Finding the Seed of Uniform Attachment Trees Alan Pereira - UFGM G - - PowerPoint PPT Presentation

Finding the Seed of Uniform Attachment Trees

Alan Pereira - UFGM G´ abor Lugosi - UPF July 26, 2019

Network Archaeology on Random Trees Setup Results Skecth of the proofs

Introduction

Studies questions about old or extinct networks.

Introduction

Studies questions about old or extinct networks. We want to find a source of a rumor/disease.

Introduction

Studies questions about old or extinct networks. We want to find a source of a rumor/disease. This problem was popularized by Shah-Zamah [6].

Setup

Setup

Sℓ is a tree with V (Sℓ) = {1, . . . , ℓ}.

Setup

Sℓ is a tree with V (Sℓ) = {1, . . . , ℓ}. A random tree Tn = Tn(Sℓ) with V (Tn) = {1, . . . , n} is a uniform attachment tree with seed Sℓ if it is generated as follows:

Setup

Sℓ is a tree with V (Sℓ) = {1, . . . , ℓ}. A random tree Tn = Tn(Sℓ) with V (Tn) = {1, . . . , n} is a uniform attachment tree with seed Sℓ if it is generated as follows: ◮ Tℓ = Sℓ ;

Setup

Sℓ is a tree with V (Sℓ) = {1, . . . , ℓ}. A random tree Tn = Tn(Sℓ) with V (Tn) = {1, . . . , n} is a uniform attachment tree with seed Sℓ if it is generated as follows: ◮ Tℓ = Sℓ ; ◮ Ti is obtained by joining vertex i to a vertex of Ti−1 chosen uniformly at random, independently of the past.

Influence of the seed

Bubeck, (Eldan), Mossel, and R´ acz studied the influence of the seed in the growth of the random tree, first in preferential attachment [3] and after in uniform attachment [2].

Influence of the seed

Bubeck, (Eldan), Mossel, and R´ acz studied the influence of the seed in the growth of the random tree, first in preferential attachment [3] and after in uniform attachment [2]. They did it by analysing δ(S1, S2) = lim

n→∞ TV (Tn(S1), Tn(S2))

Influence of the seed

Bubeck, (Eldan), Mossel, and R´ acz studied the influence of the seed in the growth of the random tree, first in preferential attachment [3] and after in uniform attachment [2]. They did it by analysing δ(S1, S2) = lim

n→∞ TV (Tn(S1), Tn(S2))

Is δ a metric?

Influence of the seed

Bubeck, (Eldan), Mossel, and R´ acz studied the influence of the seed in the growth of the random tree, first in preferential attachment [3] and after in uniform attachment [2]. They did it by analysing δ(S1, S2) = lim

n→∞ TV (Tn(S1), Tn(S2))

Is δ a metric? Curien, Duquesne, Kortchemski and Manolescu: YES. [4]

The problem

The problem

Given Tn(Sℓ) e want to find ◮ either a big set H1(Tn, ǫ) such that P(H1(Tn, ǫ) ⊂ Sℓ) ≥ 1 − ǫ;

The problem

Given Tn(Sℓ) e want to find ◮ either a big set H1(Tn, ǫ) such that P(H1(Tn, ǫ) ⊂ Sℓ) ≥ 1 − ǫ; ◮ or a small a set H2(Tn, ǫ) such that P(H2(Tn, ǫ) ⊃ Sℓ) ≥ 1 − ǫ.

Finding Adam

Finding Adam

Bubeck, Devroye, and Lugosi [1] considered the case ℓ = 1 (in UA and PA).

Finding Adam

Bubeck, Devroye, and Lugosi [1] considered the case ℓ = 1 (in UA and PA). Jog and Loh [5] considered the same problem in non-linear preferential attachment.

Finding Adam

Bubeck, Devroye, and Lugosi [1] considered the case ℓ = 1 (in UA and PA). Jog and Loh [5] considered the same problem in non-linear preferential attachment. We considered the cases ◮ Path Pℓ.

Finding Adam

Bubeck, Devroye, and Lugosi [1] considered the case ℓ = 1 (in UA and PA). Jog and Loh [5] considered the same problem in non-linear preferential attachment. We considered the cases ◮ Path Pℓ. ◮ Star Eℓ.

Finding Adam

Bubeck, Devroye, and Lugosi [1] considered the case ℓ = 1 (in UA and PA). Jog and Loh [5] considered the same problem in non-linear preferential attachment. We considered the cases ◮ Path Pℓ. ◮ Star Eℓ. ◮ UART Tℓ.

Theorem 1: Sℓ = Pℓ

Theorem 1: Sℓ = Pℓ

For ℓ ≥ max 2e2 γ log 1 ǫ , 2e2 γ log(4e2)

• we have the following:

Theorem 1: Sℓ = Pℓ

For ℓ ≥ max 2e2 γ log 1 ǫ , 2e2 γ log(4e2)

• we have the following:

Given Tn(Pℓ), n >> 1 we can find a set Hn = Hn(Tn, ε) ⊂ {1, . . . , n} with |Hn| ≥ (1 − γ)ℓ such that

Theorem 1: Sℓ = Pℓ

For ℓ ≥ max 2e2 γ log 1 ǫ , 2e2 γ log(4e2)

• we have the following:

Given Tn(Pℓ), n >> 1 we can find a set Hn = Hn(Tn, ε) ⊂ {1, . . . , n} with |Hn| ≥ (1 − γ)ℓ such that P {Hn ⊂ Pℓ} ≥ 1 − ǫ .

