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) Gn = (Vn, En)

[Pim van der Hoorn] 6/30

Some notations

G = (V , E) Gn = (Vn, En) e∗ e∗

• e

D+ D−

[Pim van der Hoorn] 6/30

Some notations

G = (V , E) Gn = (Vn, En) e∗ e∗

• e

D+ D− α, β ∈ {+, −}

[Pim van der Hoorn] 6/30

Some notations

G = (V , E) Gn = (Vn, En) e∗ e∗

• e

D+ D− α, β ∈ {+, −} Dα(e∗), Dβ(e∗)

[Pim van der Hoorn] 6/30

Some notations

G = (V , E) Gn = (Vn, En) e∗ e∗

• e

D+ D− α, β ∈ {+, −} Dα(e∗), Dβ(e∗) P(Dα > x) = Lα(x)x−γα

[Pim van der Hoorn] 6/30

Sequences of graphs

Definition

Let Gγ−γ+ denote the space of all sequences of graphs (Gn)n∈N with the following properties: G1 |Vn| = n G2 For all p ≥ γ+ or q ≥ γ−,

• v∈Vn

D+

n (v)pD− n (v)q = Θ(nmax(p/γ+,q/γ−,1)).

G3 There exist two independent regular varying random variables D+, D− such that for all p < γ+ and q < γ−, lim

n→∞

1 n

• v∈Vn

D+

n (v)pD− n (v)q = E

• (D+)p

E

• (D−)q

.

[Pim van der Hoorn] 7/30

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

General formula edges

ρβ

α(G) =

1 σα(G)σβ(G) 1 |E|

• e∈E

Dα(e∗)Dβ(e∗) − ^ ρβ

α(G) [Pim van der Hoorn] 9/30

General formula edges

ρβ

α(G) =

1 σα(G)σβ(G) 1 |E|

• e∈E

Dα(e∗)Dβ(e∗) − ^ ρβ

α(G)

^ ρβ

α(G) =

1 σα(G)σβ(G) 1 |E|2

• e∈E

Dα(e∗)

• e∈E

Dβ(e∗) σα(G) =

• 1

|E|

• e∈E

Dα(e∗)2 − 1 |E|2

• e∈E

Dα(e∗) 2 σβ(G) =

• 1

|E|

• e∈E

Dβ(e∗)2 − 1 |E|2

• e∈E

Dβ(e∗) 2

[Pim van der Hoorn] 9/30

From edges to vertices

• e∈E

Dα(e∗) =

• v∈V

D+(v)Dα(v)

• e∈E

Dα(e∗) =

• v∈V

D−(v)Dα(v)

[Pim van der Hoorn] 10/30

General formula vertices

ρβ

α(G) =

1 σασβ 1 |E|

• e∈E

Dα(e∗)Dβ(e∗) − ^ ρβ

α(G) [Pim van der Hoorn] 11/30

General formula vertices

ρβ

α(G) =

1 σασβ 1 |E|

• e∈E

Dα(e∗)Dβ(e∗) − ^ ρβ

α(G)

^ ρβ

α(G) =

1 σασβ 1 |E|2

• v∈V

D+(v)Dα(v)

• v∈V

D−(v)Dβ(v)

[Pim van der Hoorn] 11/30

General formula vertices

ρβ

α(G) =

1 σασβ 1 |E|

• e∈E

Dα(e∗)Dβ(e∗) − ^ ρβ

α(G)

^ ρβ

α(G) =

1 σασβ 1 |E|2

• v∈V

D+(v)Dα(v)

• v∈V

D−(v)Dβ(v) σα(G) =

• 1

|E|

• v∈V

D+(v)Dα(v)2 − 1 |E|2

• v∈V

D+Dα(v) 2 σβ(G) =

• 1

|E|

• v∈V

D−(v)Dβ(v)2 − 1 |E|2

• v∈V

D−(v)Dβ(v) 2

[Pim van der Hoorn] 11/30

General formula vertices

ρβ

α(G) =

1 σασβ 1 |E|

• e∈E

Dα(e∗)Dβ(e∗) − ^ ρβ

α(G)

^ ρβ

α(G) =

1 σασβ 1 |E|2

• v∈V

D+(v)Dα(v)

