The Erd os-R enyi Process Phase Transition Part I: The Coarse - PowerPoint PPT Presentation

The Erd˝ os-R´ enyi Process Phase Transition Part I: The Coarse Scaling Random Graph Processes Austin Joel Spencer May 10,2016

Working with Paul Erd˝ os was like taking a walk in the hills. Every time when I thought that we had achieved our goal and deserved a rest, Paul pointed to the top of another hill and off we would go. – Fan Chung

The Erd˝ os-R´ enyi Processes Begin with empty graph on n vertices. Each round add one randomly chosen edge enyi Time: n Erd˝ os-R´ 2 rounds is t = 1 PHASE TRANSITION at t c = 1 Modern: G ( n , p ) with p = t n .

The Erd˝ os-R´ enyi Phase Transition Subcritical t < t c = 1 | C 1 | = O (ln n ) All C simple 1 1 Simple = Tree or Unicylic

The Erd˝ os-R´ enyi Phase Transition Supercritical t > t c = 1 Subcritical GIANT COMPONENT t < t c = 1 | C 1 | = Θ( n ) | C 1 | = O (ln n ) Complex (= Not Simple) All C simple 1 All other C simple All other | C | = O (ln n ) 1 Simple = Tree or Unicylic

Phase Transition Near Criticality Caution: Double Limits! Barely Subcritical t = 1 − ǫ | C 1 | = O ( ǫ − 2 ln n ) All C simple

Phase Transition Near Criticality Caution: Double Limits! Barely Supercritical t > 1 + ǫ Barely Subcritical GIANT COMPONENT t = 1 − ǫ | C 1 | ∼ 2 ǫ n | C 1 | = O ( ǫ − 2 ln n ) Complex (= Not Simple) All C simple All other C simple | C 2 | = O ( ǫ − 2 ln n )

Galton-Watson Birth Process Begin with Eve Eve has Poisson mean λ children All children same. Final tree T . Subcritical λ < λ c = 1 T finite

Galton-Watson Birth Process Begin with Eve Eve has Poisson mean λ children All children same. Final tree T . Supercritical Subcritical λ > λ c = 1 λ < λ c = 1 INFINITE TREE T finite Pr[ T = ∞ ] > 1

Galton-Watson Near Criticality λ = λ c ± ǫ = 1 ± ǫ Barely Subcritical λ = 1 − ǫ | T | heavy tail until Θ( ǫ − 2 ) Then exponential decay

Galton-Watson Near Criticality λ = λ c ± ǫ = 1 ± ǫ Barely Supercritical Barely Subcritical λ = 1 + ǫ λ = 1 − ǫ Pr[ T = ∞ ] ∼ 2 ǫ | T | heavy tail until Θ( ǫ − 2 ) Duality Then exponential decay T finite like 1 − ǫ

Galton-Watson Near Criticality λ = λ c ± ǫ = 1 ± ǫ Barely Supercritical Barely Subcritical λ = 1 + ǫ λ = 1 − ǫ Pr[ T = ∞ ] ∼ 2 ǫ | T | heavy tail until Θ( ǫ − 2 ) Duality Then exponential decay T finite like 1 − ǫ GW λ roughly | C | at time t = λ

A Useful Non-Rigorous Argument Erd˝ os-R´ enyi Process. When C , C ′ merge, S ← S + 2 n | C | · | C ′ | C � = C ′ | C | | C ′ | S ( t + 2 n ) − S ( t ) = 2 n | C | · | C ′ | � n n

A Useful Non-Rigorous Argument Erd˝ os-R´ enyi Process. When C , C ′ merge, S ← S + 2 n | C | · | C ′ | C � = C ′ | C | | C ′ | S ( t + 2 n ) − S ( t ) = 2 n | C | · | C ′ | � n n C , C ′ n − 2 | C | 2 · | C ′ | 2 = 2 ∼ 2 n S 2 ( t ) � n

A Useful Non-Rigorous Argument Erd˝ os-R´ enyi Process. When C , C ′ merge, S ← S + 2 n | C | · | C ′ | | C ′ | C � = C ′ | C | S ( t + 2 n ) − S ( t ) = 2 n | C | · | C ′ | � n n C , C ′ n − 2 | C | 2 · | C ′ | 2 = 2 ∼ 2 n S 2 ( t ) � n S ′ ( t ) = S 2 ( t ), S (0) = 1 S ( t ) = (1 − t ) − 1

