = Θ(e−µ3/6µ3q/2−3/4) .

D., Ravelomanana Shifing the phase transition 20 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

Outline

1

2-SAT, phase transitions and degree constraints

2

Lower bound for 2-SAT

3

Saddle-point method and analytic lemma

4

Related results

D., Ravelomanana Shifing the phase transition 21 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

Trees with degree constraints

Rooted case

T0               T1

• Ω

Ω−1 Ω−1 Ω−1 Ω − k

def

= {d : d + k ∈ Ω} Example: Ω = {0, 1, 3, 6} Ω − 1 = {0, 2, 5}.

D., Ravelomanana Shifing the phase transition 22 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

Trees with degree constraints

Rooted case

T0               T1

• Ω

Ω−1 Ω−1 Ω−1        ω(z) =

d∈Ω

zd d! = zd1 d1! + zd2 d2! + . . . , ω′(z) =

d∈Ω

zd−1 (d − 1)! =

d∈Ω−1

zd d! ,      T0(z) = zω(T1(z)), T1(z) = zω′(T1(z)), T2(z) = zω′′(T1(z)).

D., Ravelomanana Shifing the phase transition 22 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

Trees with degree constraints

Unrooted case

A variant of dissymmetry theorem: (Ω) =

• +
• 1
• 1

+

• 1

T0(z)

↑ root deg. 0

= T1(z)2 2

↑ root deg. 1

+ U(z)

↑ unrooted

⇔ U(z) = T0(z) − T1(z)2 2

D., Ravelomanana Shifing the phase transition 23 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

Unicycles with degree constraints

≥ 3 V(z)

↑ unicycles

= 1 2    log

↑ cycle

1 1 − T2(z) − T2(z) − T2(z)2 2    

D., Ravelomanana Shifing the phase transition 24 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

2-core (the core) and 3-core (the kernel)

D., Ravelomanana Shifing the phase transition 25 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

2-core (the core) and 3-core (the kernel)

D., Ravelomanana Shifing the phase transition 25 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

2-core (the core) and 3-core (the kernel)

D., Ravelomanana Shifing the phase transition 25 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

2-core (the core) and 3-core (the kernel)

D., Ravelomanana Shifing the phase transition 25 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

2-core (the core) and 3-core (the kernel)

D., Ravelomanana Shifing the phase transition 25 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

2-core (the core) and 3-core (the kernel)

D., Ravelomanana Shifing the phase transition 25 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

2-core (the core) and 3-core (the kernel)

D., Ravelomanana Shifing the phase transition 25 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

2-core (the core) and 3-core (the kernel)

D., Ravelomanana Shifing the phase transition 25 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

2-core (the core) and 3-core (the kernel)

D., Ravelomanana Shifing the phase transition 25 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

2-core (the core) and 3-core (the kernel)

D., Ravelomanana Shifing the phase transition 25 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

2-core (the core) and 3-core (the kernel)

D., Ravelomanana Shifing the phase transition 25 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

2-core (the core) and 3-core (the kernel)

D., Ravelomanana Shifing the phase transition 25 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

2-core (the core) and 3-core (the kernel)

D., Ravelomanana Shifing the phase transition 25 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

2-core (the core) and 3-core (the kernel)

D., Ravelomanana Shifing the phase transition 25 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

2-core (the core) and 3-core (the kernel)

2-core of a graph

D., Ravelomanana Shifing the phase transition 25 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

2-core (the core) and 3-core (the kernel)

3-core of a graph

D., Ravelomanana Shifing the phase transition 25 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

Notion of excess

2 1 3 4 −1 4 1 3 2 1 3 4 6 5 2 1 1 2 4 3 2 Excess

def

= # edges - # vertices

D., Ravelomanana Shifing the phase transition 26 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

Kernel of a graph

Example: graphs with excess 1

1

1 4

1 2

1 4

1 2

1 6

All possible 3-core multigraphs and their compensation factors. EGF for all connected bicyclic graphs (Ω = Z≥0): W(z) = 1 4 T(z)5 (1 − T(z))2 + 1 4 T(z)6 (1 − T(z))3 + 1 6 T(z)2[3T(z)2 − 2T 3(z)] (1 − T(z))3

• inclusion-exclusion

W(z) ∼ 5 24 · 1 (1 − T(z))3 near z = e−1

D., Ravelomanana Shifing the phase transition 27 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

