Thresholds for Some Basic Properties Eight Lectures on Random Graphs: Lecture 2

Oleg Pikhurko Carnegie Mellon University

Basic Thresholds – p.1 Gn,p = random order-n graph with edge probability p. Whp = with high probability (approaching 1 as n → ∞). Markov's Inequality: for a random variable X ≥ 0 and a real a > 0 E [ X ] Pr [ X ≥ a ] ≤ . a

Graphs and Properties

Graph property = a collection of graphs. Monotone = adding edges cannot violate it.

Basic Thresholds – p.2 Whp = with high probability (approaching 1 as n → ∞). Markov's Inequality: for a random variable X ≥ 0 and a real a > 0 E [ X ] Pr [ X ≥ a ] ≤ . a

Graphs and Properties

Graph property = a collection of graphs. Monotone = adding edges cannot violate it.

Gn,p = random order-n graph with edge probability p.

Basic Thresholds – p.2 Graphs and Properties

Graph property = a collection of graphs. Monotone = adding edges cannot violate it.

Gn,p = random order-n graph with edge probability p. Whp = with high probability (approaching 1 as n → ∞). Markov's Inequality: for a random variable X ≥ 0 and a real a > 0 E [ X ] Pr [ X ≥ a ] ≤ . a

Basic Thresholds – p.2 Proof Define p0 ∈ [0, 1] by p1 + (1 − p1) p0 = p2.

Let G1 ∈ Gn,p1 and G0 ∈ Gn,p0 . Then G1 ∪ G0 ∼ Gn,p2 .

Pr [ G1 ∈ A ] ≤ Pr [ G1 ∪ G0 ∈ A ] .

Monotone Properties

Theorem For any monotone A and p1 ≤ p2

Pr [ Gn,p1 ∈ A ] ≤ Pr [ Gn,p2 ∈ A ] .

Basic Thresholds – p.3 Monotone Properties

Theorem For any monotone A and p1 ≤ p2

Pr [ Gn,p1 ∈ A ] ≤ Pr [ Gn,p2 ∈ A ] .

Proof Define p0 ∈ [0, 1] by

p1 + (1 − p1) p0 = p2.

Let G1 ∈ Gn,p1 and G0 ∈ Gn,p0 . Then G1 ∪ G0 ∼ Gn,p2 .

Pr [ G1 ∈ A ] ≤ Pr [ G1 ∪ G0 ∈ A ] .

Basic Thresholds – p.3 1 Example p0 = n is a threshold for having a cycle. Indeed, if p = o(1/n), then n n (i − 1)! Pr [ ∃ cycle ] ≤ E [ #cycles ] = pi ≤ (np)i → 0. i 2 Xi≥3   Xi≥3

2+ε n n−1 if p > n , then E [ e(G) ] = p 2 > (2 + ε) 2 . By Chernoff’s bound, whp e(G) ≥ n.

Thresholds

p0 = p0(n) is a threshold for a monotone property A if ∀p(n)

0, if p/p0 → 0, Pr [ Gn,p ∈ A ] → ( 1, if p/p0 → ∞.

Basic Thresholds – p.4 2+ε n n−1 if p > n , then E [ e(G) ] = p 2 > (2 + ε) 2 . By Chernoff’s bound, whp e(G) ≥ n.

Thresholds

p0 = p0(n) is a threshold for a monotone property A if ∀p(n)

0, if p/p0 → 0, Pr [ Gn,p ∈ A ] → ( 1, if p/p0 → ∞.

1 Example p0 = n is a threshold for having a cycle. Indeed, if p = o(1/n), then

n n (i − 1)! Pr [ ∃ cycle ] ≤ E [ #cycles ] = pi ≤ (np)i → 0. i 2 Xi≥3   Xi≥3

Basic Thresholds – p.4 Thresholds

p0 = p0(n) is a threshold for a monotone property A if ∀p(n)

0, if p/p0 → 0, Pr [ Gn,p ∈ A ] → ( 1, if p/p0 → ∞.

1 Example p0 = n is a threshold for having a cycle. Indeed, if p = o(1/n), then

n n (i − 1)! Pr [ ∃ cycle ] ≤ E [ #cycles ] = pi ≤ (np)i → 0. i 2 Xi≥3   Xi≥3 2+ε n n−1 if p > n , then E [ e(G) ] = p 2 > (2 + ε) 2 . By Chernoff’s bound, whp e(G) ≥ n.

Basic Thresholds – p.4 Proof Choose p0 = p(1/2), i.e.

Pr [ Gn,p0 ∈ A ] = 1/2.

p0 exists as f(p) = Pr [ Gn,p ∈ A ] is a polynomial with f(0) = 0 and f(1) = 1.

Monotone ⇒ ∃ Threshold

Theorem (Bollobás-Thomason’87) Every non-trivial monotone property A has a threshold.

Basic Thresholds – p.5 Monotone ⇒ ∃ Threshold

Theorem (Bollobás-Thomason’87) Every non-trivial monotone property A has a threshold.

Proof Choose p0 = p(1/2), i.e.

Pr [ Gn,p0 ∈ A ] = 1/2.

p0 exists as f(p) = Pr [ Gn,p ∈ A ] is a polynomial with f(0) = 0 and f(1) = 1.

Basic Thresholds – p.5 m As 1 − (1 − p) ≤ pm ≤ p0,

1 m ≤ Pr [ H 6∈ A ] ≤ Pr [ ∀i G 6∈ A ] = 1 − Pr [ G ∈ A ] . 2 i n,p
⇒ Pr [ Gn,p ∈ A ] < ε.

