Bootstrap Percolation on Periodic Trees Milan Bradonji work with - - PowerPoint PPT Presentation

Bootstrap Percolation on Periodic Trees

Milan Bradonjić work with Iraj Saniee Bell Labs, Alcatel-Lucent, Murray Hill, NJ 31 May ’13 AofA 2013, Spain

1 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

Bootstrap Percolation

Structure: A dynamic process defined on an underlying (deterministic or random) graph. Every node is in one of two states 0 or 1 (inactive or active).

2 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

Bootstrap Percolation

Dynamics: Random initial configuration at t = 0: Every node becomes active with prob. p i.i.d. Deterministic process for t ≥ 1: An inactive node becomes and stays active at time t if the number of its active neighbors at time t − 1 is ≥ θ, for given threshold θ.

3 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

Applications

Physics and Chemistry Some crystals modeled as lattices whose sites represent atoms with a certain spin. Ferromagnetism when a sufficient number

• f neighboring sites of each atom has the same spin.

Advertising or Rumor Spreading An individual becomes influenced when a sufficient number of its close friends have already been influenced.

4 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

Question

An interesting phenomenon to study is metastability. Do there exist 0 < p′ ≤ p′′ < 1 such that:

∀p < p′ lim

t→∞ Pp (V becomes fully active) = 0 ,

and

∀p > p′′ lim

t→∞ Pp (V becomes fully active) = 1 ?

Notice: We do not know in general if p′ = p′′.

5 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

Question

An interesting phenomenon to study is metastability. Do there exist 0 < p′ ≤ p′′ < 1 such that:

∀p < p′ lim

t→∞ Pp (V becomes fully active) = 0 ,

and

∀p > p′′ lim

t→∞ Pp (V becomes fully active) = 1 ?

Notice: We do not know in general if p′ = p′′.

5 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

Literature on Bootstrap Percolation

RRG A similar model for adoption of new communication services

• n a random regular graph. Implicit critical thresholds for

widespread adoption, Gersho and Mitra ’75. Reg.Tree Chalupa et all ’79, first formally to introduce bootstrap percolation, and derive the critical threshold on regular trees (Bethe lattices). Tree Critical threshold for non-regular trees, Balogh et all ’06. Zd Aizenman and Lebowitz ’88 studied metastability of bootstrap percolation on Zd.

6 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

Literature on Bootstrap Percolation

Z2 The existence of a sharp threshold for bootstrap percolation in Z2, Holroyd ’03. Zd Recently generalized to d-dimensional lattices Zd, Balogh et all ’11. Gn,(di) Bootstrap percolation on random graphs with a given degree sequence, Amini ’10. Gn,p Bootstrap percolation on random graphs G(n, p) by Luczak et all ’12+. Soc.Net. Formation of opinion in social networks. The percolation threshold is a fraction of the size of each neighborhood, Watts ’02.

7 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

Motivation

1 The threshold for BP on a given graph is upper bounded by

the threshold for BP on its spanning tree.

2 Bootstrap percolation has two thresholds on regular

(homogeneous) trees Fontes and Schonmann ’08.

Figure: Approximation for BP.

8 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

Bootstrap Percolation on Periodic Trees

9 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

Periodic Trees

Definition Let ℓ, m0, m1, . . . , mℓ−1 ∈ N. A periodic tree Tm0,m1,...,mℓ−1 of periodicity ℓ is recursively defined as follows. Consider a node ∅, called root. The nodes at the distance k mod ℓ from the root ∅ have degree mk + 1 for k = 0, 1, 2, . . . .

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Periodic Trees

The schematic presentation of T3,2.

• ❅

❅ ❅ ❅ ❅ ❅

• ❅

❅ ❅ ❅ ❅ ❅

• ❅

❅

• ❅

❅ ❅ ❅ ❅ ❅

• ❅

❅

• ❅

❅

• ❅

❅

• ❅

❅ ❅ ❅

• ❅

❅ ❅ ❅

• ❅

❅ r ❜ ❜ ❜ ❜ r r r r r r r r ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜

11 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

Periodic Oriented Trees

Definition Let ℓ, m0, m1, . . . , mℓ−1 ∈ N. A periodic oriented tree

• Tm0,m1,...,mℓ−1 of periodicity ℓ is recursively defined as follows.

Consider a node ∅, called root, with out-degree m0. The nodes at the distance k mod ℓ from the root ∅ have out-degree mk and in-degree 1 for k = 1, 2, 3, . . . .

12 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

Periodic Oriented Trees

The schematic presentation of T3,2.

