Degree-degree correlations in directed networks with heavy-tailed - PowerPoint PPT Presentation

Degree-degree correlations in directed networks with heavy-tailed degrees Pim van der Hoorn Stochastic Operations Research Group, University of Twente EU FP7 grant 288956, NADINE June 13, 2013

Introduction Degree-degree correlations Pearson correlation coefficients Rank correlations Results Example Future research

Introduction [Pim van der Hoorn] 3/30

Introduction ◮ Newman 2002 [Pim van der Hoorn] 3/30

Introduction ◮ Newman 2002 ◮ Nelly Litvak, Remco van de Hofstad 2013 [Pim van der Hoorn] 3/30

Introduction Degree-degree correlations Pearson correlation coefficients Rank correlations Results Example Future research

Four types of correlations [Pim van der Hoorn] 5/30

Four types of correlations [Pim van der Hoorn] 5/30

Four types of correlations • [Pim van der Hoorn] 5/30

Four types of correlations • • [Pim van der Hoorn] 5/30

Four types of correlations • • Out - In [Pim van der Hoorn] 5/30

Four types of correlations • • • • Out - In In - Out [Pim van der Hoorn] 5/30

Four types of correlations • • • • Out - In In - Out • • • • Out - Out In - In [Pim van der Hoorn] 5/30

Some notations G = ( V , E ) [Pim van der Hoorn] 6/30

Some notations G = ( V , E ) G n = ( V n , E n ) [Pim van der Hoorn] 6/30

Some notations G = ( V , E ) G n = ( V n , E n ) e ∗ e ∗ D + • • D − e [Pim van der Hoorn] 6/30

Some notations G = ( V , E ) G n = ( V n , E n ) e ∗ e ∗ D + • • D − e α, β ∈ { + , − } [Pim van der Hoorn] 6/30

Some notations G = ( V , E ) G n = ( V n , E n ) e ∗ e ∗ D + • • D − e D α ( e ∗ ) , D β ( e ∗ ) α, β ∈ { + , − } [Pim van der Hoorn] 6/30

Some notations G = ( V , E ) G n = ( V n , E n ) e ∗ e ∗ D + • • D − e P ( D α > x ) = L α ( x ) x − γ α D α ( e ∗ ) , D β ( e ∗ ) α, β ∈ { + , − } [Pim van der Hoorn] 6/30

Sequences of graphs Definition Let G γ − γ + denote the space of all sequences of graphs ( G n ) n ∈ N with the following properties: G1 | V n | = n G2 For all p ≥ γ + or q ≥ γ − , � n ( v ) q = Θ ( n max ( p /γ + , q /γ − , 1 ) ) . D + n ( v ) p D − v ∈ V n G3 There exist two independent regular varying random variables D + , D − such that for all p < γ + and q < γ − , � 1 n ( v ) q = E D + n ( v ) p D − ( D + ) p � ( D − ) q � � � lim . E n n →∞ v ∈ V n [Pim van der Hoorn] 7/30

Introduction Degree-degree correlations Pearson correlation coefficients Rank correlations Results Example Future research

General formula edges � 1 1 ρ β D α ( e ∗ ) D β ( e ∗ ) − ^ ρ β α ( G ) = α ( G ) σ α ( G ) σ β ( G ) | E | e ∈ E [Pim van der Hoorn] 9/30

General formula edges � 1 1 ρ β D α ( e ∗ ) D β ( e ∗ ) − ^ ρ β α ( G ) = α ( G ) σ α ( G ) σ β ( G ) | E | e ∈ E � � 1 1 ρ β D α ( e ∗ ) D β ( e ∗ ) ^ α ( G ) = σ α ( G ) σ β ( G ) | E | 2 e ∈ E e ∈ E � � 2 � � � � � 1 1 � D α ( e ∗ ) 2 − D α ( e ∗ ) σ α ( G ) = | E | 2 | E | e ∈ E e ∈ E � � 2 � � � � � 1 1 D β ( e ∗ ) 2 − σ β ( G ) = � D β ( e ∗ ) | E | 2 | E | e ∈ E e ∈ E [Pim van der Hoorn] 9/30