Other direction: take Gn,p0 ∪ · · · ∪ Gn,p0 .

p0 = p(1/2) is a threshold

m Given ε > 0, let (1 − ε) < 1/2. Let p < p0/m. Let G1, . . . , Gm ∈ Gn,p and

H = G1 ∪ · · · ∪ Gm ∼ Gn,1−(1−p)m .

Basic Thresholds – p.6 Other direction: take Gn,p0 ∪ · · · ∪ Gn,p0 .

p0 = p(1/2) is a threshold

m Given ε > 0, let (1 − ε) < 1/2. Let p < p0/m. Let G1, . . . , Gm ∈ Gn,p and

H = G1 ∪ · · · ∪ Gm ∼ Gn,1−(1−p)m .

m As 1 − (1 − p) ≤ pm ≤ p0,

1 m ≤ Pr [ H 6∈ A ] ≤ Pr [ ∀i G 6∈ A ] = 1 − Pr [ G ∈ A ] . 2 i n,p
⇒ Pr [ Gn,p ∈ A ] < ε.

Basic Thresholds – p.6 p0 = p(1/2) is a threshold

m Given ε > 0, let (1 − ε) < 1/2. Let p < p0/m. Let G1, . . . , Gm ∈ Gn,p and

H = G1 ∪ · · · ∪ Gm ∼ Gn,1−(1−p)m .

m As 1 − (1 − p) ≤ pm ≤ p0,

1 m ≤ Pr [ H 6∈ A ] ≤ Pr [ ∀i G 6∈ A ] = 1 − Pr [ G ∈ A ] . 2 i n,p
⇒ Pr [ Gn,p ∈ A ] < ε.

Other direction: take Gn,p0 ∪ · · · ∪ Gn,p0 .

Basic Thresholds – p.6 E [ # spanning trees ] → 0 ⇒ G 6∈ C. E [ # spanning trees ] → ∞ 6⇒ G ∈ C.

Connectivity Property C

Idea 1: Connectivity = ∃ spanning tree

E [ # spanning trees ] = nn−2 · pn−1.

The “window” is 1 p = (1 + o(1)) . n

Basic Thresholds – p.7 6⇒ G ∈ C.

Connectivity Property C

Idea 1: Connectivity = ∃ spanning tree

E [ # spanning trees ] = nn−2 · pn−1.

The “window” is 1 p = (1 + o(1)) . n

E [ # spanning trees ] → 0 ⇒ G 6∈ C. E [ # spanning trees ] → ∞

Basic Thresholds – p.7 Connectivity Property C

Idea 1: Connectivity = ∃ spanning tree

E [ # spanning trees ] = nn−2 · pn−1.

The “window” is 1 p = (1 + o(1)) . n

E [ # spanning trees ] → 0 ⇒ G 6∈ C. E [ # spanning trees ] → ∞ 6⇒ G ∈ C.

Basic Thresholds – p.7 Cuts or Isolated Components?

Ck = # k-components

Observation: G ∈ C iff Ck = 0 ∀k ∈ [1, n/2].

Idea 2: Look at Cuts

Cut:

Basic Thresholds – p.8 Ck = # k-components

Observation: G ∈ C iff Ck = 0 ∀k ∈ [1, n/2].

Idea 2: Look at Cuts

Cut:

Cuts or Isolated Components?

Basic Thresholds – p.8 Idea 2: Look at Cuts

Cut:

Cuts or Isolated Components?

Ck = # k-components

Observation: G ∈ C iff Ck = 0 ∀k ∈ [1, n/2].

Basic Thresholds – p.8 Connectivity Threshold

Theorem (Erdos-Ren˝ yi’60) Let

log n c p = + . n n

−c e−e , |c| = O(1), Then Pr [ Gn,p ∈ C ] →  0, c → −∞,  1, c → +∞.

logn In particular, p0(n) = n is a threshold for connectivity.

Basic Thresholds – p.9 n = E [ C ] ≤ (1 − p)k(n−k)kk−2pk−1 k k   X n X ≤ (en/k)k & (1 − x) ≤ e−x k   h k i ≤ n O(log n) e−np+kp → 0. X   Thus whp C2 = · · · = Cbn/2c = 0.

log n O(1) p = n + n

bn/2c Let := k=2 . P P Pr Ck ≥ 1 ≤ E Ck h X i h X i

Basic Thresholds – p.10 n ≤ (en/k)k & (1 − x) ≤ e−x k   h k i ≤ n O(log n) e−np+kp → 0. X   Thus whp C2 = · · · = Cbn/2c = 0.

log n O(1) p = n + n

bn/2c Let := k=2 . P P Pr Ck ≥ 1 ≤ E Ck h X i hnX i = E [ C ] ≤ (1 − p)k(n−k)kk−2pk−1 k k X X  

Basic Thresholds – p.10 k ≤ n O(log n) e−np+kp → 0. X   Thus whp C2 = · · · = Cbn/2c = 0.

log n O(1) p = n + n

bn/2c Let := k=2 . P P Pr Ck ≥ 1 ≤ E Ck h X i hnX i = E [ C ] ≤ (1 − p)k(n−k)kk−2pk−1 k k   X n X ≤ (en/k)k & (1 − x) ≤ e−x k h  i

Basic Thresholds – p.10 Thus whp C2 = · · · = Cbn/2c = 0.

log n O(1) p = n + n

bn/2c Let := k=2 . P P Pr Ck ≥ 1 ≤ E Ck h X i hnX i = E [ C ] ≤ (1 − p)k(n−k)kk−2pk−1 k k   X n X ≤ (en/k)k & (1 − x) ≤ e−x k   h k i ≤ n O(log n) e−np+kp → 0. X  

Basic Thresholds – p.10 log n O(1) p = n + n

bn/2c Let := k=2 . P P Pr Ck ≥ 1 ≤ E Ck h X i hnX i = E [ C ] ≤ (1 − p)k(n−k)kk−2pk−1 k k   X n X ≤ (en/k)k & (1 − x) ≤ e−x k   h k i ≤ n O(log n) e−np+kp → 0. X   Thus whp C2 = · · · = Cbn/2c = 0.

