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 S ℓ is a tree with V ( S ℓ ) = { 1 , . . . , ℓ } .
Setup S ℓ is a tree with V ( S ℓ ) = { 1 , . . . , ℓ } . A random tree T n = T n ( S ℓ ) with V ( T n ) = { 1 , . . . , n } is a uniform attachment tree with seed S ℓ if it is generated as follows:
Setup S ℓ is a tree with V ( S ℓ ) = { 1 , . . . , ℓ } . A random tree T n = T n ( S ℓ ) with V ( T n ) = { 1 , . . . , n } is a uniform attachment tree with seed S ℓ if it is generated as follows: ◮ T ℓ = S ℓ ;
Setup S ℓ is a tree with V ( S ℓ ) = { 1 , . . . , ℓ } . A random tree T n = T n ( S ℓ ) with V ( T n ) = { 1 , . . . , n } is a uniform attachment tree with seed S ℓ if it is generated as follows: ◮ T ℓ = S ℓ ; ◮ T i is obtained by joining vertex i to a vertex of T i − 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 ( S 1 , S 2 ) = lim n →∞ TV ( T n ( S 1 ) , T n ( S 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 ( S 1 , S 2 ) = lim n →∞ TV ( T n ( S 1 ) , T n ( S 2 )) 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 ( S 1 , S 2 ) = lim n →∞ TV ( T n ( S 1 ) , T n ( S 2 )) Is δ a metric? Curien, Duquesne, Kortchemski and Manolescu: YES. [4]
The problem
The problem Given T n ( S ℓ ) e want to find ◮ either a big set H 1 ( T n , ǫ ) such that P ( H 1 ( T n , ǫ ) ⊂ S ℓ ) ≥ 1 − ǫ ;
The problem Given T n ( S ℓ ) e want to find ◮ either a big set H 1 ( T n , ǫ ) such that P ( H 1 ( T n , ǫ ) ⊂ S ℓ ) ≥ 1 − ǫ ; ◮ or a small a set H 2 ( T n , ǫ ) such that P ( H 2 ( T n , ǫ ) ⊃ S ℓ ) ≥ 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 P ℓ .
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 P ℓ . ◮ Star E ℓ .
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 P ℓ . ◮ Star E ℓ . ◮ UART T ℓ .
Theorem 1: S ℓ = P ℓ
Theorem 1: S ℓ = P ℓ � 2 e 2 ǫ , 2 e 2 γ log 1 � γ log(4 e 2 ) For ℓ ≥ max we have the following:
Theorem 1: S ℓ = P ℓ � 2 e 2 ǫ , 2 e 2 γ log 1 � γ log(4 e 2 ) For ℓ ≥ max we have the following: Given T n ( P ℓ ), n >> 1 we can find a set H n = H n ( T n , ε ) ⊂ { 1 , . . . , n } with | H n | ≥ (1 − γ ) ℓ such that
Theorem 1: S ℓ = P ℓ � 2 e 2 ǫ , 2 e 2 γ log 1 � γ log(4 e 2 ) For ℓ ≥ max we have the following: Given T n ( P ℓ ), n >> 1 we can find a set H n = H n ( T n , ε ) ⊂ { 1 , . . . , n } with | H n | ≥ (1 − γ ) ℓ such that P { H n ⊂ P ℓ } ≥ 1 − ǫ .
Theorem 2: S ℓ = E ℓ � C , 8 � log 1 For ℓ ≥ max ǫ we have the following: γ Given T n ( E ℓ ), n >> 1 we can find a set H n = H n ( T n , ε ) ⊂ { 1 , . . . , n } with | H n | ≤ (1 + γ ) ℓ such that P { H n ⊃ E ℓ } ≥ 1 − ǫ .
Theorem 3: S ℓ = T ℓ There exist c 1 and c 2 such that the following holds. Let ℓ ≥ c 1 log 2 1 ǫ . Given T n ( T ℓ ), n >> 1 we can find a set H n = H n ( T n , ε ) ⊂ { 1 , . . . , n } with | H n | ≥ ℓ/ [ c 2 log( ℓ/ǫ )] such that P { H n ⊂ T ℓ } ≥ 1 − ǫ .
Theorem 4 Theorem Let ǫ ∈ (0 , e − e 2 ) . Suppose that T n is a uniform attachment tree log(1 /ǫ ) with seed S ℓ = P ℓ or S ℓ = E ℓ for ℓ ≤ log log(1 /ǫ ) . Then, for all n ≥ 2 ℓ , any seed-finding algorithm that outputs a vertex set H n of size ℓ has � | H n ∩ S ℓ | ≤ ℓ � ≥ ǫ . P 2
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 H n will be the set of the most central vertices in T n .
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 H n will be the set of the most central vertices in T n . 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 S ℓ = P ℓ
Sketch of the proof: Case S ℓ = P ℓ We will prove that old central vertices are more central than new vertices.
Sketch of the proof: Case S ℓ = P ℓ We will prove that old central vertices are more central than new vertices. More precisely � � P ℓγ/ 2 ≤ j ≤ ℓ (1 − γ/ 2) ψ ( j ) < min max ℓ< i ≤ n ψ ( i ) ≥ 1 − ǫ .
Sketch of the proof: Case S ℓ = P ℓ We will prove that old central vertices are more central than new vertices. More precisely � � P ℓγ/ 2 ≤ j ≤ ℓ (1 − γ/ 2) ψ ( j ) < min max 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 � � ℓ< i ≤ n ψ ( i ) ≤ min ℓγ/ 2 ≤ j ≤ ℓ (1 − γ/ 2) ψ ( j ) max P (1 − γ/ 2) ℓ � � � ≤ P ℓ< i ≤ n ψ ( i ) ≤ ψ ( j ) min j = γℓ/ 2 (1 − γ/ 2) ℓ ℓ � � � � ≤ ∃ v ∈ C k \{ k } : ψ ( v ) ≤ ψ ( j ) . P j = γℓ/ 2 k =1
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.
Recommend
More recommend