From edges to vertices � � D α ( e ∗ ) = D + ( v ) D α ( v ) e ∈ E v ∈ V � � D α ( e ∗ ) = D − ( v ) D α ( v ) e ∈ E v ∈ V [Pim van der Hoorn] 10/30

General formula vertices � 1 1 ρ β D α ( e ∗ ) D β ( e ∗ ) − ^ ρ β α ( G ) = α ( G ) σ α σ β | E | e ∈ E [Pim van der Hoorn] 11/30

General formula vertices � 1 1 ρ β D α ( e ∗ ) D β ( e ∗ ) − ^ ρ β α ( G ) = α ( G ) σ α σ β | E | e ∈ E � � 1 1 ρ β D + ( v ) D α ( v ) D − ( v ) D β ( v ) ^ α ( G ) = σ α σ β | E | 2 v ∈ V v ∈ V [Pim van der Hoorn] 11/30

General formula vertices � 1 1 ρ β D α ( e ∗ ) D β ( e ∗ ) − ^ ρ β α ( G ) = α ( G ) σ α σ β | E | e ∈ E � � 1 1 ρ β D + ( v ) D α ( v ) D − ( v ) D β ( v ) ^ α ( G ) = σ α σ β | E | 2 v ∈ V v ∈ V � � 2 � � � � � 1 1 � D + ( v ) D α ( v ) 2 − D + D α ( v ) σ α ( G ) = | E | 2 | E | v ∈ V v ∈ V � � 2 � � � � � 1 1 � D − ( v ) D β ( v ) 2 − σ β ( G ) = D − ( v ) D β ( v ) | E | 2 | E | v ∈ V v ∈ V [Pim van der Hoorn] 11/30

General formula vertices � 1 1 ρ β D α ( e ∗ ) D β ( e ∗ ) − ^ ρ β α ( G ) = α ( G ) σ α σ β | E | e ∈ E � � 1 1 ρ β D + ( v ) D α ( v ) D − ( v ) D β ( v ) ^ α ( G ) = σ α σ β | E | 2 v ∈ V v ∈ V � � 2 � � � � � 1 1 � D + ( v ) D α ( v ) 2 − D + D α ( v ) σ α ( G ) = | E | 2 | E | v ∈ V v ∈ V � � 2 � � � � � 1 1 � D − ( v ) D β ( v ) 2 − σ β ( G ) = D − ( v ) D β ( v ) | E | 2 | E | v ∈ V v ∈ V [Pim van der Hoorn] 11/30

General formula vertices � 1 1 ρ β D α ( e ∗ ) D β ( e ∗ ) − ^ ρ β α ( G ) = α ( G ) σ α σ β | E | e ∈ E � � 1 1 ρ β D + ( v ) D α ( v ) D − ( v ) D β ( v ) ^ α ( G ) = σ α σ β | E | 2 v ∈ V v ∈ V � � 2 � � � � � 1 1 � D + ( v ) D α ( v ) 2 − D + D α ( v ) σ α ( G ) = | E | 2 | E | v ∈ V v ∈ V � � 2 � � � � � 1 1 � D − ( v ) D β ( v ) 2 − σ β ( G ) = D − ( v ) D β ( v ) | E | 2 | E | v ∈ V v ∈ V [Pim van der Hoorn] 11/30

Convergence to a non-negative value Theorem α ⊂ R 2 such that for Let α, β ∈ { + , − } , then there exists an area A β ( γ + , γ − ) ∈ A β α and { G n } n ∈ N ∈ G γ − ,γ + ρ β n →∞ ^ lim α ( G n ) = 0 and hence n →∞ ρ β lim α ( G n ) ≥ 0 . [Pim van der Hoorn] 12/30

Convergence areas A β α [Pim van der Hoorn] 13/30

Convergence areas A β α γ − A − + 3 γ + 1 1 3 [Pim van der Hoorn] 13/30

Convergence areas A β α γ − γ − A − A + + − 3 2 γ + γ + 1 1 1 3 1 2 [Pim van der Hoorn] 13/30