Basic Thresholds – p.10 − −e c It is enough to prove Pr [ C1 = 0 ] → e because

0 ≤ Pr [ C1 = 0 ] − Pr [ C ∈ C ] ≤ Pr [ ∃i ∈ [2, n/2] Ci > 0 ] → 0.

log n O(1) p = n + n (cont.)

Thus, whp Gn,p ∈ C iff C1 = 0 (i.e. no isolated vertices).

Basic Thresholds – p.11 log n O(1) p = n + n (cont.)

Thus, whp Gn,p ∈ C iff C1 = 0 (i.e. no isolated vertices).

− −e c It is enough to prove Pr [ C1 = 0 ] → e because

0 ≤ Pr [ C1 = 0 ] − Pr [ C ∈ C ] ≤ Pr [ ∃i ∈ [2, n/2] Ci > 0 ] → 0.

Basic Thresholds – p.11 µ E [ # successes ] = n × = µ n n µi e−µ Pr [ i successes ] = pi (1 − p)n−i → . i i!  

Poisson Distribution with Mean µ

µ n independent trials, Pr [ success ] = n , constant µ. Poisson(µ) = # successes as n → ∞.

Basic Thresholds – p.12 n µi e−µ Pr [ i successes ] = pi (1 − p)n−i → . i i!  

Poisson Distribution with Mean µ

µ n independent trials, Pr [ success ] = n , constant µ. Poisson(µ) = # successes as n → ∞. µ E [ # successes ] = n × = µ n

Basic Thresholds – p.12 Poisson Distribution with Mean µ

µ n independent trials, Pr [ success ] = n , constant µ. Poisson(µ) = # successes as n → ∞. µ E [ # successes ] = n × = µ n

n µi e−µ Pr [ i successes ] = pi (1 − p)n−i → . i i!  

Basic Thresholds – p.12 The k-th Factorial Moment:

Mk[X] = E [ (X)k ] = E [ X(X − 1) . . . (X − k + 1) ] .

For fixed k k(n−1)−(k) −c k −c Mk[C1] = (n)k (1 − p) 2 → (e ) = Mk[Poisson(e )].

−c This is known to imply that C1 → Poisson(e ). In particular,

− −e c Pr [ C1 = 0 ] → e .

Isolated Vertices

n−1 −c E [ C1 ] = n(1 − p) → e .

Basic Thresholds – p.13 For fixed k k(n−1)−(k) −c k −c Mk[C1] = (n)k (1 − p) 2 → (e ) = Mk[Poisson(e )].

−c This is known to imply that C1 → Poisson(e ). In particular,

− −e c Pr [ C1 = 0 ] → e .

Isolated Vertices

n−1 −c E [ C1 ] = n(1 − p) → e .

The k-th Factorial Moment:

Mk[X] = E [ (X)k ] = E [ X(X − 1) . . . (X − k + 1) ] .

Basic Thresholds – p.13 −c This is known to imply that C1 → Poisson(e ). In particular,

− −e c Pr [ C1 = 0 ] → e .

Isolated Vertices

n−1 −c E [ C1 ] = n(1 − p) → e .

The k-th Factorial Moment:

Mk[X] = E [ (X)k ] = E [ X(X − 1) . . . (X − k + 1) ] .

For fixed k

k(n−1)−(k) −c k −c Mk[C1] = (n)k (1 − p) 2 → (e ) = Mk[Poisson(e )].

Basic Thresholds – p.13 Isolated Vertices

n−1 −c E [ C1 ] = n(1 − p) → e .

The k-th Factorial Moment:

Mk[X] = E [ (X)k ] = E [ X(X − 1) . . . (X − k + 1) ] .

For fixed k

k(n−1)−(k) −c k −c Mk[C1] = (n)k (1 − p) 2 → (e ) = Mk[Poisson(e )].

−c This is known to imply that C1 → Poisson(e ). In particular,

− −e c Pr [ C1 = 0 ] → e .

Basic Thresholds – p.13 Examples, connectivity: sharp, having a triangle: not sharp, having a cycle: ‘one-sided sharp’.

Sharp Threshold

Connectivity Threshold

p0 is a sharp threshold for a monotone A if ∀ ε > 0 whp

Gn,(1−ε)p0 6∈ A and Gn,(1+ε)p0 ∈ A.

Basic Thresholds – p.14 Sharp Threshold

Connectivity Threshold p0 is a sharp threshold for a monotone A if ∀ ε > 0 whp

Gn,(1−ε)p0 6∈ A and Gn,(1+ε)p0 ∈ A. Examples, connectivity: sharp, having a triangle: not sharp, having a cycle: ‘one-sided sharp’.

