The Brownian Rabbit and scaling limits of preferential attachment trees

The Brownian Rabbit and

scaling limits of preferential attachment trees

Igor Kortchemski (joint work with N. Curien, T. Duquesne & I. Manolescu)

DMA – École Normale Supérieure – Paris

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Outline

I. Preferential attachment and influence of the seed II. Looptrees and preferential attachment

III. Extensions and conjectures

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Outline

I. Preferential attachment and influence of the seed

II. Looptrees and preferential attachment III. Extensions and conjectures

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Trees built by preferential attachment

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Trees built by preferential attachment

Let(Tn^(S))n>k be a sequence of random trees grown recursively at random:

I T_k^(S)=Sis a tree withkvertices (the seed),

I for everyn>1,T_n+1^(S) is obtained fromTn^(S)by adding an edge to a vertex ofTn^(S)

chosen at randomproportionally to its degree.

(animation of preferential attachment here)

This is the preferential attachement model (Szymánski ’87; Albert & Barabási

’99; Bollobás, Riordan, Spencer & Tusnády ’01).

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Trees built by preferential attachment

Let(Tn^(S))n>k be a sequence of random trees grown recursively at random:

I T_k^(S)=Sis a tree withkvertices (the seed),

I for everyn>1,T_n+1^(S) is obtained fromTn^(S)by adding an edge to a vertex ofTn^(S)

chosen at randomproportionally to its degree.

(animation of preferential attachment here)

This is the preferential attachement model (Szymánski ’87; Albert & Barabási

’99; Bollobás, Riordan, Spencer & Tusnády ’01).

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Trees built by preferential attachment

Let(Tn^(S))n>k be a sequence of random trees grown recursively at random:

I T_k^(S)=Sis a tree withkvertices (the seed),

I for everyn>1,T_n+1^(S) is obtained fromTn^(S)by adding an edge to a vertex ofTn^(S)

chosen at randomproportionally to its degree.

(animation of preferential attachment here)

This is the preferential attachement model (Szymánski ’87; Albert & Barabási

’99; Bollobás, Riordan, Spencer & Tusnády ’01).

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Trees built by preferential attachment

Let(Tn^(S))n>k be a sequence of random trees grown recursively at random:

I T_k^(S)=Sis a tree withkvertices (the seed),

I for everyn>1,T_n+1^(S) is obtained fromTn^(S)by adding an edge to a vertex ofTn^(S) chosen at random

proportionally to its degree.

(animation of preferential attachment here)

This is the preferential attachement model (Szymánski ’87; Albert & Barabási

’99; Bollobás, Riordan, Spencer & Tusnády ’01).

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Trees built by preferential attachment

Let(Tn^(S))n>k be a sequence of random trees grown recursively at random:

I T_k^(S)=Sis a tree withkvertices (the seed),

I for everyn>1,T_n+1^(S) is obtained fromTn^(S)by adding an edge to a vertex ofTn^(S) chosen at randomproportionally to its degree.

(animation of preferential attachment here)

This is the preferential attachement model (Szymánski ’87; Albert & Barabási

’99; Bollobás, Riordan, Spencer & Tusnády ’01).

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Trees built by preferential attachment

Let(Tn^(S))n>k be a sequence of random trees grown recursively at random:

I T_k^(S)=Sis a tree withkvertices (the seed),

I for everyn>1,T_n+1^(S) is obtained fromTn^(S)by adding an edge to a vertex ofTn^(S) chosen at randomproportionally to its degree.

(animation of preferential attachment here)

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Influence of the seed

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Question(Bubeck, Mossel & Rácz): What is the influence of the seed tree?

Can one distinguish asymptotically between different seeds?

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Question(Bubeck, Mossel & Rácz): What is the influence of the seed tree?

Can one distinguish asymptotically between different seeds?

