Scaling limits and influence of the seed graph in preferential attachment trees Ioan Manolescu joint work with Nicolas Curien, Thomas Duquesne and Igor Kortchemski University of Geneva 9th December 2014 Journ´ ee cartes al´ eatoires - IHES Ioan Manolescu (University of Geneva) LPAM 9th December 2014 1 / 14
Linear preferential attachment: the model Initial tree: T ( S ) = S (where n 0 = | S | ) n 0 T ( S ) n +1 is obtained from T ( S ) by adding an edge to a random vertex v ∈ T ( S ) , n n chosen proportionally to its degree. Ioan Manolescu (University of Geneva) LPAM 9th December 2014 2 / 14
Linear preferential attachment: the model Initial tree: T ( S ) = S (where n 0 = | S | ) n 0 T ( S ) n +1 is obtained from T ( S ) by adding an edge to a random vertex v ∈ T ( S ) , n n chosen proportionally to its degree. Ioan Manolescu (University of Geneva) LPAM 9th December 2014 2 / 14
Linear preferential attachment: the model Initial tree: T ( S ) = S (where n 0 = | S | ) n 0 T ( S ) n +1 is obtained from T ( S ) by adding an edge to a random vertex v ∈ T ( S ) , n n chosen proportionally to its degree. Ioan Manolescu (University of Geneva) LPAM 9th December 2014 2 / 14
Linear preferential attachment: the model Initial tree: T ( S ) = S (where n 0 = | S | ) n 0 T ( S ) n +1 is obtained from T ( S ) by adding an edge to a random vertex v ∈ T ( S ) , n n chosen proportionally to its degree. Ioan Manolescu (University of Geneva) LPAM 9th December 2014 2 / 14
Linear preferential attachment: the model Initial tree: T ( S ) = S (where n 0 = | S | ) n 0 T ( S ) n +1 is obtained from T ( S ) by adding an edge to a random vertex v ∈ T ( S ) , n n chosen proportionally to its degree. Ioan Manolescu (University of Geneva) LPAM 9th December 2014 2 / 14
Linear preferential attachment: the model Initial tree: T ( S ) = S (where n 0 = | S | ) n 0 T ( S ) n +1 is obtained from T ( S ) by adding an edge to a random vertex v ∈ T ( S ) , n n chosen proportionally to its degree. Ioan Manolescu (University of Geneva) LPAM 9th December 2014 2 / 14
Linear preferential attachment: the model Initial tree: T ( S ) = S (where n 0 = | S | ) n 0 T ( S ) n +1 is obtained from T ( S ) by adding an edge to a random vertex v ∈ T ( S ) , n n chosen proportionally to its degree. Ioan Manolescu (University of Geneva) LPAM 9th December 2014 2 / 14
Linear preferential attachment: the model Initial tree: T ( S ) = S (where n 0 = | S | ) n 0 T ( S ) n +1 is obtained from T ( S ) by adding an edge to a random vertex v ∈ T ( S ) , n n chosen proportionally to its degree. Ioan Manolescu (University of Geneva) LPAM 9th December 2014 2 / 14
Linear preferential attachment: the model Initial tree: T ( S ) = S (where n 0 = | S | ) n 0 T ( S ) n +1 is obtained from T ( S ) by adding an edge to a random vertex v ∈ T ( S ) , n n chosen proportionally to its degree. Ioan Manolescu (University of Geneva) LPAM 9th December 2014 2 / 14