Convergence areas A β α γ − γ − A − A + + − 3 2 γ + γ + 1 1 1 3 1 2 γ − γ − A + A − + − 3 γ + γ + 1 1 3 [Pim van der Hoorn] 13/30

Outline of the proof [Pim van der Hoorn] 14/30

Outline of the proof � 2 � � 2 � � � n ( v ) D β 1 v ∈ V D + n ( v ) D α 1 v ∈ V D − n ( v ) n ( v ) | E n | | E n | α ( G n ) 2 = ρ β ^ σ α ( G n ) 2 σ β ( G n ) 2 [Pim van der Hoorn] 14/30

Outline of the proof � 2 � � 2 � � � n ( v ) D β 1 v ∈ V D + n ( v ) D α 1 v ∈ V D − n ( v ) n ( v ) | E n | | E n | α ( G n ) 2 = ρ β ^ σ α ( G n ) 2 σ β ( G n ) 2 a n = a n + b n − c n − d n [Pim van der Hoorn] 14/30

Outline of the proof � 2 � � 2 � � � n ( v ) D β 1 v ∈ V D + n ( v ) D α 1 v ∈ V D − n ( v ) n ( v ) | E n | | E n | α ( G n ) 2 = ρ β ^ σ α ( G n ) 2 σ β ( G n ) 2 a n = a n + b n − c n − d n � n c + n d � n a � � a n c n + d n = Θ = Θ n b n b b n b n n a n b � � � � a n b n = Θ = Θ n c + n d n c + n d c n + d n c n + d n [Pim van der Hoorn] 14/30

Outline of the proof continued... [Pim van der Hoorn] 15/30

Outline of the proof continued... ( a < b ∧ max ( c , d ) < b ) ∨ ( a < max ( c , d ) ∧ b < max ( c , d )) [Pim van der Hoorn] 15/30

Outline of the proof continued... ( a < b ∧ max ( c , d ) < b ) ∨ ( a < max ( c , d ) ∧ b < max ( c , d )) a n c n + d n lim = 0 and lim = 0 b n b n n →∞ n →∞ or a n b n lim = 0 and lim = 0 c n + d n c n + d n n →∞ n →∞ [Pim van der Hoorn] 15/30

Outline of the proof continued... ( a < b ∧ max ( c , d ) < b ) ∨ ( a < max ( c , d ) ∧ b < max ( c , d )) a n c n + d n lim = 0 and lim = 0 b n b n n →∞ n →∞ or a n b n lim = 0 and lim = 0 c n + d n c n + d n n →∞ n →∞ a n ⇒ lim = 0 a n + b n − c n − d n n →∞ [Pim van der Hoorn] 15/30

Outline of the proof continued... ( a < b ∧ max ( c , d ) ≤ b ) ∨ ( a < max ( c , d ) ∧ b ≤ max ( c , d )) a n c n + d n lim = 0 and lim = 0 b n b n n →∞ n →∞ or a n b n lim = 0 and lim = 0 c n + d n c n + d n n →∞ n →∞ a n ⇒ lim = 0 a n + b n − c n − d n n →∞ [Pim van der Hoorn] 15/30

Outline of the proof continued... ( a < b ∧ max ( c , d ) ≤ b ) ∨ ( a < max ( c , d ) ∧ b ≤ max ( c , d )) a n c n + d n lim = 0 and lim = 0 b n b n n →∞ n →∞ or a n b n lim = 0 and lim = 0 c n + d n c n + d n n →∞ n →∞ a n ⇒ lim = 0 a n + b n − c n − d n n →∞ [Pim van der Hoorn] 15/30

Issues [Pim van der Hoorn] 16/30

Issues ◮ Graph model with heavy tails have non-negative degree-degree correlation limit [Pim van der Hoorn] 16/30

Issues ◮ Graph model with heavy tails have non-negative degree-degree correlation limit ◮ Degree-degree correlations cannot be compared for different sizes [Pim van der Hoorn] 16/30

Introduction Degree-degree correlations Pearson correlation coefficients Rank correlations Results Example Future research