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Four simulations ofTn^(S1) forn=5000:

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Four simulations ofTn^(S2) forn=5000:

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Referendum

Four simulations ofTn^(S1),Tn^(S2) forn=5000:

Do we haveS1=S2?

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Referendum

Four simulations ofTn^(S1),Tn^(S2) forn=5000:

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Referendum

Four simulations ofTn^(S1),Tn^(S2) forn=5000:

S1 S₂

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Four simulations ofTn^(S10) forn=5000:

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Four simulations ofTn^(S20) forn=5000:

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Referendum

Four simulations ofTn^(S10),Tn^(S20) forn=5000:

Do we haveS₁⁰ =S₂⁰?

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Referendum

Four simulations ofTn^(S10),Tn^(S20) forn=5000:

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Referendum

Four simulations ofTn^(S10),Tn^(S20) forn=5000:

S’₁ S’₂

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Influence of the seed graph

For finite treesS1andS2, set

d(S1,S2) = lim

n→∞dTV(T_n^(S1),T_n^(S2)),

where dT V denotes the total variation distance for random variables taking values in the space of finite trees

(dT V(X,Y) =sup_A|P(X∈A) −P(Y∈A)|).

Proposition (Bubeck, Mossel & Rácz ’14)

The function dis a pseudo-metric.

Conjecture (Bubeck, Mossel & Rácz ’14)

The function dis a metric on trees with at least3vertices.

Bubeck, Mossel & Rácz showed that this is true whenS1andS2do not have the same degree sequence by studying the maximal degree ofTn^(S).

The conjecture is true.

Theorem(Curien, Duquesne, K. & Manolescu ’14).

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Influence of the seed graph

For finite treesS1andS2, set

d(S1,S2) = lim

n→∞dTV(T_n^(S1),T_n^(S2)),

where dT V denotes the total variation distance for random variables taking values in the space of finite trees (dT V(X,Y) =sup_A|P(X∈A) −P(Y∈A)|).

Proposition (Bubeck, Mossel & Rácz ’14)

The function dis a pseudo-metric.

Conjecture (Bubeck, Mossel & Rácz ’14)

The function dis a metric on trees with at least3vertices.

Bubeck, Mossel & Rácz showed that this is true whenS1andS2do not have the same degree sequence by studying the maximal degree ofTn^(S).

The conjecture is true.

Theorem(Curien, Duquesne, K. & Manolescu ’14).

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Influence of the seed graph

For finite treesS1andS2, set

d(S1,S2) = lim

n→∞dTV(T_n^(S1),T_n^(S2)),

where dT V denotes the total variation distance for random variables taking values in the space of finite trees (dT V(X,Y) =sup_A|P(X∈A) −P(Y∈A)|).

Proposition (Bubeck, Mossel & Rácz ’14)

The function dis a pseudo-metric.

Conjecture (Bubeck, Mossel & Rácz ’14)

The function dis a metric on trees with at least3vertices.

Bubeck, Mossel & Rácz showed that this is true whenS1andS2do not have the same degree sequence by studying the maximal degree ofTn^(S).

The conjecture is true.

Theorem(Curien, Duquesne, K. & Manolescu ’14).

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Influence of the seed graph

For finite treesS1andS2, set

d(S1,S2) = lim

n→∞dTV(T_n^(S1),T_n^(S2)),

where dT V denotes the total variation distance for random variables taking values in the space of finite trees (dT V(X,Y) =sup_A|P(X∈A) −P(Y∈A)|).

Proposition (Bubeck, Mossel & Rácz ’14)

The function dis a pseudo-metric.

Conjecture (Bubeck, Mossel & Rácz ’14)

The function dis a metric on trees with at least3vertices.

Bubeck, Mossel & Rácz showed that this is true whenS1andS2do not have the same degree sequence by studying the maximal degree ofTn^(S).

The conjecture is true.

Theorem(Curien, Duquesne, K. & Manolescu ’14).

