Phase Transitions in Random Discrete Structures Mihyun Kang - - PowerPoint PPT Presentation

Phase Transitions in Random Discrete Structures

Mihyun Kang Institute of Optimization and Discrete Mathematics Graz University of Technology

Phase Transition in Computer Science

Random k-SAT problem

To decide whether or not a random k-CNF formula Fk(n, m) with n variables and m clauses is satisfiable. Phase transition from satisfiability to unsatisfiability of Fk(n, m): m n ∼ 2kln 2 Computational time required to find a satisfying truth assignment increases drastically: m n ∼ 2k k

Mihyun Kang Phase Transitions in Random Discrete Structures

Phase Transition in Statistical Physics

Ising model (mathematical model of ferromagnetism)

(up or down) Spins are arranged in lattice which interact with nearest neighbours

Mihyun Kang Phase Transitions in Random Discrete Structures

Phase Transition in Statistical Physics

Ising model (mathematical model of ferromagnetism)

(up or down) Spins are arranged in lattice which interact with nearest neighbours

Ordered phase at low temperatures Disordered phase at high temperatures

Mihyun Kang Phase Transitions in Random Discrete Structures

Percolation in Geography, Materials Science and Physics

the passage of fluid or gas going through porous or disordered media

Mihyun Kang Phase Transitions in Random Discrete Structures

Mathematical Models of Percolation

Bond percolation: each bond (or edge) is either open with prob. p

• r closed with prob. 1 − p, independently

Site percolation: each site (or vertex) is either occupied with prob. p

• r empty with prob. 1 − p, independently

p < pc p > pc Bond Percolation on Square Lattice Site Percolation on Hexagonal Lattice

Mihyun Kang Phase Transitions in Random Discrete Structures

Percolation on Complete Graph Kn

Binomial random graph G(n, p)

each edge of the complete graph Kn is open with probability p, independently of each other

• cf. G(n, m): a graph sampled uniformly at random among all graphs
• n n vertices and m edges

Alfréd Rényi (1921 – 1970) Paul Erd˝

• s (1913 – 1996)

Mihyun Kang Phase Transitions in Random Discrete Structures

Phase Transition in Random Graphs

• I. Binomial Random Graph G(n, p)

⊲ Galton-Watson Tree

• II. Random Planar Graphs

⊲ Internal Structure – Kernel

• III. Random Graph Processes

⊲ Differential Equations Method

Mihyun Kang Phase Transitions in Random Discrete Structures

• I. Binomial Random Graph G(n, p)

Cycle threshold

P [G(n, p) contains a cycle ] →

• if

p ≪ 1

n

1 if p ≫ 1

n p cycles empty complete connected p=1 p=1/n p=0 p=log n /n

Threshold Evolution of G(n, p)

Mihyun Kang Phase Transitions in Random Discrete Structures

Phase Transition of Largest Component

Binomial random graph G(n, p)

[ ERD ˝

OS–RÉNYI 60 ]

Let p = t/n for a constant t > 0. If t < 1, with probability tending to 1 as n → ∞ (whp) all the components have O(log n) vertices. If t > 1, whp there is a unique largest component of order Θ(n), while every other component has O(log n) vertices. ⊲ Component exposure: Breath-First Search & Galton-Watson Tree [ KARP 90 ]

Mihyun Kang Phase Transitions in Random Discrete Structures

Galton-Watson Tree

Branching Process

The number of children is given by i.i.d. random variable ∼ Po(t). If t < 1, the process dies out with probability 1. If t > 1, with positive probability ρ the process continues forever.

„small” component in G(n, p) „giant” component of size ρn + o(n) „small” component in G(n, p) in G(n, p) where 1 − ρ = e−tρ

Mihyun Kang Phase Transitions in Random Discrete Structures

Extinction Probability of Galton-Watson process

Let T be the total number of nodes created in the process. Suppose t > 1. Consider the probability generating function q(z) :=

• i<∞ P(T = i)zi.

It satisfies q(z) = z

• k

P(Po(t) = k)q(z)k = z

• k

e−t tk k!q(z)k = zet(q(z)−1).

k

The extinction probability q(1) =

i<∞ P(T = i) satisfies q(1) = et(q(1)−1).

Since q(1) = 1 − ρ we have 1 − ρ = e−tρ.

Mihyun Kang Phase Transitions in Random Discrete Structures

Largest Component in G(n, p)

Let L(n) be the number of vertices in the largest component in G(n, p).