• v∈V

D−(v)Dβ(v) σα(G) =

• 1

|E|

• v∈V

D+(v)Dα(v)2 − 1 |E|2

• v∈V

D+Dα(v) 2 σβ(G) =

• 1

|E|

• v∈V

D−(v)Dβ(v)2 − 1 |E|2

• v∈V

D−(v)Dβ(v) 2

[Pim van der Hoorn] 11/30

General formula vertices

ρβ

α(G) =

1 σασβ 1 |E|

• e∈E

Dα(e∗)Dβ(e∗) − ^ ρβ

α(G)

^ ρβ

α(G) =

1 σασβ 1 |E|2

• v∈V

D+(v)Dα(v)

• v∈V

D−(v)Dβ(v) σα(G) =

• 1

|E|

• v∈V

D+(v)Dα(v)2 − 1 |E|2

• v∈V

D+Dα(v) 2 σβ(G) =

• 1

|E|

• v∈V

D−(v)Dβ(v)2 − 1 |E|2

• v∈V

D−(v)Dβ(v) 2

[Pim van der Hoorn] 11/30

Convergence to a non-negative value

Theorem

Let α, β ∈ {+, −}, then there exists an area Aβ

α ⊂ R2 such that for

(γ+, γ−) ∈ Aβ

α and {Gn}n∈N ∈ Gγ−,γ+

lim

n→∞ ^

ρβ

α(Gn) = 0

and hence lim

n→∞ ρβ α(Gn) ≥ 0. [Pim van der Hoorn] 12/30

Convergence areas Aβ

α

[Pim van der Hoorn] 13/30

Convergence areas Aβ

α

γ− γ+ 1 1 3 3 A−

+ [Pim van der Hoorn] 13/30

Convergence areas Aβ

α

γ− γ+ 1 1 3 3 A−

+

γ− γ+ 2 2 1 1 A+

− [Pim van der Hoorn] 13/30

Convergence areas Aβ

α

γ− γ+ 1 1 3 3 A−

+

γ− γ+ 2 2 1 1 A+

−

γ− γ+ 1 3 A+

+

γ− γ+ 1 3 A−

− [Pim van der Hoorn] 13/30

Outline of the proof

[Pim van der Hoorn] 14/30

Outline of the proof

^ ρβ

α(Gn)2 =

• 1

|En|

• v∈V D+

n (v)Dα n (v)

2

1 |En|

• v∈V D−

n (v)Dβ n (v)

2 σα(Gn)2σβ(Gn)2

[Pim van der Hoorn] 14/30

Outline of the proof

^ ρβ

α(Gn)2 =

• 1

|En|

• v∈V D+

n (v)Dα n (v)

2

1 |En|

• v∈V D−

n (v)Dβ n (v)

2 σα(Gn)2σβ(Gn)2 = an an + bn − cn − dn

[Pim van der Hoorn] 14/30

Outline of the proof

^ ρβ

α(Gn)2 =

• 1

|En|

• v∈V D+

n (v)Dα n (v)

2

1 |En|

• v∈V D−

n (v)Dβ n (v)

2 σα(Gn)2σβ(Gn)2 = an an + bn − cn − dn an bn = Θ na nb

• cn + dn

bn = Θ nc + nd nb

• an

cn + dn = Θ

• na

nc + nd

• bn

cn + dn = Θ

• nb

nc + nd

• [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)) lim

n→∞

an bn = 0 and lim

n→∞

cn + dn bn = 0

• r

lim

n→∞

an cn + dn = 0 and lim

n→∞

bn cn + dn = 0

[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)) lim

n→∞

an bn = 0 and lim

n→∞

cn + dn bn = 0

• r

lim

n→∞

an cn + dn = 0 and lim

n→∞

bn cn + dn = 0 ⇒ lim

n→∞

an an + bn − cn − dn = 0

[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)) lim

n→∞

an bn = 0 and lim

n→∞

cn + dn bn = 0

• r

lim

n→∞

an cn + dn = 0 and lim

n→∞

bn cn + dn = 0 ⇒ lim

n→∞

an an + bn − cn − dn = 0

[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)) lim

n→∞

an bn = 0 and lim

n→∞

cn + dn bn = 0

• r

lim

n→∞

an cn + dn = 0 and lim

n→∞

bn cn + dn = 0 ⇒ lim

n→∞

an an + bn − cn − dn = 0

[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

Spearman’s Rho

[Pim van der Hoorn] 18/30

Spearman’s Rho

{Xi}1≤i≤n , {Yi}1≤i≤n, i.i.d. samples of X, Y rX

i , rY i

ranks of Xi, Yi.