A Useful Non-Rigorous Argument Erd˝ os-R´ enyi Process. When C , C ′ merge, S ← S + 2 n | C | · | C ′ | | C ′ | C � = C ′ | C | S ( t + 2 n ) − S ( t ) = 2 n | C | · | C ′ | � n n C , C ′ n − 2 | C | 2 · | C ′ | 2 = 2 ∼ 2 n S 2 ( t ) � n S ′ ( t ) = S 2 ( t ), S (0) = 1 S ( t ) = (1 − t ) − 1 Critical t c = 1 when S ( t ) → ∞

Fictitious Continutation X 1 , X 2 , . . . mutually independent, X i ∼ Pois ( λ ) i -th node has X i children and dies Y t = number of living children, Y 0 = 1, Y t = Y t − 1 + X t − 1 Example: 2 , 1 , 0 , 1 , 0 , 2 , . . . Alanna has Brenda and Colleen ( X 1 = 2, Y 1 = 2)

Fictitious Continutation X 1 , X 2 , . . . mutually independent, X i ∼ Pois ( λ ) i -th node has X i children and dies Y t = number of living children, Y 0 = 1, Y t = Y t − 1 + X t − 1 Example: 2 , 1 , 0 , 1 , 0 , 2 , . . . Alanna has Brenda and Colleen ( X 1 = 2, Y 1 = 2) Brenda has Deidra( X 2 = 1, Y 2 = 2)

Fictitious Continutation X 1 , X 2 , . . . mutually independent, X i ∼ Pois ( λ ) i -th node has X i children and dies Y t = number of living children, Y 0 = 1, Y t = Y t − 1 + X t − 1 Example: 2 , 1 , 0 , 1 , 0 , 2 , . . . Alanna has Brenda and Colleen ( X 1 = 2, Y 1 = 2) Brenda has Deidra( X 2 = 1, Y 2 = 2) Colleen has no children ( X 3 = 0, Y 3 = 1)

Fictitious Continutation X 1 , X 2 , . . . mutually independent, X i ∼ Pois ( λ ) i -th node has X i children and dies Y t = number of living children, Y 0 = 1, Y t = Y t − 1 + X t − 1 Example: 2 , 1 , 0 , 1 , 0 , 2 , . . . Alanna has Brenda and Colleen ( X 1 = 2, Y 1 = 2) Brenda has Deidra( X 2 = 1, Y 2 = 2) Colleen has no children ( X 3 = 0, Y 3 = 1) Deidra has Erin ( X 4 = 1, Y 4 = 1)

Fictitious Continutation X 1 , X 2 , . . . mutually independent, X i ∼ Pois ( λ ) i -th node has X i children and dies Y t = number of living children, Y 0 = 1, Y t = Y t − 1 + X t − 1 Example: 2 , 1 , 0 , 1 , 0 , 2 , . . . Alanna has Brenda and Colleen ( X 1 = 2, Y 1 = 2) Brenda has Deidra( X 2 = 1, Y 2 = 2) Colleen has no children ( X 3 = 0, Y 3 = 1) Deidra has Erin ( X 4 = 1, Y 4 = 1) Erin has no children ( X 5 = 0, Y 5 = 0) T = 5

Fictitious Continutation X 1 , X 2 , . . . mutually independent, X i ∼ Pois ( λ ) i -th node has X i children and dies Y t = number of living children, Y 0 = 1, Y t = Y t − 1 + X t − 1 Example: 2 , 1 , 0 , 1 , 0 , 2 , . . . Alanna has Brenda and Colleen ( X 1 = 2, Y 1 = 2) Brenda has Deidra( X 2 = 1, Y 2 = 2) Colleen has no children ( X 3 = 0, Y 3 = 1) Deidra has Erin ( X 4 = 1, Y 4 = 1) Erin has no children ( X 5 = 0, Y 5 = 0) T = 5 Fictitous Continuation (convenient!) Fiona (no parent) has two children ( X 6 = 2, Y 6 = 1)) Never Ends. T = min t with X t = 0 (or T = ∞ ) History ( X 1 , . . . , X t ).