[ PITTEL–WORMALD 05; BEHRISCH–COJA-OGHLAN–K. 09; BOLLOBAS AND RIORDAN 12+]

Local Limit Theorem

Let p = t/n with t > 1. Then E(L(n)) = ρn and σ2 := V(L(n)) = (ρ(1 − ρ)/(1 − t(1 − ρ))2)n. For any integer k with k = ρn + x where x = O(√n ) = O(σ) P(L(n) = k) ∼ 1 σ √ 2π exp

• − x2

2σ2

• .

⌊ρn − t√n⌋ ⌊ρn + t√n⌋ ⌊ρn⌋ t√n t√n

Mihyun Kang Phase Transitions in Random Discrete Structures

Critical Phase

How big is the largest component in G(n, p), when pn = 1 + ε for ε = o(1) ?

Béla Bollobás Tomasz Łuczak

Mihyun Kang Phase Transitions in Random Discrete Structures

Critical Phase

How big is the largest component in G(n, p), when pn = 1 + ε for ε = o(1) ?

[ BOLLOBÁS 84; ŁUCZAK 90; JANSON–KNUTH–ŁUCZAK–PITTEL 93; BOLLOBÁS–RIORDAN 13+]

If ε n1/3 → −∞, whp L(n) = o(n2/3). If ε n1/3 → λ, a constant, whp L(n) = Θ(n2/3). If ε n1/3 → ∞, whp L(n) = (1 + o(1)) 2εn.

2/3 << 2/3 2/3

~

>>

n n n

⊲ Uniform random graph G(n, m): m = n/2 + s, s n−2/3 = ε n1/3

Mihyun Kang Phase Transitions in Random Discrete Structures

• II. Random Planar Graphs

Planar graphs

A planar graph is a graph that can be embedded in the plane (without crossing edges).

5

K K3,3 non−planar Random planar graphs

[ FRIEZE 87; MCDIARMID–STEGER–WELSH 05 ]

Let P(n, m) be a uniform random planar graph with n vertices and m edges.

Mihyun Kang Phase Transitions in Random Discrete Structures

Phase Transition in Random Planar Graphs

Let L(n) denote the number of vertices in the largest component in P(n, m).

Two critical periods

[ K.– ŁUCZAK 12 ]

Let m = n/2 + s. If s n−2/3 → −∞, whp L(n) ≪ n2/3. If s n−2/3 → ∞, whp L(n) = (2 + o(1))s ≫ n2/3. Let m = n + r. If r n−3/5 → −∞, whp n − L(n) ≫ n3/5. If r n−3/5 → ∞, whp n − L(n) = Θ(n3/2r −3/2) ≪ n3/5.

2 s

~

3/5

n

<<

n−L(n) L(n) Mihyun Kang Phase Transitions in Random Discrete Structures

Random Planar Graphs

Look into internal structure

complex com.

• unicyc. com.

trees

• Mihyun Kang

Phase Transitions in Random Discrete Structures

Random Planar Graphs

Look into internal structure

complex com.

• unicyc. com.

trees

⇒ Kernel of complex components

[ BOLLOBÁS 84; ŁUCZAK 90 ]

• Mihyun Kang

Phase Transitions in Random Discrete Structures

Random Planar Graphs

Look into internal structure

complex com.

• unicyc. com.

trees

⇒ Kernel of complex components

[ BOLLOBÁS 84; ŁUCZAK 90 ]

• Mihyun Kang

Phase Transitions in Random Discrete Structures

Random Planar Graphs

Look into internal structure

complex com.

• unicyc. com.

trees

⇒ Kernel of complex components

[ BOLLOBÁS 84; ŁUCZAK 90 ]

• Mihyun Kang

Phase Transitions in Random Discrete Structures

Random Planar Graphs

Look into internal structure

complex com.

• unicyc. com.

trees

⇒ Kernel of complex components

[ BOLLOBÁS 84; ŁUCZAK 90 ]

• Mihyun Kang

Phase Transitions in Random Discrete Structures

Random Planar Graphs

Look into internal structure

complex com.

• unicyc. com.

trees

⇒ Kernel of complex components

[ BOLLOBÁS 84; ŁUCZAK 90 ]

• Mihyun Kang

Phase Transitions in Random Discrete Structures

Random Planar Graphs

Look into internal structure

complex com.

• unicyc. com.

trees

⇒ Kernel of complex components

[ BOLLOBÁS 84; ŁUCZAK 90 ]

• Mihyun Kang

Phase Transitions in Random Discrete Structures

Random Planar Graphs

Look into internal structure

complex com.

• unicyc. com.

trees

⇒ Kernel of complex components

[ BOLLOBÁS 84; ŁUCZAK 90 ]

• Mihyun Kang

Phase Transitions in Random Discrete Structures

Random Planar Graphs

Look into internal structure

complex com.