Linear preferential attachment: the model Initial tree: T ( S ) = S (where n 0 = | S | ) n 0 T ( S ) n +1 is obtained from T ( S ) by adding an edge to a random vertex v ∈ T ( S ) , n n chosen proportionally to its degree. Ioan Manolescu (University of Geneva) LPAM 9th December 2014 2 / 14
Linear preferential attachment: the model Initial tree: T ( S ) = S (where n 0 = | S | ) n 0 T ( S ) n +1 is obtained from T ( S ) by adding an edge to a random vertex v ∈ T ( S ) , n n chosen proportionally to its degree. Questions: Does the process mix? For S 1 � = S 2 , does d TV ( T ( S 1 ) ; T ( S 2 ) ) → 0 as n → ∞ ? n n How does T ( S ) look like when n is very large? Scaling limit? n Ioan Manolescu (University of Geneva) LPAM 9th December 2014 2 / 14
Recognise the seed Question: For S 1 � = S 2 , does d TV ( T ( S 1 ) ; T ( S 2 ) ) → 0 as n → ∞ ? n n Ioan Manolescu (University of Geneva) LPAM 9th December 2014 3 / 14
Recognise the seed Question: For S 1 � = S 2 , does d TV ( T ( S 1 ) ; T ( S 2 ) ) → 0 as n → ∞ ? It may. . . n n Example: T ( S 1 ) T ( S 2 ) = S 1 = S 2 2 3 Ioan Manolescu (University of Geneva) LPAM 9th December 2014 3 / 14
Recognise the seed Question: For S 1 � = S 2 , does d TV ( T ( S 1 ) ; T ( S 2 ) ) → 0 as n → ∞ ? It may. . . n n Example: T ( S 1 ) T ( S 2 ) = S 2 = S 2 3 3 Ioan Manolescu (University of Geneva) LPAM 9th December 2014 3 / 14
Recognise the seed Question: For S 1 � = S 2 , does d TV ( T ( S 1 ) ; T ( S 2 ) ) → 0 as n → ∞ ? It may. . . n n Example: T ( S 1 ) T ( S 2 ) = S 2 = S 2 3 3 Ioan Manolescu (University of Geneva) LPAM 9th December 2014 3 / 14
Recognise the seed Question: For S 1 � = S 2 , with | S 1 | = | S 2 | ≥ 3 does d TV ( T ( S 1 ) ; T ( S 2 ) ) → 0 as n → ∞ ? NO! n n Ioan Manolescu (University of Geneva) LPAM 9th December 2014 3 / 14
Recognise the seed Question: For S 1 � = S 2 , with | S 1 | = | S 2 | ≥ 3 does d TV ( T ( S 1 ) ; T ( S 2 ) ) → 0 as n → ∞ ? NO! n n Idea of Bubeck, Mossel, R´ acz: Study the degree sequence of T n : Deg ( T n ) = { deg( v ) : v ∈ T n } . Ioan Manolescu (University of Geneva) LPAM 9th December 2014 3 / 14
Recognise the seed Question: For S 1 � = S 2 , with | S 1 | = | S 2 | ≥ 3 does d TV ( T ( S 1 ) ; T ( S 2 ) ) → 0 as n → ∞ ? NO! n n Idea of Bubeck, Mossel, R´ acz: Study the degree sequence of T n : Deg ( T n ) = { deg( v ) : v ∈ T n } . Theorem (Bubeck, Mossel, R´ acz) For seeds S 1 � = S 2 with | S 1 | = | S 2 | but Deg ( S 1 ) � = Deg ( S 2 ) , d TV ( Deg ( T ( S 1 ) ); Deg ( T ( S 2 ) )) / − → 0 . n n Ioan Manolescu (University of Geneva) LPAM 9th December 2014 3 / 14
Recognise the seed Question: For S 1 � = S 2 , with | S 1 | = | S 2 | ≥ 3 does d TV ( T ( S 1 ) ; T ( S 2 ) ) → 0 as n → ∞ ? NO! n n Idea of Bubeck, Mossel, R´ acz: Study the degree sequence of T n : Deg ( T n ) = { deg( v ) : v ∈ T n } . Theorem (Bubeck, Mossel, R´ acz) For seeds S 1 � = S 2 with | S 1 | = | S 2 | but Deg ( S 1 ) � = Deg ( S 2 ) , d TV ( Deg ( T ( S 1 ) ); Deg ( T ( S 2 ) )) / − → 0 . n n Proof: The degree sequence is given by a P´ olya urn. The tail of max Deg ( T ( S ) ) depends on Deg ( S ). n Ioan Manolescu (University of Geneva) LPAM 9th December 2014 3 / 14