Theorem 2: Sℓ = Eℓ

For ℓ ≥ max

• C, 8

γ

• log 1

ǫ we have the following: Given Tn(Eℓ), n >> 1 we can find a set Hn = Hn(Tn, ε) ⊂ {1, . . . , n} with |Hn| ≤ (1 + γ)ℓ such that P {Hn ⊃ Eℓ} ≥ 1 − ǫ .

Theorem 3: Sℓ = Tℓ

There exist c1 and c2 such that the following holds. Let ℓ ≥ c1 log2 1 ǫ . Given Tn(Tℓ), n >> 1 we can find a set Hn = Hn(Tn, ε) ⊂ {1, . . . , n} with |Hn| ≥ ℓ/[c2 log(ℓ/ǫ)] such that P {Hn ⊂ Tℓ} ≥ 1 − ǫ .

Theorem 4

Theorem

Let ǫ ∈ (0, e−e2). Suppose that Tn is a uniform attachment tree with seed Sℓ = Pℓ or Sℓ = Eℓ for ℓ ≤

log(1/ǫ) log log(1/ǫ). Then, for all

n ≥ 2ℓ, any seed-finding algorithm that outputs a vertex set Hn of size ℓ has P

• |Hn ∩ Sℓ| ≤ ℓ

2

• ≥ ǫ .

How can we prove it?

How can we prove it?

The main idea is to prove that old vertices are more central than the new vertices (in some sense).

How can we prove it?

The main idea is to prove that old vertices are more central than the new vertices (in some sense). The set Hn will be the set of the most central vertices in Tn.

How can we prove it?

The main idea is to prove that old vertices are more central than the new vertices (in some sense). The set Hn will be the set of the most central vertices in Tn. Let us define what means be more central.

Rooted tree and Induced subtree

Rooted tree and Induced subtree

A rooted tree (T, v) is the tree T with a distinguished vertex v ∈ V (T).

Rooted tree and Induced subtree

A rooted tree (T, v) is the tree T with a distinguished vertex v ∈ V (T). The subtree induced by u (T, v)u↓ is the subtree of (T, v) which grows from u in the opposite direction of v.

Rooted tree and Induced subtree

A rooted tree (T, v) is the tree T with a distinguished vertex v ∈ V (T). The subtree induced by u (T, v)u↓ is the subtree of (T, v) which grows from u in the opposite direction of v.

Centrality: Definition

Given a tree T, the anti-centrality of a vertex v ∈ V (T) is defined by ψ(v) = max

u∈N(v) |(T, v)u↓| .

Centrality: Definition

Given a tree T, the anti-centrality of a vertex v ∈ V (T) is defined by ψ(v) = max

u∈N(v) |(T, v)u↓| .

Centrality

Given v, we denote v′ to be some vertex in N(v) such that ψ(v) =

• (T, v)v′↓
• .

Comparing Centrality

Comparing Centrality

Case 1: When v is between v′ and j we have ψ(v) ≤ ψ(j)

Comparing Centrality

Case 1: When v is between v′ and j we have ψ(v) ≤ ψ(j)

Comparing Centrality

Case 2: When v′ is between v and j we have ψ(v) ≤ ψ(j) if

Comparing Centrality

Case 2: When v′ is between v and j we have ψ(v) ≤ ψ(j) if ◮ |(T, j)v↓| ≥ |(T, v)j↓|;

Sketch of the proof: Case Sℓ = Pℓ

Sketch of the proof: Case Sℓ = Pℓ

We will prove that

• ld central vertices are more central than new vertices.

Sketch of the proof: Case Sℓ = Pℓ

We will prove that

• ld central vertices are more central than new vertices.

More precisely P

• max

ℓγ/2≤j≤ℓ(1−γ/2) ψ(j)< min ℓ<i≤n ψ(i)

• ≥ 1 − ǫ .

Sketch of the proof: Case Sℓ = Pℓ

We will prove that

• ld central vertices are more central than new vertices.

More precisely P

• max

ℓγ/2≤j≤ℓ(1−γ/2) ψ(j)< min ℓ<i≤n ψ(i)

• ≥ 1 − ǫ .

Let us prove that the complement has small probability.

Sketch of the proof

By Union Bound we have P

• min

ℓ<i≤n ψ(i) ≤

max

ℓγ/2≤j≤ℓ(1−γ/2) ψ(j)

• ≤

(1−γ/2)ℓ

• j=γℓ/2

P

• min

ℓ<i≤n ψ(i) ≤ ψ(j)

• ≤

(1−γ/2)ℓ

• j=γℓ/2

ℓ

• k=1

P

• ∃v ∈ Ck\{k} : ψ(v) ≤ ψ(j)
• .

First case: (v ′, v, j)

Second case: (v, v ′, j)

Second case: (v, v ′, j)

The value of ℓ arise from a optimization of the bounds.

S´ ebastien Bubeck, Luc Devroye, and G´ abor Lugosi. Finding Adam in random growing trees. Random Structures & Algorithms, 50(2):158–172, 2017. S´ ebastien Bubeck, Ronen Eldan, Elchanan Mossel, and Mikl´

• s

R´ acz. From trees to seeds: on the inference of the seed from large trees in the uniform attachment model. Bernoulli, 23(4A):2887–2916, 2017. S´ ebastien Bubeck, Elchanan Mossel, and Mikl´

• s Z R´

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