• unicyc. com.

trees Typical kernel

[ BODIRSKY–K.–LÖFFLER–MCDIARMID 07; K.– ŁUCZAK 12 ]

⊲ Cubic planar weighted multigraphs ⊲ Tutte’s decomposition & singularity analysis of generating functions

• Mihyun Kang

Phase Transitions in Random Discrete Structures

Asymptotic Number of ‘Sparse’ Planar Graphs

Let p(n, m) be the number of planar graphs with n vertices and m edges.

When m = n/2 + s, s = o(n)

[ K.– ŁUCZAK 12 ]

p(n, n/2 + s) ∼

1 √eπ nn+2s (n + 2s)−n/2−s−1/2 en/2+s

if s n−2/3 → −∞ ∼ Θ(1) nn+2s (n + 2s)−n/2−s−1/2 en/2+s if s n−2/3 → λ ∼ α nn+11/6 s−7/2 (n − 2s)−n/2+s en/2−s+β s n−2/3 if s n−2/3 → ∞

Recursive method for random sampling

Generate a typical kernel recursively Relpace edges of the kernel by paths Plant rooted forest

Mihyun Kang Phase Transitions in Random Discrete Structures

• III. Random Graph Processes

Achlioptas process, Bohman-Frieze process: power of two choices

Achlioptas Bohman Frieze

In each step, two random edges are present

if the first edge would join two isolated vertices, it is added to a graph

• therwise the second edge is added

⊲ it delays the appearance of the giant component

[ BOHMAN-FRIEZE 01 ] Mihyun Kang Phase Transitions in Random Discrete Structures

Bohman-Frieze Process

Phase transition

[ SPENCER–WORMALD 07; JANSON–SPENCER 12; RIORDAN–WARNKE 12 ]

Susceptibility (= average component size): let t = 2 # edges /n. S(t) = 1 n

• 1≤i≤n |C(vi)| = 1

n

• 1≤k≤n k Xk(t, n).

Here Xk(t, n) is the number of vertices in components of size k at time t.

Janson Spencer Wormald Mihyun Kang Phase Transitions in Random Discrete Structures

Bohman-Frieze Process

Phase transition

[ SPENCER–WORMALD 07; JANSON–SPENCER 12; RIORDAN–WARNKE 12 ]

Susceptibility (= average component size): let t = 2 # edges /n. S(t) = 1 n

• 1≤i≤n |C(vi)| = 1

n

• 1≤k≤n k Xk(t, n).

Here Xk(t, n) is the number of vertices in components of size k at time t. Differential equations method: ∃ a deterministic function xk(t) s.t. whp Xk(t, n) n ∼ xk(t) Smoluchowski coagulation equation: x′

1(t) = −x1(t) − x2 1 (t) + x3 1 (t)

x′

2(t) = 2x2 1(t) − x4 1 (t) − 2(1 − x2 1 (t))x2(t)

x′

i (t)= −i(1 − x2 1 (t))xi(t) + i

2(1 − x2

1 (t))

• k<i xk(t)xi−k(t)

Mihyun Kang Phase Transitions in Random Discrete Structures

Critical Phase in Bohman-Frieze Process

Small components

[ K.–PERKINS–SPENCER 13 ]

Let tc be the critical point of the phase transition and t = tc ± ǫ for ǫ small. Vertices in components of size k at time t: ∃ constants a, b > 0 s.t. xk(t) ∼ a k −3/2 exp

• − ǫ2k b
• .

Quasi-linear partial differential equation

[ K.–PERKINS–SPENCER 13 ]

Susceptibility (= average component size): S(t) ∼

k≥1 k xk(t)

The moment generating function G(t, z) =

k≥1 xk(t) zk satisfies

∂G(t, z) ∂t − z(1 − x1(t)2)(G(t, z) − 1)∂G(t, z) ∂z = z(z − 1)x1(t)2.

Mihyun Kang Phase Transitions in Random Discrete Structures

Concluding Remarks

Ubiquitous phase transitions

Computer science Statistical physics Percolation theory Random graphs

Universal behaviour in random graphs

Phase transition with same critical exponents Small components containing a random vertex vs rooted trees ⊲ Probabilistic and analytic combinatorics

Mihyun Kang Phase Transitions in Random Discrete Structures

Call for Participation

Fall School “Phase Transitions in Random Discrete Structures”

When: 2-20 September 2013 Where: TU Graz, Austria Lecturers: Amin Coja-Oghlan (University of Frankfurt) Konstantinos Panagiotou (University of Munich) More info: http://www.math.tugraz.at/discrete/fallschool Application deadline: 5 July 2013 Supported by European Science Foundation and Austrian Science Fund

Mihyun Kang Phase Transitions in Random Discrete Structures