Basic Thresholds – p.14 Theorem (Friedgut’99) If for a monotone A

∂ 1 Pr [ G ∈ A ] = O , ∂p n,p p   then ∀ ε > 0 there is a finite family F of graphs such that ∀ n, p Pr [ Gn,p ∈ A 4 {an F-subgraph} ] ≤ ε.

Friedgut's Theorem

∂ 1 Note: Sharp threshold ⇒ ∂pPr [ Gn,p ∈ A ] =6 O p .  

Basic Thresholds – p.15 Friedgut's Theorem

∂ 1 Note: Sharp threshold ⇒ ∂pPr [ Gn,p ∈ A ] =6 O p .   Theorem (Friedgut’99) If for a monotone A

∂ 1 Pr [ G ∈ A ] = O , ∂p n,p p   then ∀ ε > 0 there is a finite family F of graphs such that ∀ n, p Pr [ Gn,p ∈ A 4 {an F-subgraph} ] ≤ ε.

Basic Thresholds – p.15 Theorem (Achlioptas-Friedgut’99) For fixed k ≥ 3 k-colorability has a sharp threshold.

Applying Friedgut's Theorem

Difficult to apply: the type of A ∪ B depends on which one appears ‘earlier’.

Basic Thresholds – p.16 Applying Friedgut's Theorem

Difficult to apply: the type of A ∪ B depends on which one appears ‘earlier’.

Theorem (Achlioptas-Friedgut’99) For fixed k ≥ 3 k-colorability has a sharp threshold.

Basic Thresholds – p.16 Gn,p: edge probability p.

Theorem For a monotone A,

Pr Gn,M ∈ A is a non-decreasing function of M. 

Model Gn,M

Gn,M : random M edges.

Basic Thresholds – p.17 Theorem For a monotone A,

Pr Gn,M ∈ A is a non-decreasing function of M. 

Model Gn,M

Gn,M : random M edges.

Gn,p: edge probability p.

Basic Thresholds – p.17 Model Gn,M

Gn,M : random M edges.

Gn,p: edge probability p.

Theorem For a monotone A,

Pr Gn,M ∈ A is a non-decreasing function of M. 

Basic Thresholds – p.17 Theorem For any non-trivial monotone A, there is a threshold M0, that is,

0, if M/M0 → 0, Pr Gn,M ∈ A → ( 1, if M/M0 → ∞.   log n+c n Theorem Let M = n 2 . Then

−c e−e , |c| = O(1), Pr Gn,M ∈ C →  0, c → −∞,     1, c → +∞.  In particular, M0 = n log n isa threshold for connectivity.

Relating Gn,p and Gn,M

Basic Thresholds – p.18 log n+c n Theorem Let M = n 2 . Then

−c e−e , |c| = O(1), Pr Gn,M ∈ C →  0, c → −∞,     1, c → +∞.  In particular, M0 = n log n isa threshold for connectivity.

Relating Gn,p and Gn,M

Theorem For any non-trivial monotone A, there is a threshold M0, that is,

0, if M/M0 → 0, Pr Gn,M ∈ A → ( 1, if M/M0 → ∞.  

Basic Thresholds – p.18 Relating Gn,p and Gn,M

Theorem For any non-trivial monotone A, there is a threshold M0, that is,

0, if M/M0 → 0, Pr Gn,M ∈ A → ( 1, if M/M0 → ∞.  

log n+c n Theorem Let M = n 2 . Then

−c e−e , |c| = O(1), Pr Gn,M ∈ C →  0, c → −∞,     1, c → +∞.  In particular, M0 = n log n isa threshold for connectivity.

Basic Thresholds – p.18 By Chernoff’s bound, Pr [ e(G) > M ] → 0. Hence,

− −e c+ε Pr Gn,M ∈ C ≥ Pr [ G ∈ C ] − Pr [ e(G) > M ] ≥ e − o(1).   log n+c+ε Upper bound: remove edges from Gn,p, p = n .

Connectivity of Gn,M

Proof Enough to consider |c| = O(1). Take small ε > 0. Let

log n + c − ε p = . n

Take G ∈ Gn,p. Let l = M − e(G). If l ≥ 0, let

H = G + l random edges; H ∼ Gn,M .

Basic Thresholds – p.19 log n+c+ε Upper bound: remove edges from Gn,p, p = n .

Connectivity of Gn,M

Proof Enough to consider |c| = O(1). Take small ε > 0. Let

log n + c − ε p = . n

Take G ∈ Gn,p. Let l = M − e(G). If l ≥ 0, let

H = G + l random edges; H ∼ Gn,M .

By Chernoff’s bound, Pr [ e(G) > M ] → 0. Hence,

− −e c+ε Pr Gn,M ∈ C ≥ Pr [ G ∈ C ] − Pr [ e(G) > M ] ≥ e − o(1).  

Basic Thresholds – p.19 Connectivity of Gn,M

Proof Enough to consider |c| = O(1). Take small ε > 0. Let

log n + c − ε p = . n

Take G ∈ Gn,p. Let l = M − e(G). If l ≥ 0, let

H = G + l random edges; H ∼ Gn,M .

By Chernoff’s bound, Pr [ e(G) > M ] → 0. Hence,

− −e c+ε Pr Gn,M ∈ C ≥ Pr [ G ∈ C ] − Pr [ e(G) > M ] ≥ e − o(1).   log n+c+ε Upper bound: remove edges from Gn,p, p = n .