The Queue Queue size Y 0 = 1; Y t = Y t − 1 + X t − 1 Tree size T = T λ : minimal t , Y t = 0. (Maybe T = ∞ .) Theorem: (Proof later!) Pr[ T λ = k ] = e − λ k ( λ k ) k − 1 k ! Critical: Pr[ T 1 = k ] ∼ (2 π ) − 1 / 2 k − 3 / 2 . Heavy Tail. E [ T 1 ] = ∞ . Comparing: Pr[ T λ = k ] = Pr[ T 1 = k ] λ − 1 ( λ e 1 − λ ) k NonCritical: λ e 1 − λ < 1. Exponential tail.

Immortality x = Pr[ T = ∞ ] The Amazing Property: If Po ( λ ) children, each type σ with probability p σ – equivalently Po ( λ x σ ) children of type σ , independently. Infinite iff at least one child has infinite tree. x = Pr[ Po ( λ x ) � = 0] = 1 − e − λ x Subcritical. λ < 1. x = Pr[ T = ∞ ] = 0. Critical. λ = 1. x = Pr[ T = ∞ ] = 0, E [ T ] = ∞ . SuperCritical. λ > 1. x = Pr[ T = ∞ ] is positive solution to equation.

Creating a Component Initial t = 0: Queue Y 0 = 1; Neutral N 0 = n − 1. BFS finds X t new vertices and adds them to queue X t = BIN [ N t − 1 , p ]; Y t = Y t − 1 + X t − 1; N t = N t − N t − 1 − X t Fictional Continuation T = min t with Y t = 0. (Always T ≤ n .) Component C ( v ) has size T . N t ∼ BIN [ n − 1 , (1 − p ) t ] (BFS backwards ) History ( X 1 , . . . , X T )

Graphs Components & Galton-Watson T GR := size of C ( v ) in G ( n , λ n ) T PO := size of tree in Galton-Watson process Poisson Property: For any constants c , k n →∞ Pr[ BIN [ n − c , λ lim n ] = k ] = Pr[ Po ( λ ) = k ] Theorem: For any possible history H = ( x 1 , . . . , x t ) the limit, as n → ∞ of the probability of history H in the graph process is the probability of history H is the Galton-Watson process. Corollary: For any fixed λ, k n →∞ Pr[ T GR = k ] = Pr[ T PO = k ] lim

An Unusual Proof Theorem: Pr[ T λ = k ] = e − λ k ( λ k ) k − 1 k ! Proof: In G ( n , p ) with p = λ n � n � (1 − p ) k ( n − k ) Pr[ G ( k , p ) connected] Pr[ | C ( v ) | = k ] = k − 1 For k fixed, p → 0 Pr[ G ( k , p ) connected] ∼ k k − 2 p k − 1 via Cayley’s Theorem. (1 − p ) k ( n − k ) k k − 2 p k − 1 = e − λ k ( λ k ) k − 1 � � n lim k − 1 k ! n →∞ and Pr[ T λ = k ] = lim n →∞ Pr[ | C ( v ) | = k ] gives the theorem!

Duality d < 1 < c dual if de − d = ce − c T PO conditioned on being finite is T PO . c d Roughly: G ( n , c n with giant component removed is G ( m , d m ).

A Convenience In the graph process: To avoid technical calculations we shall replace BIN [ N t − 1 , p ] with Po [ N t − 1 p ] in finding the number of “new” vertices.

The Subcritical Regime λ < 1 T = | C ( v ) | is stochastically dominated by taking BIN [ n − 1 , p ] ∼ Po [ λ ] new vertices at each step. Pr[ | C ( v ) | ≥ k ] ≤ Pr[ T PO ≥ k ] Exponential decay. For k = K ln n , λ Pr[ | C ( v ) | ≥ k ] = o ( n − 1 ) so that | C MAX | ≤ K ln n

The Supercritical Regime λ > 1 The GIANT Component

The Supercritical Regime λ > 1 The GIANT Component EXISTENCE

The Supercritical Regime λ > 1 The GIANT Component EXISTENCE UNIQUENESS