Recognise the seed Question: For S 1 � = S 2 , with | S 1 | = | S 2 | ≥ 3 does d TV ( T ( S 1 ) ; T ( S 2 ) ) → 0 as n → ∞ ? NO! n n Idea of Bubeck, Mossel, R´ acz: Study the degree sequence of T n : Deg ( T n ) = { deg( v ) : v ∈ T n } . Theorem (Bubeck, Mossel, R´ acz) For seeds S 1 � = S 2 with | S 1 | = | S 2 | but Deg ( S 1 ) � = Deg ( S 2 ) , d TV ( Deg ( T ( S 1 ) ); Deg ( T ( S 2 ) )) / → 0 . − n n Proof: The degree sequence is given by a P´ olya urn. The tail of max Deg ( T ( S ) ) depends on Deg ( S ). n Problem: This strategy can not distinguish between seeds with same degree sequences. S 1 S 2 Ioan Manolescu (University of Geneva) LPAM 9th December 2014 3 / 14
Recognise the seed Question: For S 1 � = S 2 , with | S 1 | = | S 2 | ≥ 3 does d TV ( T ( S 1 ) ; T ( S 2 ) ) → 0 as n → ∞ ? NO! n n Idea of Bubeck, Mossel, R´ acz: Study the degree sequence of T n : Deg ( T n ) = { deg( v ) : v ∈ T n } . Theorem (Bubeck, Mossel, R´ acz) For seeds S 1 � = S 2 with | S 1 | = | S 2 | but Deg ( S 1 ) � = Deg ( S 2 ) , d TV ( Deg ( T ( S 1 ) ); Deg ( T ( S 2 ) )) / → 0 . − n n Proof: The degree sequence is given by a P´ olya urn. The tail of max Deg ( T ( S ) ) depends on Deg ( S ). n Conclusion: Need some geometric observable to distinguish T ( S 1 ) form T ( S 2 ) n n Ioan Manolescu (University of Geneva) LPAM 9th December 2014 3 / 14
How to define a limit of T n ? Convergence in Gromov–Hausdorff topology? Ioan Manolescu (University of Geneva) LPAM 9th December 2014 4 / 14
How to define a limit of T n ? Convergence in Gromov–Hausdorff topology? Maximal degree of T n : √ n . Diameter of T n : log n . No non-trivial compact scaling limit! Ioan Manolescu (University of Geneva) LPAM 9th December 2014 4 / 14
How to define a limit of T n ? Convergence in Gromov–Hausdorff topology? Solution: consider the loop tree Loop( T n ) Ioan Manolescu (University of Geneva) LPAM 9th December 2014 4 / 14
How to define a limit of T n ? Convergence in Gromov–Hausdorff topology? Solution: consider the loop tree Loop( T n ) Ioan Manolescu (University of Geneva) LPAM 9th December 2014 4 / 14
How to define a limit of T n ? Convergence in Gromov–Hausdorff topology? → diameter: √ n . Solution: consider the loop tree Loop( T n ) − Ioan Manolescu (University of Geneva) LPAM 9th December 2014 4 / 14
How to define a limit of T n ? Convergence in Gromov–Hausdorff topology? → diameter: √ n . Solution: consider the loop tree Loop( T n ) − Theorem n − 1 / 2 · Loop( T ( S ) a.s. for G.H. L ( S ) , ) − − − − − − − → n n →∞ where L ( S ) is a random compact metric space called the ”Bownian looptree”. Ioan Manolescu (University of Geneva) LPAM 9th December 2014 4 / 14
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