Kernel of a graph

Example: graphs with excess 1

1

1 4

1 2

1 4

1 2

1 6

All possible 3-core multigraphs and their compensation factors. EGF for all connected bicyclic graphs (arbitrary Ω): WΩ(z) = 1 4 T4(z)T2(z)4 (1 − T2(z))2 +1 4 T3(z)2T2(z)4 (1 − T2(z))3 +1 6 T3(z)2[3T2(z)2 − 2T2(z)3] (1 − T2(z))3

• inclusion-exclusion

WΩ(z) ∼ (???) · T3(z)2(???) (1 − T2(z))3

D., Ravelomanana Shifing the phase transition 27 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

Role of cubic graphs

EGF for all (not necessary connected) complex multigraphs with excess r, WΩ,r(z) ∼ er0 T3(z)2r (1 − T2(z))3r

↑ comes from cubic graphs

, er0 = (6r)! 25r32r(3r)!(2r)!

↑ can be shown combinatorially

.

D., Ravelomanana Shifing the phase transition 28 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

Local summary

1 EGF for unrooted trees with degree constraints 2 EGF for unicycles with degree constraints 3 EGF for graphs of fixed excess (main asymptotics)

D., Ravelomanana Shifing the phase transition 29 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

Local summary

1 EGF for unrooted trees with degree constraints 2 EGF for unicycles with degree constraints 3 EGF for graphs of fixed excess (main asymptotics)

D., Ravelomanana Shifing the phase transition 29 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

Local summary

1 EGF for unrooted trees with degree constraints 2 EGF for unicycles with degree constraints 3 EGF for graphs of fixed excess (main asymptotics)

D., Ravelomanana Shifing the phase transition 29 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

Desired probability

Subcritical phase

P (graph g ∈ G(n, m, Ω) consists only of trees and unicycles) = # graphs from G(n, m, Ω) whose components are trees and unicycles # graphs from G(n, m, Ω)

D., Ravelomanana Shifing the phase transition 30 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

Number of graphs with degree constraints

1 Ω = Z≥0. Stirling approximation:

n! (n − m)! (n

2)

m

∼ √ 4πnα · 2mnnmm n2m(n − m)n−m × exp

• − n + m

n + m2 n2

• 3/4
• 2 Arbitrary Ω ([de Panafieu, Ramos ’16])

n! (n − m)!|Gn,m,Ω| ∼ √ 4πnα p · 2mnnmm n2m(n − m)n−m × exp

• − n log ω(

z) + 2m log z + 1 2φ0( z) + 1 4φ2

0(

z)

• 3/4
• D., Ravelomanana

Shifing the phase transition 31 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

Number of graphs with degree constraints

1 Ω = Z≥0. Stirling approximation:

n! (n − m)! (n

2)

m

∼ √ 4πnα · 2mnnmm n2m(n − m)n−m × exp

• − n + m

n + m2 n2

• 3/4
• 2 Arbitrary Ω ([de Panafieu, Ramos ’16])

n! (n − m)!|Gn,m,Ω| ∼ √ 4πnα p · 2mnnmm n2m(n − m)n−m × exp

• − n log ω(

z) + 2m log z + 1 2φ0( z) + 1 4φ2

0(

z)

• 3/4
• D., Ravelomanana

Shifing the phase transition 31 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

Number of graphs with degree constraints

1 Ω = Z≥0. Stirling approximation:

n! (n − m)! (n

2)

m

∼ √ 4πnα · 2mnnmm n2m(n − m)n−m × exp

• − n + m

n + m2 n2

• 3/4
• 2 Arbitrary Ω ([de Panafieu, Ramos ’16])

n! (n − m)!|Gn,m,Ω| ∼ √ 4πnα p · 2mnnmm n2m(n − m)n−m × exp

• − n log ω(

z) + 2m log z + 1 2φ0( z) + 1 4φ2

0(

z)

• 3/4
• D., Ravelomanana

Shifing the phase transition 31 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

Number of graphs with degree constraints

1 Ω = Z≥0. Stirling approximation:

n! (n − m)! (n

2)

m

∼ √ 4πnα · 2mnnmm n2m(n − m)n−m × exp

• − n + m

n + m2 n2

• 3/4
• 2 Arbitrary Ω ([de Panafieu, Ramos ’16])

n! (n − m)!|Gn,m,Ω| ∼ √ 4πnα p · 2mnnmm n2m(n − m)n−m × exp

• − n log ω(

z) + 2m log z + 1 2φ0( z) + 1 4φ2

0(

z)

• 3/4
• D., Ravelomanana

Shifing the phase transition 31 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