✲ ✻ ✛ ❄

• ✒

❅ ❅ ❅ ❘ ❅ ❅ ❅ ■

• ✠
• ✒

❅ ❅ ❅ ■ ❅ ❅ ❅ ❘

• ✠

❅ ❅ ❘

• ✒

❅ ❅ ■ ❅ ❅ ■ ❅ ❅ ❘

• ✒
• ✠❅

❅ ❘

• ✒
• ✠

❅ ❅ ■

• ✒
• ✠

❅ ❅ ■

• ✒
• ✠

❅ ❅ ■ ❅ ❅ ❘

• ✠

❅ ❅ ■ ❅ ❅ ❘

• ✠
• ✒

❅ ❅ ❘ r ❜ ❜ ❜ ❜ r r r r r r r r ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜

13 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

Main Results

The existence of and a closed form system of equations for the critical threshold pf (θ) ∈ (0, 1) such that a.a.s.

1 if p > pf (θ) then the periodic tree becomes (eventually) fully

active,

2 if p < pf (θ) then a periodic tree does not become

(eventually) fully active.

14 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

Lemma Given n, θ ∈ N such that 2 ≤ θ ≤ n − 1 and x ∈ [0, 1] let Φn,p,θ(x) := p + (1 − p)

n

• k=θ
• n

k

• xk(1 − x)n−k − x .

There exists pc ∈ (0, 1) such that the equation Φn,p,θ(x) = 0 (i) does not have real roots in (0, 1) for any p > pc, (ii) has either one or two real roots in (0, 1) for any p ≤ pc. x = 1 is always a solution of Φn,p,θ(x) = 0 for any p ∈ (0, 1).

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Lemma Given n, θ ∈ N such that 2 ≤ θ ≤ n − 1 and x ∈ [0, 1] let Φn,p,θ(x) := p + (1 − p)

n

• k=θ
• n

k

• xk(1 − x)n−k − x .

There exists pc ∈ (0, 1) such that the equation Φn,p,θ(x) = 0 (i) does not have real roots in (0, 1) for any p > pc, (ii) has either one or two real roots in (0, 1) for any p ≤ pc. x = 1 is always a solution of Φn,p,θ(x) = 0 for any p ∈ (0, 1).

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Figure: The curve Φn,p,θ(x) for n = 7, θ = 5, p = 0.4 (no roots).

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Figure: The curve Φn,p,θ(x) for n = 7, θ = 5, p = 0.3 (two roots).

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Main result

First concentrate on a tree of periodicity ℓ = 2, then generalize. Theorem Given a, b ∈ N and 2 ≤ θ < a, b consider BP on periodic tree Ta,b with the initial probability p. There exists pf ∈ (0, 1) such that a.a.s. (i) Ta,b becomes completely active for p ≥ pf , (ii) Ta,b does not become completely active for p < pf .

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Main result

First concentrate on a tree of periodicity ℓ = 2, then generalize. Theorem Given a, b ∈ N and 2 ≤ θ < a, b consider BP on periodic tree Ta,b with the initial probability p. There exists pf ∈ (0, 1) such that a.a.s. (i) Ta,b becomes completely active for p ≥ pf , (ii) Ta,b does not become completely active for p < pf .

18 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

Proof Outline

1 Derive the percolation threshold

pf on Ta,b.

2 Relate this to the percolation threshold pf on Ta,b. 19 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

BP on (oriented) Ta,b

1 Call Va (resp. Vb) the set of nodes of degree a + 1 (resp.

b + 1).

2 Dynamics of BP on

Ta,b captured by ζt(v), ηt(u) ∈ {0, 1} for every v ∈ Va and u ∈ Vb at time t = 0, 1, . . . .

3 Given symmetry and dynamical rules the marginal distribution

that a node v ∈ Va at time t is active is the same for all nodes in Va.

20 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

BP on (oriented) Ta,b

1 Call Va (resp. Vb) the set of nodes of degree a + 1 (resp.

b + 1).

2 Dynamics of BP on

Ta,b captured by ζt(v), ηt(u) ∈ {0, 1} for every v ∈ Va and u ∈ Vb at time t = 0, 1, . . . .

3 Given symmetry and dynamical rules the marginal distribution

that a node v ∈ Va at time t is active is the same for all nodes in Va.

20 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

BP on (oriented) Ta,b

1 Call Va (resp. Vb) the set of nodes of degree a + 1 (resp.

b + 1).

2 Dynamics of BP on

Ta,b captured by ζt(v), ηt(u) ∈ {0, 1} for every v ∈ Va and u ∈ Vb at time t = 0, 1, . . . .

3 Given symmetry and dynamical rules the marginal distribution

that a node v ∈ Va at time t is active is the same for all nodes in Va.