Basic Thresholds – p.19 Theorem (Erdos-Ren˝ yi’60) Whp τ [ δ ≥ 1 ] = τ [ C ].

Proof Let B = {H : δ ≥ 1 & H 6∈ C}.

Idea 1:

Pr [ ∃M : GM ∈ B ] ≤ Pr Gn,M ∈ B 6→ 0. M X  

Hitting Time Version

Random graph process:

G0 = n isolated vertices; GM+1 = GM + a random edge.

Hitting time τ [ A ] = min{M : GM ∈ A}.

Basic Thresholds – p.20 Proof Let B = {H : δ ≥ 1 & H 6∈ C}.

Idea 1:

Pr [ ∃M : GM ∈ B ] ≤ Pr Gn,M ∈ B 6→ 0. M X  

Hitting Time Version

Random graph process:

G0 = n isolated vertices; GM+1 = GM + a random edge.

Hitting time τ [ A ] = min{M : GM ∈ A}.

Theorem (Erdos-Ren˝ yi’60) Whp τ [ δ ≥ 1 ] = τ [ C ].

Basic Thresholds – p.20 Hitting Time Version

Random graph process:

G0 = n isolated vertices; GM+1 = GM + a random edge.

Hitting time τ [ A ] = min{M : GM ∈ A}.

Theorem (Erdos-Ren˝ yi’60) Whp τ [ δ ≥ 1 ] = τ [ C ].

Proof Let B = {H : δ ≥ 1 & H 6∈ C}.

Idea 1:

Pr [ ∃M : GM ∈ B ] ≤ Pr Gn,M ∈ B 6→ 0. M X   Basic Thresholds – p.20 log nc Fix large c > 0. Let m = b n c.

Pr [ ∃M : GM ∈ B ] ≤ Pr [ ∃M ≤ m− δ(GM ) ≥ 1 ] + Pr [ ∃M ∈ (m−, m+) GM ∈ B ] + Pr [ ∃M ≥ m+ GM 6∈ C ] = p− + p0 + p+ Now,

− −e c p+ = Pr [ Gn,m+ 6∈ C ] = 1 − e + o(1), −ec p− = Pr [ δ(Gn,m− ) ≥ 1 ] = e + o(1).

Idea 2: Using GM ⊂ GM+1

Basic Thresholds – p.21 Now,

− −e c p+ = Pr [ Gn,m+ 6∈ C ] = 1 − e + o(1), −ec p− = Pr [ δ(Gn,m− ) ≥ 1 ] = e + o(1).

Idea 2: Using GM ⊂ GM+1

log nc Fix large c > 0. Let m = b n c.

Pr [ ∃M : GM ∈ B ] ≤ Pr [ ∃M ≤ m− δ(GM ) ≥ 1 ] + Pr [ ∃M ∈ (m−, m+) GM ∈ B ] + Pr [ ∃M ≥ m+ GM 6∈ C ] = p− + p0 + p+

Basic Thresholds – p.21 Idea 2: Using GM ⊂ GM+1

log nc Fix large c > 0. Let m = b n c.

Pr [ ∃M : GM ∈ B ] ≤ Pr [ ∃M ≤ m− δ(GM ) ≥ 1 ] + Pr [ ∃M ∈ (m−, m+) GM ∈ B ] + Pr [ ∃M ≥ m+ GM 6∈ C ] = p− + p0 + p+ Now,

− −e c p+ = Pr [ Gn,m+ 6∈ C ] = 1 − e + o(1), −ec p− = Pr [ δ(Gn,m− ) ≥ 1 ] = e + o(1).

Basic Thresholds – p.21 The Old Trick

Recall: B = {H : δ ≥ 1 & H 6∈ C}.

Aim: Pr [ ∃M ∈ (m−, m+) GM ∈ B ] → 0.

Let log n − c − ε p = , n G ∈ Gn,p.

Basic Thresholds – p.22 Proof n−1 c+ε E [ C1 ] = n(1 − p) ≤ e + o(1) = O(1).

E[ C1 ] So Pr [ C1 > log n ] < log n → 0.

We already proved that whp C2 = · · · = Cbn/2c = 0.

Counting Components (Again)

Lemma Whp C1 ≤ log n and C2 = · · · = Cbn/2c = 0, i.e. G consists of at most log n isolated vertices and one component.

Basic Thresholds – p.23 Counting Components (Again)

Lemma Whp C1 ≤ log n and C2 = · · · = Cbn/2c = 0, i.e. G consists of at most log n isolated vertices and one component.

Proof

n−1 c+ε E [ C1 ] = n(1 − p) ≤ e + o(1) = O(1).

E[ C1 ] So Pr [ C1 > log n ] < log n → 0.

We already proved that whp C2 = · · · = Cbn/2c = 0.

Basic Thresholds – p.23 Putting all together: whp τ [ C ] = τ [ δ ≥ 1 ].

Process between m− and m+

Pr [ ∃M ∈ (m−, m+) : GM ∈ B ] ≤ Pr [ e(G) > m− ] + Pr [ C1 > log n ] + Pr [ ∃k ∈ [2, n/2] Ck = 0 ] log n + m 2 → 0. + n 2 2 − o(
n )

Basic Thresholds – p.24 Process between m− and m+

Pr [ ∃M ∈ (m−, m+) : GM ∈ B ] ≤ Pr [ e(G) > m− ] + Pr [ C1 > log n ] + Pr [ ∃k ∈ [2, n/2] Ck = 0 ] log n + m 2 → 0. + n 2 2 − o(
n )
Putting all together: whp τ [ C ] = τ [ δ ≥ 1 ].