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Influence of the seed graph

For finite treesS1andS2, set

d(S1,S2) = lim

n→∞dTV(T_n^(S1),T_n^(S2)),

where dT V denotes the total variation distance for random variables taking values in the space of finite trees (dT V(X,Y) =sup_A|P(X∈A) −P(Y∈A)|).

Proposition (Bubeck, Mossel & Rácz ’14)

The function dis a pseudo-metric.

Conjecture (Bubeck, Mossel & Rácz ’14)

The function dis a metric on trees with at least3vertices.

Bubeck, Mossel & Rácz showed that this is true whenS1andS2do not have the same degree sequence by studying the maximal degree ofTn^(S).

The conjecture is true.

Theorem(Curien, Duquesne, K. & Manolescu ’14).

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Influence of the seed graph

For finite treesS1andS2, set

d(S1,S2) = lim

n→∞dTV(T_n^(S1),T_n^(S2)),

where dT V denotes the total variation distance for random variables taking values in the space of finite trees (dT V(X,Y) =sup_A|P(X∈A) −P(Y∈A)|).

Proposition (Bubeck, Mossel & Rácz ’14)

The function dis a pseudo-metric.

Conjecture (Bubeck, Mossel & Rácz ’14)

The function dis a metric on trees with at least3vertices.

Bubeck, Mossel & Rácz showed that this is true whenS1andS2do not have the same degree sequence by studying the maximal degree ofTn^(S).

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Plane trees built by preferential attachment

Let(Tn^(S))n>k be a sequence of randomplanetrees grown recursively at random:

I T_k^(S)=Sis a planetree withk vertices (the seed),

I for everyn>1,T_n+1^(S) is obtained fromTn^(S)by adding an edge into a corner ofTn^(S)

chosenuniformlyat random.

y The graph structure ofTn^(S)is that of preferential attachment.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Plane trees built by preferential attachment

Let(Tn^(S))n>k be a sequence of randomplanetrees grown recursively at random:

I T_k^(S)=Sis a planetree withkvertices (the seed),

I for everyn>1,T_n+1^(S) is obtained fromTn^(S)by adding an edge into a corner ofTn^(S)

chosenuniformlyat random.

y The graph structure ofTn^(S)is that of preferential attachment.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Plane trees built by preferential attachment

Let(Tn^(S))n>k be a sequence of randomplanetrees grown recursively at random:

I T_k^(S)=Sis a planetree withkvertices (the seed),

I for everyn>1,T_n+1^(S) is obtained fromTn^(S)by adding an edge into a corner ofTn^(S)

chosenuniformlyat random.

y The graph structure ofTn^(S)is that of preferential attachment.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Plane trees built by preferential attachment

Let(Tn^(S))n>k be a sequence of randomplanetrees grown recursively at random:

I T_k^(S)=Sis a planetree withkvertices (the seed),

I for everyn>1,T_n+1^(S) is obtained fromTn^(S)by adding an edge into a corner ofTn^(S) chosenuniformly at random.

y The graph structure ofTn^(S)is that of preferential attachment.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Plane trees built by preferential attachment

Let(Tn^(S))n>k be a sequence of randomplanetrees grown recursively at random:

I T_k^(S)=Sis a planetree withkvertices (the seed),

I for everyn>1,T_n+1^(S) is obtained fromTn^(S)by adding an edge into a corner ofTn^(S) chosenuniformly at random.

y The graph structure ofTn^(S)is that of preferential attachment.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Our observables: embeddings of decorated trees

A decorated treeτis a treeτwith positive integer labels(`(u),u∈τ).

We imagine that there are`(u)non-distinguishable arrows pointing to each vertex u∈τ.

IfT is a plane tree,Dτ(T)denotes the number of decorated embeddings ofτin T.

I.e. D_τ(T)is the number of ways to embedτinT s.t. each arrow pointing to a vertex ofτis associated with a corner of T adjacent to the corresponding vertex (distinct arrows associated with distinct corners).