20 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

1 Choose any node v0 ∈ Va, and independently u0 ∈ Vb. 2 Denote

xt := P ( ηt(v0) = 1) and yt := P

• ζt(u0) = 1
• .

3 The probability

xt that a node v0 is active at time t is: P ( ηt(v0) = 1) = P ( η0(v0) = 1) + P ( η0(v0) = 0) P

uv0

• ζt(u) ≥ θ :

η0(v0) = 0

• .

4

• ζt(u) are independent Bernoulli random variables, and

independent of η0(v0):

• xt = p + (1 − p)

a

• k=θ
• a

k

• yk

t−1 (1 −

yt−1)a−k .

21 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

1 Choose any node v0 ∈ Va, and independently u0 ∈ Vb. 2 Denote

xt := P ( ηt(v0) = 1) and yt := P

• ζt(u0) = 1
• .

3 The probability

xt that a node v0 is active at time t is: P ( ηt(v0) = 1) = P ( η0(v0) = 1) + P ( η0(v0) = 0) P

uv0

• ζt(u) ≥ θ :

η0(v0) = 0

• .

4

• ζt(u) are independent Bernoulli random variables, and

independent of η0(v0):

• xt = p + (1 − p)

a

• k=θ
• a

k

• yk

t−1 (1 −

yt−1)a−k .

21 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

1 Choose any node v0 ∈ Va, and independently u0 ∈ Vb. 2 Denote

xt := P ( ηt(v0) = 1) and yt := P

• ζt(u0) = 1
• .

3 The probability

xt that a node v0 is active at time t is: P ( ηt(v0) = 1) = P ( η0(v0) = 1) + P ( η0(v0) = 0) P

uv0

• ζt(u) ≥ θ :

η0(v0) = 0

• .

4

• ζt(u) are independent Bernoulli random variables, and

independent of η0(v0):

• xt = p + (1 − p)

a

• k=θ
• a

k

• yk

t−1 (1 −

yt−1)a−k .

21 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

Analogously for the nodes from u0 ∈ Vb,

• yt = p + (1 − p)

b

• k=θ
• b

k

• xk

t−1 (1 −

xt−1)b−k .

22 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

Recurrence

• xt = Φa,p,θ(

yt−1) ,

• yt = Φb,p,θ(

xt−1) , equivalently

• xt = Φa,p,θ (Φb,p,θ(

xt−1)) ,

• yt = Φb,p,θ (Φa,p,θ(

yt−1)) .

23 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

Recurrence

Consider Φa,p,θ ◦ Φb,p,θ : [0, 1] → [0, 1]. There exist limits x∞, y∞ in [0, 1]:

• x∞

= Φa,p,θ (Φb,p,θ( x∞)) ,

• y∞

= Φb,p,θ (Φa,p,θ( y∞)) . Notice ( x∞, y∞) = (1, 1) is a solution for any p. Exists pf , s.t. for p ≥ pf the only solution is ( x∞, y∞) = (1, 1).

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Recurrence

Consider Φa,p,θ ◦ Φb,p,θ : [0, 1] → [0, 1]. There exist limits x∞, y∞ in [0, 1]:

• x∞

= Φa,p,θ (Φb,p,θ( x∞)) ,

• y∞

= Φb,p,θ (Φa,p,θ( y∞)) . Notice ( x∞, y∞) = (1, 1) is a solution for any p. Exists pf , s.t. for p ≥ pf the only solution is ( x∞, y∞) = (1, 1).

24 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

Recurrence

Consider Φa,p,θ ◦ Φb,p,θ : [0, 1] → [0, 1]. There exist limits x∞, y∞ in [0, 1]:

• x∞

= Φa,p,θ (Φb,p,θ( x∞)) ,

• y∞

= Φb,p,θ (Φa,p,θ( y∞)) . Notice ( x∞, y∞) = (1, 1) is a solution for any p. Exists pf , s.t. for p ≥ pf the only solution is ( x∞, y∞) = (1, 1).

24 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

Recurrence

Consider Φa,p,θ ◦ Φb,p,θ : [0, 1] → [0, 1]. There exist limits x∞, y∞ in [0, 1]:

• x∞

= Φa,p,θ (Φb,p,θ( x∞)) ,

• y∞

= Φb,p,θ (Φa,p,θ( y∞)) . Notice ( x∞, y∞) = (1, 1) is a solution for any p. Exists pf , s.t. for p ≥ pf the only solution is ( x∞, y∞) = (1, 1).

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BP on an unoriented tree Ta,b

Let x∞, y∞ be the probabilities that v0 ∈ Va, u0 ∈ Vb become eventually active in BP on unoriented Ta,b (resp.). Theorem The probabilities x∞, x∞, y∞, y∞ satisfy x∞ = p + (1 − p)

a+1

• k=θ
• a + 1

k

• yk

∞(1 −

y∞)b+1−k , and y∞ = p + (1 − p)

b+1

• k=θ
• b + 1

k

• xk

∞(1 −

x∞)b+1−k .