Basic Thresholds – p.24 Some Spanning Subgraphs

0, c → −∞, Pr [ Gn,p ∈ A ] → ( 1, c → +∞, Erdos˝ & Renyi’66: log n+c A = {perfect matching}, n even, p = n . Korshunov’83, Komlós & Szemerédi’83: log n+log log n+c A = {Hamiltonian}, p = n . Riordan’00: d 1 log d A = {d-dimensional cube}, n = 2 , p = 4 + c d .

Basic Thresholds – p.25 log n+c Theorem (Erdos˝ & Renyi’64) Let p = n and G ∈ Gn,n,p. Then −c Pr [ G has a matching ] → e−2e . log n In particular, p0 = n is a sharp threshold.

Perfect Matchings

Gn,n,p: random subgraph of Kn,n, Pr [ edge ] = p (or random n × n 0/1-matrix).

Basic Thresholds – p.26 Perfect Matchings

Gn,n,p: random subgraph of Kn,n, Pr [ edge ] = p (or random n × n 0/1-matrix).

log n+c Theorem (Erdos˝ & Renyi’64) Let p = n and G ∈ Gn,n,p. Then −c Pr [ G has a matching ] → e−2e . log n In particular, p0 = n is a sharp threshold.

Basic Thresholds – p.26 Pr [ ∃ such S : |S| ≥ 2 ] ≤ E [ # such S ] dn/2e n n s s−1 ≤ 2 p2s−2(1 − p)s(n−s+1) = o(ne−pn). s s − 1 2 Xs=2     n −c −c E [ C1 ] = 2n(1 − p) → 2 e . As before C1 → Poisson(2e ).

Using Hall's Theorem

Proof No matching ⇔ ∃ S s.t. |S| = |Γ(S)| + 1, |S| ≤ dn/2e, ∀x ∈ Γ(S) |Γ(x) ∩ S| ≥ 2.

Basic Thresholds – p.27 n −c −c E [ C1 ] = 2n(1 − p) → 2 e . As before C1 → Poisson(2e ).

Using Hall's Theorem

Proof No matching ⇔ ∃ S s.t. |S| = |Γ(S)| + 1, |S| ≤ dn/2e, ∀x ∈ Γ(S) |Γ(x) ∩ S| ≥ 2.

Pr [ ∃ such S : |S| ≥ 2 ] ≤ E [ # such S ] dn/2e n n s s−1 ≤ 2 p2s−2(1 − p)s(n−s+1) = o(ne−pn). s s − 1 2 Xs=2    

Basic Thresholds – p.27 Using Hall's Theorem

Proof No matching ⇔ ∃ S s.t. |S| = |Γ(S)| + 1, |S| ≤ dn/2e, ∀x ∈ Γ(S) |Γ(x) ∩ S| ≥ 2.

Pr [ ∃ such S : |S| ≥ 2 ] ≤ E [ # such S ] dn/2e n n s s−1 ≤ 2 p2s−2(1 − p)s(n−s+1) = o(ne−pn). s s − 1 2 Xs=2     n −c −c E [ C1 ] = 2n(1 − p) → 2 e . As before C1 → Poisson(2e ).

Basic Thresholds – p.27 Gn,n,M : random M edges of Kn,n. log n+c 2 Theorem Let M = n n and G ∈ Gn,n,p. Then

−c Pr [ G has a matching ] → e−2e .

In particular, M0 = n log n is a sharp threshold.

Model Gn,n,M

Basic Thresholds – p.28 log n+c 2 Theorem Let M = n n and G ∈ Gn,n,p. Then

−c Pr [ G has a matching ] → e−2e .

In particular, M0 = n log n is a sharp threshold.

Model Gn,n,M

Gn,n,M : random M edges of Kn,n.

Basic Thresholds – p.28 Model Gn,n,M

Gn,n,M : random M edges of Kn,n.

log n+c 2 Theorem Let M = n n and G ∈ Gn,n,p. Then

−c Pr [ G has a matching ] → e−2e .

In particular, M0 = n log n is a sharp threshold.

Basic Thresholds – p.28 log nc Proof Take large c and m = b n c.

Interesting interval: M ∈ (m−, m+)

Pr [ ∃|S| : |S| ≥ 3 ] ≤ m+ E [ # such S ] → 0.

Pr [ ∃|S| : |S| = 2 ] ≤ Pr [ Gm− has such 2-element S ]

+ Pr [ C1(Gm− ) > log n ] + log n Pr [ such 2-element S is created ] → 0.

Random Graph Process GM ⊂ Kn,n

Hitting time version: τ [ matching ] = τ [ δ ≥ 1 ].

Basic Thresholds – p.29 Pr [ ∃|S| : |S| ≥ 3 ] ≤ m+ E [ # such S ] → 0.

Pr [ ∃|S| : |S| = 2 ] ≤ Pr [ Gm− has such 2-element S ]

+ Pr [ C1(Gm− ) > log n ] + log n Pr [ such 2-element S is created ] → 0.

Random Graph Process GM ⊂ Kn,n

Hitting time version: τ [ matching ] = τ [ δ ≥ 1 ].

log nc Proof Take large c and m = b n c.

Interesting interval: M ∈ (m−, m+)

Basic Thresholds – p.29 Pr [ ∃|S| : |S| = 2 ] ≤ Pr [ Gm− has such 2-element S ]

+ Pr [ C1(Gm− ) > log n ] + log n Pr [ such 2-element S is created ] → 0.