1 1

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Our observables: embeddings of decorated trees

A decorated treeτis a treeτwith positive integer labels(`(u),u∈τ). We imagine that there are`(u)non-distinguishable arrows pointing to each vertex u∈τ.

IfT is a plane tree,Dτ(T)denotes the number of decorated embeddings ofτin T.

I.e. D_τ(T)is the number of ways to embedτinT s.t. each arrow pointing to a vertex ofτis associated with a corner of T adjacent to the corresponding vertex (distinct arrows associated with distinct corners).

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Our observables: embeddings of decorated trees

A decorated treeτis a treeτwith positive integer labels(`(u),u∈τ). We imagine that there are`(u)non-distinguishable arrows pointing to each vertex u∈τ.

IfT is a plane tree,Dτ(T)denotes the number of decorated embeddings ofτin T.

I.e. Dτ(T)is the number of ways to embedτinT s.t. each arrow pointing to a vertex ofτis associated with a corner of T adjacent to the corresponding vertex (distinct arrows associated with distinct corners).

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Our observables: embeddings of decorated trees

A decorated treeτis a treeτwith positive integer labels(`(u),u∈τ). We imagine that there are`(u)non-distinguishable arrows pointing to each vertex u∈τ.

IfT is a plane tree,Dτ(T)denotes the number of decorated embeddings ofτin T.

I.e. Dτ(T)is the number of ways to embedτinT s.t. each arrow pointing to a vertex ofτis associated with a corner ofT adjacent to the corresponding vertex (distinct arrows associated with distinct corners).

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Our observables: embeddings of decorated trees

Hereτ_k:=unique vertex with labelk.

- D_τ1(T_n^(S))=

2n−2(the number of corners of a tree withnvertices). - What aboutD_τ2(Tn^(S))?

E h

Dτ₂(T_n+1^(S)) T_n^(S)i

=

1+ 2

2n−2

Dτ₂(T_n^(S))+1.

Hence there exist constantsαn,βn such that M2(n)=αnDτ₂(T_n^(S))−βn

is a martingale.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Our observables: embeddings of decorated trees

Hereτ_k:=unique vertex with labelk.

- D_τ1(T_n^(S))=

2n−2(the number of corners of a tree withnvertices). - What aboutD_τ2(Tn^(S))?

E h

Dτ₂(T_n+1^(S)) T_n^(S)i

=

1+ 2

2n−2

Dτ₂(T_n^(S))+1.

Hence there exist constantsαn,βn such that M2(n)=αnDτ₂(T_n^(S))−βn

is a martingale.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Our observables: embeddings of decorated trees

Hereτ_k:=unique vertex with labelk.

- D_τ1(T_n^(S))=2n−2

(the number of corners of a tree withnvertices). - What aboutD_τ2(Tn^(S))?

E h

Dτ₂(T_n+1^(S)) T_n^(S)i

=

1+ 2

2n−2

Dτ₂(T_n^(S))+1.

Hence there exist constantsαn,βn such that M2(n)=αnDτ₂(T_n^(S))−βn

is a martingale.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Our observables: embeddings of decorated trees

Hereτ_k:=unique vertex with labelk.

- D_τ1(T_n^(S))=2n−2(the number of corners of a tree withnvertices).

- What aboutD_τ2(Tn^(S))? E

h

Dτ₂(T_n+1^(S)) T_n^(S)i

=

1+ 2

2n−2

Dτ₂(T_n^(S))+1.

Hence there exist constantsαn,βn such that M2(n)=αnDτ₂(T_n^(S))−βn

is a martingale.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Our observables: embeddings of decorated trees

Hereτ_k:=unique vertex with labelk.

- D_τ1(T_n^(S))=2n−2(the number of corners of a tree withnvertices).

- What aboutD_τ2(T_n^(S))?