[Pim van der Hoorn] 18/30

Spearman’s Rho

{Xi}1≤i≤n , {Yi}1≤i≤n, i.i.d. samples of X, Y rX

i , rY i

ranks of Xi, Yi. ρ[n]

S

• rX

i , rY i

• :=

n

i=1(rX i

− n+1

2 )(rY i

− n+1

2 )

n

i=1(rX i

− n+1

2 )2 n i=1(rY i

− n+1

2 ) [Pim van der Hoorn] 18/30

Spearman’s Rho

{Xi}1≤i≤n , {Yi}1≤i≤n, i.i.d. samples of X, Y rX

i , rY i

ranks of Xi, Yi. ρ[n]

S

• rX

i , rY i

• :=

n

i=1(rX i

− n+1

2 )(rY i

− n+1

2 )

n

i=1(rX i

− n+1

2 )2 n i=1(rY i

− n+1

2 )

ρS(X, Y ) = E [FX(X)FY (Y )] − E [FX(X)] E [FY (Y )]

• E [FX(X)2] − E [FX(X)]2

E [FY (Y )2] − E [FY (Y )]2

[Pim van der Hoorn] 18/30

Spearman’s Rho

{Xi}1≤i≤n , {Yi}1≤i≤n, i.i.d. samples of X, Y rX

i , rY i

ranks of Xi, Yi. ρ[n]

S

• rX

i , rY i

• :=

n

i=1(rX i

− n+1

2 )(rY i

− n+1

2 )

n

i=1(rX i

− n+1

2 )2 n i=1(rY i

− n+1

2 )

ρS(X, Y ) = E [FX(X)FY (Y )] − E [FX(X)] E [FY (Y )]

• E [FX(X)2] − E [FX(X)]2

E [FY (Y )2] − E [FY (Y )]2 = E [FX(X)FY (Y )] − 1

4

1/12

[Pim van der Hoorn] 18/30

Spearman’s Rho

{Xi}1≤i≤n , {Yi}1≤i≤n, i.i.d. samples of X, Y rX

i , rY i

ranks of Xi, Yi. ρ[n]

S

• rX

i , rY i

• :=

n

i=1(rX i

− n+1

2 )(rY i

− n+1

2 )

n

i=1(rX i

− n+1

2 )2 n i=1(rY i

− n+1

2 )

ρS(X, Y ) = E [FX(X)FY (Y )] − E [FX(X)] E [FY (Y )]

• E [FX(X)2] − E [FX(X)]2

E [FY (Y )2] − E [FY (Y )]2 = E [FX(X)FY (Y )] − 1

4

1/12 FX(X) := FX ◦ X is a uniform random variable on (0,1).

[Pim van der Hoorn] 18/30

Resloving ties, uniform at random

[Pim van der Hoorn] 19/30

Resloving ties, uniform at random

Turn discrete random variables X, Y into continuous random variables ˜ X, ˜ Y

[Pim van der Hoorn] 19/30

Resloving ties, uniform at random

Turn discrete random variables X, Y into continuous random variables ˜ X, ˜ Y Two uniform random variables U, V on (0,1) ˜ X := X + U ˜ Y := Y + V ˜ Xi := Xi + Ui ˜ Yi := Yi + Vi

[Pim van der Hoorn] 19/30

Resloving ties, uniform at random

Turn discrete random variables X, Y into continuous random variables ˜ X, ˜ Y Two uniform random variables U, V on (0,1) ˜ X := X + U ˜ Y := Y + V ˜ Xi := Xi + Ui ˜ Yi := Yi + Vi ˜ rX

i , ˜

rY

i

ranking.

[Pim van der Hoorn] 19/30

Resloving ties, uniform at random

Turn discrete random variables X, Y into continuous random variables ˜ X, ˜ Y Two uniform random variables U, V on (0,1) ˜ X := X + U ˜ Y := Y + V ˜ Xi := Xi + Ui ˜ Yi := Yi + Vi ˜ rX

i , ˜

rY

i

ranking. ρ[n]

S

• ˜

rX

i ,˜

rY

i

• [Pim van der Hoorn]

19/30

Resolving ties, take average

[Pim van der Hoorn] 20/30

Resolving ties, take average

rX

i

= 1 |{k|Xk = Xi}|

• j:Xj=Xi

|{k|Xk > Xi}| + j (1, 2, 1, 3, 3) → (1.5, 3, 1.5, 4.5, 4.5)