Random Graph Process GM ⊂ Kn,n

Hitting time version: τ [ matching ] = τ [ δ ≥ 1 ].

log nc Proof Take large c and m = b n c.

Interesting interval: M ∈ (m−, m+)

Pr [ ∃|S| : |S| ≥ 3 ] ≤ m+ E [ # such S ] → 0.

Basic Thresholds – p.29 Random Graph Process GM ⊂ Kn,n

Hitting time version: τ [ matching ] = τ [ δ ≥ 1 ].

log nc Proof Take large c and m = b n c.

Interesting interval: M ∈ (m−, m+)

Pr [ ∃|S| : |S| ≥ 3 ] ≤ m+ E [ # such S ] → 0.

Pr [ ∃|S| : |S| = 2 ] ≤ Pr [ Gm− has such 2-element S ]

+ Pr [ C1(Gm− ) > log n ] + log n Pr [ such 2-element S is created ] → 0.

Basic Thresholds – p.29 Proof Let F = H2; ∆(F ) < D.

Lemma (Hajnal-Szemerédi’70): n ∃ F -stable sets V1 ∪ · · · ∪ VD = V (F ), each |Vi| = D  1.

Take V (G) = U1 ∪ · · · ∪ UD with |Ui| = |Vi|.

General Spanning Subgraphs

Theorem (Alon-Füredi’92) Let v(H) = n, ∆(H) ≤ d 2 D = d + 1 and G ∈ Gn,p. npd If D − log n → ∞, then whp H ⊂ G.

Basic Thresholds – p.30 General Spanning Subgraphs

Theorem (Alon-Füredi’92) Let v(H) = n, ∆(H) ≤ d 2 D = d + 1 and G ∈ Gn,p. npd If D − log n → ∞, then whp H ⊂ G.

Proof Let F = H2; ∆(F ) < D.

Lemma (Hajnal-Szemerédi’70): n ∃ F -stable sets V1 ∪ · · · ∪ VD = V (F ), each |Vi| = D  1.

Take V (G) = U1 ∪ · · · ∪ UD with |Ui| = |Vi|.

Basic Thresholds – p.30 Observe F ∼ Gm,m,≥pd .

Pr [ FAIL ] = Pr [ no matching ] = O(m e−pm) = o(1/D).

So, whp fi exists ∀ i ∈ [D], i.e. H ⊂ G.

Partial H-Embeddings

Build fi : V1 ∪ · · · ∪ Vi → U1 ∪ · · · ∪ Ui inductively. Let m = |Vi+1| = |Ui+1| and

F = { (u, v) : u ∈ Ui+1, v ∈ Vi+1 ΓH (v) ⊂ fi(ΓG(u)) }.

Basic Thresholds – p.31 Partial H-Embeddings

Build fi : V1 ∪ · · · ∪ Vi → U1 ∪ · · · ∪ Ui inductively. Let m = |Vi+1| = |Ui+1| and

F = { (u, v) : u ∈ Ui+1, v ∈ Vi+1 ΓH (v) ⊂ fi(ΓG(u)) }.

Observe F ∼ Gm,m,≥pd .

Pr [ FAIL ] = Pr [ no matching ] = O(m e−pm) = o(1/D).

So, whp fi exists ∀ i ∈ [D], i.e. H ⊂ G.

Basic Thresholds – p.31 ∞ −3 Let ζ(3) = i=1 i . Theorem (FriezP e’85) 1. E [ w(T ) ] → ζ(3). 2. ∀ε > 0 Pr [ |w(T ) − ζ(3)| > ε ] → 0.

Proof of 1. Let Gp = [n], {e : we ≤ p} ∼ Gn,p.

Random Edge-Weights

[n] The model: Random independent weights we, e ∈ 2 , each uniformly distributed in (0, 1).
T = the minimal spanning tree.

Basic Thresholds – p.32 Proof of 1. Let Gp = [n], {e : we ≤ p} ∼ Gn,p.

Random Edge-Weights

[n] The model: Random independent weights we, e ∈ 2 , each uniformly distributed in (0, 1).
T = the minimal spanning tree.

∞ −3 Let ζ(3) = i=1 i . Theorem (FriezP e’85) 1. E [ w(T ) ] → ζ(3). 2. ∀ε > 0 Pr [ |w(T ) − ζ(3)| > ε ] → 0.

Basic Thresholds – p.32 Random Edge-Weights

[n] The model: Random independent weights we, e ∈ 2 , each uniformly distributed in (0, 1).
T = the minimal spanning tree.

∞ −3 Let ζ(3) = i=1 i . Theorem (FriezP e’85) 1. E [ w(T ) ] → ζ(3). 2. ∀ε > 0 Pr [ |w(T ) − ζ(3)| > ε ] → 0.

Proof of 1. Let Gp = [n], {e : we ≤ p} ∼ Gn,p.

Basic Thresholds – p.32 1 1 = we>p dp p=0 Xe∈T Z 1 1 = we>p dp p=0 Z Xe∈T 1 = {e ∈ T : e 6∈ Gp} dp p=0 Z 1

= (κ(Gp) − 1)) dp. Zp=0 where κ(G) = # components of G.

An Expression for w(T )

w(T ) = we Xe∈T

Basic Thresholds – p.33 1 1 = we>p dp p=0 Z Xe∈T 1 = {e ∈ T : e 6∈ Gp} dp p=0 Z 1

= (κ(Gp) − 1)) dp. Zp=0 where κ(G) = # components of G.