E h

Dτ₂(T_n+1^(S)) T_n^(S)i

=

1+ 2

2n−2

Dτ₂(T_n^(S))+1.

Hence there exist constantsαn,βn such that M2(n)=αnDτ₂(T_n^(S))−βn

is a martingale.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Our observables: embeddings of decorated trees

Hereτ_k:=unique vertex with labelk.

- D_τ1(T_n^(S))=2n−2(the number of corners of a tree withnvertices).

- What aboutD_τ2(T_n^(S))?

E h

D_τ2(T_n+1^(S)) T_n^(S)i

=

Dτ₂(T_n^(S))

+ +.

E

hD_τ2(T_n+1^(S)) T_n^(S)i

=

1+ 2

2n−2

D_τ2(T_n^(S))+1.

Hence there exist constantsα_n,β_n such that M2(n)=αnDτ₂(T_n^(S))−βn

is a martingale.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Our observables: embeddings of decorated trees

Hereτ_k:=unique vertex with labelk.

- D_τ1(T_n^(S))=2n−2(the number of corners of a tree withnvertices).

- What aboutD_τ2(T_n^(S))?

E h

D_τ2(T_n+1^(S)) T_n^(S)i

= Dτ₂(T_n^(S)) +

+.

E

hD_τ2(T_n+1^(S)) T_n^(S)i

=

1+ 2

2n−2

D_τ2(T_n^(S))+1.

Hence there exist constantsα_n,β_n such that M2(n)=αnDτ₂(T_n^(S))−βn

is a martingale.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Our observables: embeddings of decorated trees

Hereτ_k:=unique vertex with labelk.

- D_τ1(T_n^(S))=2n−2(the number of corners of a tree withnvertices).

- What aboutD_τ2(T_n^(S))?

E h

D_τ2(T_n+1^(S)) T_n^(S)i

= Dτ₂(T_n^(S))

+(one arrow points to a newly created corner)

+(the two arrows point to the two newly created corners).

E

hD_τ2(T_n+1^(S)) T_n^(S)i

=

1+ 2

2n−2

D_τ2(T_n^(S))+1.

Hence there exist constantsα_n,β_n such that M2(n)=αnDτ₂(T_n^(S))−βn

is a martingale.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Our observables: embeddings of decorated trees

Hereτ_k:=unique vertex with labelk.

- D_τ1(T_n^(S))=2n−2(the number of corners of a tree withnvertices).

- What aboutD_τ2(T_n^(S))?

E h

D_τ2(T_n+1^(S)) T_n^(S)i

= Dτ₂(T_n^(S))

+(one arrow points to a newly created corner) + 1

2n−2Dτ₁(T_n^(S)).

E h

Dτ₂(T_n+1^(S)) T_n^(S)i

=

1+ 2

2n−2

Dτ₂(T_n^(S))+1.

Hence there exist constantsαn,βn such that M2(n)=αnDτ₂(T_n^(S))−βn

is a martingale.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Our observables: embeddings of decorated trees

Hereτ_k:=unique vertex with labelk.

- D_τ1(T_n^(S))=2n−2(the number of corners of a tree withnvertices).

- What aboutD_τ2(T_n^(S))?

E h

D_τ2(T_n+1^(S)) T_n^(S)i

= Dτ₂(T_n^(S))

+(one arrow points to a newly created corner) +1.

E

hD_τ2(T_n+1^(S)) T_n^(S)i

=

1+ 2

2n−2

D_τ2(T_n^(S))+1.

Hence there exist constantsα_n,β_n such that M2(n)=αnDτ₂(T_n^(S))−βn

is a martingale.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Our observables: embeddings of decorated trees

Hereτ_k:=unique vertex with labelk.

- D_τ1(T_n^(S))=2n−2(the number of corners of a tree withnvertices).

- What aboutD_τ2(T_n^(S))?