[Pim van der Hoorn] 20/30

Resolving ties, take average

rX

i

= 1 |{k|Xk = Xi}|

• j:Xj=Xi

|{k|Xk > Xi}| + j (1, 2, 1, 3, 3) → (1.5, 3, 1.5, 4.5, 4.5)

◮ Average unchanged [Pim van der Hoorn] 20/30

Resolving ties, take average

rX

i

= 1 |{k|Xk = Xi}|

• j:Xj=Xi

|{k|Xk > Xi}| + j (1, 2, 1, 3, 3) → (1.5, 3, 1.5, 4.5, 4.5)

◮ Average unchanged ◮ No randomness [Pim van der Hoorn] 20/30

Resolving ties, take average

rX

i

= 1 |{k|Xk = Xi}|

• j:Xj=Xi

|{k|Xk > Xi}| + j (1, 2, 1, 3, 3) → (1.5, 3, 1.5, 4.5, 4.5)

◮ Average unchanged ◮ No randomness ◮ Variance? [Pim van der Hoorn] 20/30

Kendall Tau

[Pim van der Hoorn] 21/30

Kendall Tau

{(Xi, Yi)}1≤i≤n observations

[Pim van der Hoorn] 21/30

Kendall Tau

{(Xi, Yi)}1≤i≤n observations number of concordant pairs − number of disconcordant pairs

1 2n(n − 1) [Pim van der Hoorn] 21/30

Kendall Tau

{(Xi, Yi)}1≤i≤n observations number of concordant pairs − number of disconcordant pairs

1 2n(n − 1)

{Xi}1≤i≤n, {Yi}1≤i≤n i.i.d. samples of X and Y

[Pim van der Hoorn] 21/30

Kendall Tau

{(Xi, Yi)}1≤i≤n observations number of concordant pairs − number of disconcordant pairs

1 2n(n − 1)

{Xi}1≤i≤n, {Yi}1≤i≤n i.i.d. samples of X and Y τ :=

n

• i=1,j=i

P((Xi − Xj)(Yi − Yj) > 0) − P((Xi − Xj)(Yi − Yj) < 0)

[Pim van der Hoorn] 21/30

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

Results for Wikipedia

Graph Exponent1 Assortativity Pearson Spearman’s Rho Kendall Tau γ− γ+ Average Uniform DE wiki 1.7 1.9 +/-

• 0.0552
• 0.1435
• 0.1434
• 0.0986
• /+

0.0154 0.0484 0.0481 0.0326 +/+

• 0.0323
• 0.0640
• 0.0640
• 0.0446
• /-
• 0.0123

0.0120 0.0119 0.0074 EN wiki 1.9 2.5 +/-

• 0.0557
• 0.1999
• 0.1999
• 0.1364
• /+
• 0.0007

0.0240 0.0239 0.0163 +/+

• 0.0713
• 0.0855
• 0.0855
• 0.0581
• /-
• 0.0074
• 0.0666
• 0.0664
• 0.0457

ES wiki 1.4 2.5 +/-

• 0.1031
• 0.1429
• 0.1429
• 0.0972
• /+
• 0.0033
• 0.0417
• 0.0407
• 0.0294

+/+

• 0.0272

0.0178 0.0178 0.0119

• /-
• 0.0262
• 0.1669
• 0.1627
• 0.1174

FR wiki 1.5 2.6 +/-

• 0.0536
• 0.1065
• 0.1065
• 0.0720
• /+

0.0048 0.0121 0.0119 0.0085 +/+

• 0.0512
• 0.0126
• 0.0126
• 0.0087
• /-
• 0.0094
• 0.0267
• 0.0262
• 0.0186

Table: Results on the wikipedia graphs obtained from the

http://law.di.unimi.it/ database

1 determined using Hill’s estimator

[Pim van der Hoorn] 23/30

Graph Exponent1 Assortativity Pearson Spearman’s Rho Kendall Tau γ− γ+ Average Uniform HU wiki 1.3 2.2 +/-