An Expression for w(T )

w(T ) = we Xe∈T 1 1 = we>p dp p=0 Xe∈T Z

Basic Thresholds – p.33 1 = {e ∈ T : e 6∈ Gp} dp p=0 Z 1

= (κ(Gp) − 1)) dp. Zp=0 where κ(G) = # components of G.

An Expression for w(T )

w(T ) = we Xe∈T 1 1 = we>p dp p=0 Xe∈T Z 1 1 = we>p dp p=0 Z Xe∈T

Basic Thresholds – p.33 1 = (κ(Gp) − 1)) dp. Zp=0 where κ(G) = # components of G.

An Expression for w(T )

w(T ) = we Xe∈T 1 1 = we>p dp p=0 Xe∈T Z 1 1 = we>p dp p=0 Z Xe∈T 1 = {e ∈ T : e 6∈ Gp} dp p=0 Z

Basic Thresholds – p.33 An Expression for w(T )

w(T ) = we Xe∈T 1 1 = we>p dp p=0 Xe∈T Z 1 1 = we>p dp p=0 Z Xe∈T 1 = {e ∈ T : e 6∈ Gp} dp p=0 Z 1

= (κ(Gp) − 1)) dp. Zp=0 where κ(G) = # components of G.

Basic Thresholds – p.33 3 log n n → E [ κ(Gp) ] dp Zp=0 3 log n n1/3 n n → kk−2 pk−1 (1 − p)k(n−k) (1 + O(k2p)) dp p=0 k Z Xk≥1   n1/3 nkkk−2 1 → pk−1 (1 − p)k(n−k) dp k! p=0 Xk≥1 Z n1/3 nkkk−2 (k − 1)! (k(n − k))! 1 = × → . k! (k(n − k + 1))! k3 Xk≥1 Xk≥1

Computing E [ w(T ) ]

1 E [ w(T ) ] = E [ κ(Gp) − 1) ] dp Zp=0

Basic Thresholds – p.34 3 log n n1/3 n n → kk−2 pk−1 (1 − p)k(n−k) (1 + O(k2p)) dp p=0 k Z Xk≥1   n1/3 nkkk−2 1 → pk−1 (1 − p)k(n−k) dp k! p=0 Xk≥1 Z n1/3 nkkk−2 (k − 1)! (k(n − k))! 1 = × → . k! (k(n − k + 1))! k3 Xk≥1 Xk≥1

Computing E [ w(T ) ]

1 E [ w(T ) ] = E [ κ(Gp) − 1) ] dp Zp=0 3 log n n → E [ κ(Gp) ] dp Zp=0

Basic Thresholds – p.34 n1/3 nkkk−2 1 → pk−1 (1 − p)k(n−k) dp k! p=0 Xk≥1 Z n1/3 nkkk−2 (k − 1)! (k(n − k))! 1 = × → . k! (k(n − k + 1))! k3 Xk≥1 Xk≥1

Computing E [ w(T ) ]

1 E [ w(T ) ] = E [ κ(Gp) − 1) ] dp Zp=0 3 log n n → E [ κ(Gp) ] dp Zp=0 3 log n n1/3 n n → kk−2 pk−1 (1 − p)k(n−k) (1 + O(k2p)) dp p=0 k Z Xk≥1  

Basic Thresholds – p.34 n1/3 nkkk−2 (k − 1)! (k(n − k))! 1 = × → . k! (k(n − k + 1))! k3 Xk≥1 Xk≥1

Computing E [ w(T ) ]

1 E [ w(T ) ] = E [ κ(Gp) − 1) ] dp Zp=0 3 log n n → E [ κ(Gp) ] dp Zp=0 3 log n n1/3 n n → kk−2 pk−1 (1 − p)k(n−k) (1 + O(k2p)) dp p=0 k Z Xk≥1   n1/3 nkkk−2 1 → pk−1 (1 − p)k(n−k) dp k! p=0 Xk≥1 Z

Basic Thresholds – p.34 Computing E [ w(T ) ]

1 E [ w(T ) ] = E [ κ(Gp) − 1) ] dp Zp=0 3 log n n → E [ κ(Gp) ] dp Zp=0 3 log n n1/3 n n → kk−2 pk−1 (1 − p)k(n−k) (1 + O(k2p)) dp p=0 k Z Xk≥1   n1/3 nkkk−2 1 → pk−1 (1 − p)k(n−k) dp k! p=0 Xk≥1 Z n1/3 nkkk−2 (k − 1)! (k(n − k))! 1 = × → . k! (k(n − k + 1))! k3 Xk≥1 Xk≥1

Basic Thresholds – p.34 Proof As before, we argue that bn/2c −pn E [ Ck ] = O(n e ) = o(1/n). Xk=1

Hence, E w(T \ G3 log n/n) ≤ n × o(1/n) → 0.  

Above the Connectivity Threshold

3 log n Lemma Let p = n . Then

Pr [ Gn,p 6∈ C ] = o(1/n).

Basic Thresholds – p.35 Above the Connectivity Threshold

3 log n Lemma Let p = n . Then

Pr [ Gn,p 6∈ C ] = o(1/n).

Proof As before, we argue that

bn/2c −pn E [ Ck ] = O(n e ) = o(1/n). Xk=1

Hence, E w(T \ G3 log n/n) ≤ n × o(1/n) → 0.  

Basic Thresholds – p.35 References

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