E h

D_τ2(T_n+1^(S)) T_n^(S)i

= Dτ₂(T_n^(S)) + 2

2n−2Dτ₂(T_n^(S)) +1.

E h

Dτ₂(T_n+1^(S)) T_n^(S)i

=

1+ 2

2n−2

Dτ₂(T_n^(S))+1.

Hence there exist constantsα_n,β_n such that M2(n)=αnDτ₂(T_n^(S))−βn

is a martingale.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Our observables: embeddings of decorated trees

Hereτ_k:=unique vertex with labelk.

- D_τ1(T_n^(S))=2n−2(the number of corners of a tree withnvertices).

- What aboutD_τ2(T_n^(S))?

E h

Dτ₂(T_n+1^(S)) T_n^(S)i

=

1+ 2

2n−2

Dτ₂(T_n^(S))+1.

Hence there exist constantsαn,βn such that M2(n)=αnDτ₂(T_n^(S))−βn

is a martingale.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Our observables: embeddings of decorated trees

Hereτ_k:=unique vertex with labelk.

- D_τ1(T_n^(S))=2n−2(the number of corners of a tree withnvertices).

- What aboutD_τ2(T_n^(S))?

E h

Dτ₂(T_n+1^(S)) T_n^(S)i

=

1+ 2

2n−2

Dτ₂(T_n^(S))+1.

Hence there exist constantsαn,βn such that M2(n)=αnDτ₂(T_n^(S))−βn

is a martingale.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Our observables: embeddings of decorated trees

Hereτ_k:=unique vertex with labelk.

- What aboutD_τ3(T_n^(S))? E

h

Dτ3(T_n+1^(S)) T_n^(S)i

= Dτ₃(T_n^(S)) +

+

E h

Dτ₂(T_n+1^(S)) T_n^(S)i

=

1+ 3

2n−2

Dτ₃(T_n^(S))+ 2

2n−2Dτ₂(T_n^(S)).

Hence there exist constantsa_n,b_n,c_n such that

M3(n)=anDτ₃(T_n^(S))+bnDτ₂(T_n^(S))−cn

is a martingale.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Our observables: embeddings of decorated trees

Hereτ_k:=unique vertex with labelk.

- What aboutD_τ3(T_n^(S))? E

h

Dτ3(T_n+1^(S)) T_n^(S)i

= Dτ₃(T_n^(S))

+(one arrow points to a newly created corner) +(two arrows point to the two newly created corners)

E h

Dτ₂(T_n+1^(S)) T_n^(S)i

=

1+ 3

2n−2

Dτ₃(T_n^(S))+ 2

2n−2Dτ₂(T_n^(S)).

Hence there exist constantsa_n,b_n,c_n such that

M3(n)=anDτ₃(T_n^(S))+bnDτ₂(T_n^(S))−cn

is a martingale.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Our observables: embeddings of decorated trees

Hereτ_k:=unique vertex with labelk.

- What aboutD_τ3(T_n^(S))? E

h

Dτ3(T_n+1^(S)) T_n^(S)i

= Dτ₃(T_n^(S)) + 3

2n−2Dτ₃(T_n^(S))

+(two arrows point to the two newly created corners)

E h

Dτ₂(T_n+1^(S)) T_n^(S)i

=

1+ 3

2n−2

Dτ₃(T_n^(S))+ 2

2n−2Dτ₂(T_n^(S)).

Hence there exist constantsan,bn,cn such that

M₃(n)=a_nDτ₃(T_n^(S))+b_nDτ₂(T_n^(S))−c_n is a martingale.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Our observables: embeddings of decorated trees

Hereτ_k:=unique vertex with labelk.

- What aboutD_τ3(T_n^(S))? E

h

Dτ3(T_n+1^(S)) T_n^(S)i

= Dτ₃(T_n^(S)) + 3

2n−2Dτ₃(T_n^(S)) + 2

2n−2Dτ₂(T_n^(S)).