• 0.1048
• 0.1280
• 0.1280
• 0.0877
• /+

0.0120 0.0595 0.0525 0.0442 +/+

• 0.0579
• 0.0207
• 0.0207
• 0.0140
• /-
• 0.0279

0.0060 0.0051 0.0050 IT wiki 1.4 2.5 +/-

• 0.0711
• 0.0964
• 0.0964
• 0.0653
• /+

0.0048 0.0469 0.0468 0.0319 +/+

• 0.0704
• 0.0277
• 0.0277
• 0.0189
• /-
• 0.0115
• 0.0429
• 0.0428
• 0.0296

NL wiki 1.3 1.8 +/-

• 0.0585
• 0.3018
• 0.3017
• 0.2089
• /+

0.0100 0.0730 0.0727 0.0504 +/+

• 0.0628

0.0016 0.0016 0.0015

• /-
• 0.0233
• 0.1505
• 0.1498
• 0.1048

KO wiki

• +/-
• 0.0805
• 0.2733
• 0.2696
• 0.1985
• /+

0.0157 0.2323 0.1760 0.1902 +/+

• 0.1697

0.0191 0.0175 0.0170

• /-
• 0.0138
• 0.0618
• 0.0493
• 0.0463

RU wiki

• +/-
• 0.0911
• 0.1084
• 0.1080
• 0.0755
• /+

0.0398 0.2200 0.1977 0.1655 +/+ 0.0082 0.2480 0.2472 0.1736

• /-
• 0.0242

0.0255 0.0236 0.0187

Table: Results on the wikipedia graphs obtained from the

http://law.di.unimi.it/ database

1 determined using Hill’s estimator

[Pim van der Hoorn] 24/30

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

In-Out correlation

[Pim van der Hoorn] 26/30

In-Out correlation

• n

n Gn

• [Pim van der Hoorn]

26/30

In-Out correlation

• n

n Gn

• ρ+

−(Gn) =

2n3 − 3n2 2n3 − n2 + 1

[Pim van der Hoorn] 26/30

In-Out correlation

• n

n Gn

• ρ+

−(Gn) =

2n3 − 3n2 2n3 − n2 + 1 → 1

[Pim van der Hoorn] 26/30

Convergence to random variable

[Pim van der Hoorn] 27/30

Convergence to random variable

• X1

Y1

• Xn

Yn

[Pim van der Hoorn] 27/30

Convergence to random variable

• X1

Y1

• Xn

Yn Xi := Wi + W ′

i

Yi := Wi + aW ′

i

Wi, W ′

i i.i.d samples W , W ′

heavy tailed, same exponent. a > 0

[Pim van der Hoorn] 27/30

Convergence to random variable

• X1

Y1

• Xn

Yn Xi := Wi + W ′

i

Yi := Wi + aW ′

i

Wi, W ′

i i.i.d samples W , W ′

heavy tailed, same exponent. a > 0 a >> 1 ⇒ ρ+

− → −1 [Pim van der Hoorn] 27/30

Convergence to random variable, continued...

[Pim van der Hoorn] 28/30

Convergence to random variable, continued...

ρ+

− →

Z1 + aZ2 √Z1 + Z2

• Z1 + a2Z2

Z1, Z2 stable random variables

[Pim van der Hoorn] 28/30

Convergence to random variable, continued...

ρ+

− →

Z1 + aZ2 √Z1 + Z2

• Z1 + a2Z2

Z1, Z2 stable random variables T := Z2 Z1

[Pim van der Hoorn] 28/30

Convergence to random variable, continued...

ρ+

− →

Z1 + aZ2 √Z1 + Z2

• Z1 + a2Z2

Z1, Z2 stable random variables T := Z2 Z1 ρ+

− →

1 + aT √ 1 + T √ 1 + a2T

[Pim van der Hoorn] 28/30

Convergence to random variable, continued...

ρ+

− →

Z1 + aZ2 √Z1 + Z2

• Z1 + a2Z2

Z1, Z2 stable random variables T := Z2 Z1 ρ+

− →

1 + aT √ 1 + T √ 1 + a2T 0 < ε < 1 a = 2(1 ± √ 1 − ε2) ε2 − 1 ρ+

− has support on (ε, 1). [Pim van der Hoorn] 28/30

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

Possible topics

[Pim van der Hoorn] 30/30

Possible topics

◮ Investigate null model [Pim van der Hoorn] 30/30

Possible topics

◮ Investigate null model ◮ Include null model in code framework [Pim van der Hoorn] 30/30

Possible topics

◮ Investigate null model ◮ Include null model in code framework ◮ Lower bounds on average ties [Pim van der Hoorn] 30/30

Possible topics

◮ Investigate null model ◮ Include null model in code framework ◮ Lower bounds on average ties ◮ Directed models [Preferential Attachment] [Pim van der Hoorn] 30/30

Possible topics

◮ Investigate null model ◮ Include null model in code framework ◮ Lower bounds on average ties ◮ Directed models [Preferential Attachment] ◮ Angular measure (dependence between large nodes) [Pim van der Hoorn] 30/30