Number of graphs with degree constraints

1 Ω = Z≥0. Stirling approximation:

n! (n − m)! (n

2)

m

∼ √ 4πnα · 2mnnmm n2m(n − m)n−m × exp

• − n + m

n + m2 n2

• 3/4
• 2 Arbitrary Ω ([de Panafieu, Ramos ’16])

n! (n − m)!|Gn,m,Ω| ∼ √ 4πnα p · 2mnnmm n2m(n − m)n−m × exp

• − n log ω(

z) + 2m log z + 1 2φ0( z) + 1 4φ2

0(

z)

• 3/4
• D., Ravelomanana

Shifing the phase transition 31 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

Number of graphs with degree constraints

1 Ω = Z≥0. Stirling approximation:

n! (n − m)! (n

2)

m

∼ √ 4πnα · 2mnnmm n2m(n − m)n−m × exp

• − n + m

n + m2 n2

• 3/4
• 2 Arbitrary Ω ([de Panafieu, Ramos ’16])

n! (n − m)!|Gn,m,Ω| ∼ √ 4πnα p · 2mnnmm n2m(n − m)n−m × exp

• − n log ω(

z) + 2m log z + 1 2φ0( z) + 1 4φ2

0(

z)

• 3/4
• D., Ravelomanana

Shifing the phase transition 31 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

Example: contour integral for subcritical phase

n! |Gn,m,Ω| 1 2πi U(z)n−m (n − m)!

↑ trees

eV(z)

↑ unicycles

dz zn+1 = 1 − O(µ−3) near the critical point m = αn:        2α = φ0( z)

def

= z ω′( z) ω( z) , 1 = φ1( z)

def

= z ω′′( z) ω′( z) .

D., Ravelomanana Shifing the phase transition 32 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

Contour integral: Erdős–Rényi case

Picture from Flajolet’s book

D., Ravelomanana Shifing the phase transition 33 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Generating functions of ingredients Desired probability Contour integrals

Contour integral: graphs with degree constraints

Not a single mountain but dangerous expedition!

D., Ravelomanana Shifing the phase transition 34 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Diameter, circumference and longest path Planarity

Outline

1

2-SAT, phase transitions and degree constraints

2

Lower bound for 2-SAT

3

Saddle-point method and analytic lemma

4

Related results

D., Ravelomanana Shifing the phase transition 35 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Diameter, circumference and longest path Planarity

Diameter, circumference and longest path of complex component

All of order Θ(n1/3)

D., Ravelomanana Shifing the phase transition 36 / 40

2-SAT, phase transitions and degree constraints Lower bound for 2-SAT Saddle-point method and analytic lemma Related results Diameter, circumference and longest path Planarity

Planarity

Let p(µ) be the probability that Gn,m,Ω is planar.

1 p(µ) = 1 − Θ(|µ|−3), as µ → −∞; 2 p(µ) → constant ∈ (0, 1), as |µ| = O(1), and p(µ) is

computable;

3 p(µ) → 0, as µ → +∞.

D., Ravelomanana Shifing the phase transition 37 / 40

Summary Open problems

Summary

1 Analytic description of phase transition in model with degree

constraints

2

1 2 proof of 2-SAT phase transition

3 Study of distribution of parameters.

D., Ravelomanana Shifing the phase transition 38 / 40

Summary Open problems

Summary

1 Analytic description of phase transition in model with degree

constraints

2

1 2 proof of 2-SAT phase transition

3 Study of distribution of parameters.

D., Ravelomanana Shifing the phase transition 38 / 40

Summary Open problems

Summary

1 Analytic description of phase transition in model with degree

constraints

2

1 2 proof of 2-SAT phase transition

3 Study of distribution of parameters.

D., Ravelomanana Shifing the phase transition 38 / 40

Summary Open problems

Open problems

1 The case 1 /

∈ Ω.

2 Upper bound for 2-SAT. 3 Size of the largest component.

D., Ravelomanana Shifing the phase transition 39 / 40

Summary Open problems

Open problems

1 The case 1 /

∈ Ω.

2 Upper bound for 2-SAT. 3 Size of the largest component.

D., Ravelomanana Shifing the phase transition 39 / 40

Summary Open problems

Open problems

1 The case 1 /

∈ Ω.

2 Upper bound for 2-SAT. 3 Size of the largest component.

D., Ravelomanana Shifing the phase transition 39 / 40

Summary Open problems

That’s all!

Thank you for your atention. Good flight back home.

D., Ravelomanana Shifing the phase transition 40 / 40