E h

Dτ₂(T_n+1^(S)) T_n^(S)i

=

1+ 3

2n−2

Dτ₃(T_n^(S))+ 2

2n−2Dτ₂(T_n^(S)).

Hence there exist constantsan,bn,cn such that

M3(n)=anDτ₃(T_n^(S))+bnDτ₂(T_n^(S))−cn

is a martingale.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Our observables: embeddings of decorated trees

Hereτ_k:=unique vertex with labelk.

- What aboutD_τ3(T_n^(S))? E

h

Dτ₂(T_n+1^(S)) T_n^(S)i

=

1+ 3

2n−2

Dτ₃(T_n^(S))+ 2

2n−2Dτ₂(T_n^(S)).

Hence there exist constantsan,bn,cn such that

M3(n)=anDτ₃(T_n^(S))+bnDτ₂(T_n^(S))−cn

is a martingale.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

Our observables: embeddings of decorated trees

Hereτ_k:=unique vertex with labelk.

- What aboutD_τ3(T_n^(S))? E

h

Dτ₂(T_n+1^(S)) T_n^(S)i

=

1+ 3

2n−2

Dτ₃(T_n^(S))+ 2

2n−2Dτ₂(T_n^(S)).

Hence there exist constantsan,bn,cn such that

M3(n)=anDτ₃(T_n^(S))+bnDτ₂(T_n^(S))−cn

is a martingale.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

More generally:

There exists a partial order 4 on decorated trees

, such that for every decorated tree τ, there exist constants{cn(τ,τ⁰) :τ⁰ 4τ,n>2}such that, for every seed S,

M^(S)τ (n)= X

τ⁰4τ

cn(τ,τ⁰)·D_τ⁰(T_n^(S))

is a martingale and is bounded inL².

Proposition.

Proof of the theorem, i.e. lim

d

(T

, T

) > 0 .

IfS₁6=S₂, there exists a decorated treeτandn₀such that E

h

M^(Sτ ¹⁾(n0)i 6=E

h

M^(Sτ²⁾(n0)i

.

Hence

n→lim∞dTV(T_n^(S1),T_n^(S2))>lim inf

n→∞ dTV(M^(Sτ¹⁾(n),M^(Sτ²⁾(n))

>0.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

More generally:

There exists a partial order 4 on decorated trees, such that for every decorated tree τ, there exist constants{cn(τ,τ⁰) :τ⁰4τ,n>2}such that, for every seed S,

M^(S)τ (n)= X

τ⁰4τ

cn(τ,τ⁰)·D_τ⁰(T_n^(S))

is a martingale and is bounded inL².

Proposition.

Proof of the theorem, i.e. lim

d

(T

, T

) > 0 .

IfS₁6=S₂, there exists a decorated treeτandn₀such that E

h

M^(Sτ ¹⁾(n0)i 6=E

h

M^(Sτ²⁾(n0)i

.

Hence

n→lim∞dTV(T_n^(S1),T_n^(S2))>lim inf

n→∞ dTV(M^(Sτ¹⁾(n),M^(Sτ²⁾(n))

>0.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

More generally:

There exists a partial order 4 on decorated trees, such that for every decorated tree τ, there exist constants{cn(τ,τ⁰) :τ⁰4τ,n>2}such that, for every seed S,

M^(S)τ (n)= X

τ⁰4τ

cn(τ,τ⁰)·D_τ⁰(T_n^(S))

is a martingale

and is bounded inL².

Proposition.

Proof of the theorem, i.e. lim

d

(T

, T

) > 0 .

IfS₁6=S₂, there exists a decorated treeτandn₀such that E

h

M^(Sτ ¹⁾(n0)i 6=E

h

M^(Sτ²⁾(n0)i

.