25 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

Proof Outline

Goal: Relate the critical thresholds for BP on Ta,b and Ta,b, i.e.

• pf = pf .

Consider a root node u0 of dergeee b + 1 and subtrees T1, . . . , Tb+1 incident to the root.

v0 v1 v2 vb+1 1 2 a 1 2 a 1 2 a T1 T2 Tb+1

Figure: Periodic tree with the root node u0 and subtrees T1, . . . , Tb+1 incident to the root.

26 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

Proof Outline

Goal: Relate the critical thresholds for BP on Ta,b and Ta,b, i.e.

• pf = pf .

Consider a root node u0 of dergeee b + 1 and subtrees T1, . . . , Tb+1 incident to the root.

v0 v1 v2 vb+1 1 2 a 1 2 a 1 2 a T1 T2 Tb+1

Figure: Periodic tree with the root node u0 and subtrees T1, . . . , Tb+1 incident to the root.

26 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

1 Πt is BP which runs on the entire Ta,b. 2 ζ(i)

t

is BP which starts and runs on Ti only.

3 Then Π∞(u0) = 1 if only if:

(i) Π0(u0) = 1, or (ii) b+1

i=1 ζ(i) t (ui) ≥ θ, given Π0(u0) = 0.

P (Π∞(u0) = 1) = P (Π0(u0) = 1) + P (Π0(u0) = 0) P

b+1

• i=1

ζ(i)

t (ui) ≥ θ

• = p + (1 − p)

b+1

• k=θ
• b + 1

k

• P
• ζ(1)

∞ (u1) = 1

k

×

• 1 − P
• ζ(1)

∞ (u1) = 1

b+1−k .

27 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

Analogously P (Π∞(v0) = 1) = P (Π0(v0) = 1) + P (Π0(v0) = 0) P

a+1

• i=1

κ(i)

t (vi) ≥ θ

• = p + (1 − p)

a+1

• k=θ
• a + 1

k

• P
• κ(1)

∞ (v1) = 1

k

×

• 1 − P
• κ(1)

∞ (v1) = 1

a+1−k .

28 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

Comparison: BP on T1 and T1

First (from the stochastic dominance) P

• ζ(1)

∞ (v1) = 1

• ≤ P
• ζ(1)

∞ (v1) = 1

• .

Second (more involved)

• ζ(1)

∞ (v1) = 0

• ⊆
• ζ(1)

∞ (v1) = 0

• , i.e.

P

• ζ(1)

∞ (v1) = 0

• ≤ P
• ζ(1)

∞ (v1) = 0

• .

29 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

x∞ = p + (1 − p)

a+1

• k=θ
• a + 1

k

• yk

∞ (1 −

y∞)a+1−k , y∞ = p + (1 − p)

b+1

• k=θ
• b + 1

k

• xk

∞ (1 −

x∞)b+1−k . (x∞, y∞) = (1, 1) if and only if ( x∞, y∞) = (1, 1) hence pf = pf . By the same method the proof generalizes for trees of periodicity ℓ > 2.

30 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

x∞ = p + (1 − p)

a+1

• k=θ
• a + 1

k

• yk

∞ (1 −

y∞)a+1−k , y∞ = p + (1 − p)

b+1

• k=θ
• b + 1

k

• xk

∞ (1 −

x∞)b+1−k . (x∞, y∞) = (1, 1) if and only if ( x∞, y∞) = (1, 1) hence pf = pf . By the same method the proof generalizes for trees of periodicity ℓ > 2.

30 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

x∞ = p + (1 − p)

a+1

• k=θ
• a + 1

k

• yk

∞ (1 −

y∞)a+1−k , y∞ = p + (1 − p)

b+1

• k=θ
• b + 1

k

• xk

∞ (1 −

x∞)b+1−k . (x∞, y∞) = (1, 1) if and only if ( x∞, y∞) = (1, 1) hence pf = pf . By the same method the proof generalizes for trees of periodicity ℓ > 2.

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Figure: Numerical evaluation of the critical probability pf for n = m and 2 ≤ θ ≤ 9.

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Figure: Numerical evaluation of the critical probability pf for n = m + 1 and 2 ≤ θ ≤ 9.

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Figure: Numerical evaluation of the critical probability pf for n = m + 2 and 2 ≤ θ ≤ 9.

33 / 34 Milan Bradonjić Bootstrap Percolation on Periodic Trees

Thank You!

ect.bell-labs.com/who/milan milan@research.bell-labs.com

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