Hence

n→lim∞dTV(T_n^(S1),T_n^(S2))>lim inf

n→∞ dTV(M^(Sτ¹⁾(n),M^(Sτ²⁾(n))

>0.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

More generally:

There exists a partial order 4 on decorated trees, such that for every decorated tree τ, there exist constants{cn(τ,τ⁰) :τ⁰4τ,n>2}such that, for every seed S,

M^(S)τ (n)= X

τ⁰4τ

cn(τ,τ⁰)·D_τ⁰(T_n^(S))

is a martingale and is bounded inL². Proposition.

Proof of the theorem, i.e. lim

d

(T

, T

) > 0 .

IfS₁6=S₂, there exists a decorated treeτandn₀such that E

h

M^(Sτ ¹⁾(n0)i 6=E

h

M^(Sτ²⁾(n0)i

.

Hence

n→lim∞dTV(T_n^(S1),T_n^(S2))>lim inf

n→∞ dTV(M^(Sτ¹⁾(n),M^(Sτ²⁾(n))

>0.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

More generally:

There exists a partial order 4 on decorated trees, such that for every decorated tree τ, there exist constants{cn(τ,τ⁰) :τ⁰4τ,n>2}such that, for every seed S,

M^(S)τ (n)= X

τ⁰4τ

cn(τ,τ⁰)·D_τ⁰(T_n^(S))

is a martingale and is bounded inL². Proposition.

Proof of the theorem, i.e. lim

d

(T

, T

) > 0 .

IfS₁6=S₂, there exists a decorated treeτandn₀such that E

h

M^(Sτ ¹⁾(n0)i 6=E

h

M^(Sτ²⁾(n0)i

.

Hence

n→lim∞dTV(T_n^(S1),T_n^(S2))>lim inf

n→∞ dTV(M^(Sτ¹⁾(n),M^(Sτ²⁾(n))

>0.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

More generally:

There exists a partial order 4 on decorated trees, such that for every decorated tree τ, there exist constants{cn(τ,τ⁰) :τ⁰4τ,n>2}such that, for every seed S,

M^(S)τ (n)= X

τ⁰4τ

cn(τ,τ⁰)·D_τ⁰(T_n^(S))

is a martingale and is bounded inL². Proposition.

Proof of the theorem, i.e. lim

d

(T

, T

) > 0 .

IfS₁6=S₂, there exists a decorated treeτandn₀such that h

M^(S1)(n )i 6= h

M^(S2)(n )i .

Hence

n→lim∞dTV(T_n^(S1),T_n^(S2))>lim inf

n→∞ dTV(M^(Sτ¹⁾(n),M^(Sτ²⁾(n))

>0.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

More generally:

There exists a partial order 4 on decorated trees, such that for every decorated tree τ, there exist constants{cn(τ,τ⁰) :τ⁰4τ,n>2}such that, for every seed S,

M^(S)τ (n)= X

τ⁰4τ

cn(τ,τ⁰)·D_τ⁰(T_n^(S))

is a martingale and is bounded inL². Proposition.

Proof of the theorem, i.e. lim

d

(T

, T

) > 0 .

IfS₁6=S₂, there exists a decorated treeτandn₀such that h

M^(S1)(n )i 6= h

M^(S2)(n )i . Hence

>0.

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

More generally:

There exists a partial order 4 on decorated trees, such that for every decorated tree τ, there exist constants{cn(τ,τ⁰) :τ⁰4τ,n>2}such that, for every seed S,

M^(S)τ (n)= X

τ⁰4τ

cn(τ,τ⁰)·D_τ⁰(T_n^(S))

is a martingale and is bounded inL². Proposition.

Proof of the theorem, i.e. lim

d

(T

, T

) > 0 .

IfS₁6=S₂, there exists a decorated treeτandn₀such that h

M^(S1)(n )i 6= h

M^(S2)(n )i . Hence

Preferential attachment and influence of the seed Looptrees and preferential attachment Extensions and conjectures

I. Preferential attachment and influence of the seed II. Looptrees and preferential attachment

III. Extensions and conjectures