Matthew effects via team semantics (work in progress) Fan Yang - - PowerPoint PPT Presentation

Matthew effects via team semantics (work in progress)

Fan Yang Delft University of Technology, The Netherlands ————————————————————

Joint work with S. Frittella1, G. Greco2, M. Piazzai2,3 and N. Wijnberg3

1 University of Orléans, France 2 TU Delft 3 Amsterdam Business School

1/15

The Matthew effect (Merton 1968)

For whoever has will be given more, and they will have an abundance. Whoever does not have, even what they have will be taken from them. Matthew, 25:29

2/15

The Matthew effect (Merton 1968)

For whoever has will be given more, and they will have an abundance. Whoever does not have, even what they have will be taken from them. Matthew, 25:29

2/15

The Matthew effect (Merton 1968)

For whoever has will be given more, and they will have an abundance. Whoever does not have, even what they have will be taken from them. Matthew, 25:29

2/15

The Matthew effect (Merton 1968)

For whoever has will be given more, and they will have an abundance. Whoever does not have, even what they have will be taken from them. Matthew, 25:29

2/15

The Matthew effect (Merton 1968)

For whoever has will be given more, and they will have an abundance. Whoever does not have, even what they have will be taken from them. Matthew, 25:29

“The rich get richer; the poor get poorer.”

2/15

Matthew effects

sociology of science: (Merton 1968) business: social network:

3/15

Matthew effects

sociology of science: (Merton 1968)

reputation citations

business: social network:

3/15

Matthew effects

sociology of science: (Merton 1968)

reputation citations +

business: social network:

3/15

Matthew effects

sociology of science: (Merton 1968)

reputation citations + +

business: social network:

3/15

Matthew effects

sociology of science: (Merton 1968)

reputation citations + +

business:

reviews sales

social network:

3/15

Matthew effects

sociology of science: (Merton 1968)

reputation citations + +

business:

reviews sales +

social network:

3/15

Matthew effects

sociology of science: (Merton 1968)

reputation citations + +

business:

reviews sales + +

social network:

3/15

Matthew effects

sociology of science: (Merton 1968)

reputation citations + +

business:

reviews sales + +

social network:

3/15

Matthew effects

sociology of science: (Merton 1968)

reputation citations + +

business:

reviews sales + +

social network:

3/15

Matthew effects

sociology of science: (Merton 1968)

reputation citations + +

business:

reviews sales + +

social network:

3/15

Matthew effects

sociology of science: (Merton 1968)

reputation citations + +

business:

reviews sales + +

social network:

+

3/15

Data sets and regression analysis

Sales Reviews Time 2 2010 3 5 2011 6 8 2012 8 10 2013 9 14 2014 12 15 2015

4/15

Data sets and regression analysis

Sales Reviews Time 2 2010 3 5 2011 6 8 2012 8 10 2013 9 14 2014 12 15 2015

2 4 6 8 10 12 14 2 4 6 8 10 12 reviews at time t − ℓ sales at time t

4/15

Data sets and regression analysis

Sales Reviews Time 2 2010 3 5 2011 6 8 2012 8 10 2013 9 14 2014 12 15 2015

2 4 6 8 10 12 14 2 4 6 8 10 12 reviews at time t − ℓ sales at time t s(t) = β0 + β1r(t−ℓ) + ǫ, where β1 > 0.

4/15

Dependence relation

x1 ... xn w1 w2 y t 1 ... 2 3 2000 3 ... 5 ⋮ ⋮ 6 2001 6 ... 7 10 2002 7 ... 8 ⋮ ⋮ 15 2003 12 ... 8 21 2004 Independent variables Dependent variable

y(t) = β0 + β1(x1)(t−ℓ) + β2(x2)(t−ℓ) + ⋅⋅⋅ + βn(xn)(t−ℓ) + ǫ

5/15

Dependence relation

x1 ... xn w1 w2 y t 1 ... 2 3 2000 3 ... 5 ⋮ ⋮ 6 2001 6 ... 7 10 2002 7 ... 8 ⋮ ⋮ 15 2003 12 ... 8 21 2004 Independent variables Dependent variable Time variable

y(t) = β0 + β1(x1)(t−ℓ) + β2(x2)(t−ℓ) + ⋅⋅⋅ + βn(xn)(t−ℓ) + ǫ

5/15

Dependence relation

x1 ... xn w1 w2 y t 1 ... 2 3 2000 3 ... 5 ⋮ ⋮ 6 2001 6 ... 7 10 2002 7 ... 8 ⋮ ⋮ 15 2003 12 ... 8 21 2004 Independent variables Dependent variable Time variable

y(t) = β0 + β1(x1)(t−ℓ) + β2(x2)(t−ℓ) + ⋅⋅⋅ + βn(xn)(t−ℓ) + ǫ y(t−ℓ) = β0 + β1(x1)(t−2ℓ) + β2(x2)(t−2ℓ) + ⋅⋅⋅ + βn(xn)(t−2ℓ) + ǫ y(t) − y(t−ℓ) = β1((x1)(t−ℓ) − (x1)(t−2ℓ)) + ǫ

If β1 > 0: xt−2ℓ xt−ℓ yt−ℓ yt

5/15

Dependence relation

x1 ... xn w1 w2 y t 1 ... 2 3 2000 3 ... 5 ⋮ ⋮ 6 2001 6 ... 7 10 2002 7 ... 8 ⋮ ⋮ 15 2003 12 ... 8 21 2004 Independent variables Dependent variable Time variable

y(t) = β0 + β1(x1)(t−ℓ) + β2(x2)(t−ℓ) + ⋅⋅⋅ + βn(xn)(t−ℓ) + ǫ y(t−ℓ) = β0 + β1(x1)(t−2ℓ) + β2(x2)(t−2ℓ) + ⋅⋅⋅ + βn(xn)(t−2ℓ) + ǫ y(t) − y(t−ℓ) = β1((x1)(t−ℓ) − (x1)(t−2ℓ)) + ǫ

If β1 > 0: xt−2ℓ xt−ℓ yt−ℓ yt x ℓ y

5/15

Team semantics & (In)dependence logic

⃗ u ⃗ w x y t 1 3 2000 ⋮ ⋮ 3 6 2002 6 10 2004 ⋮ ⋮ 7 15 2006 12 21 2008

Team semantics (Hodges 1997) Dependence logic (Väänänen 2007): FO+ =(t,y) Independence logic (Grädel, Väänänen 2013): FO + x ⊥ y

6/15

Team semantics & (In)dependence logic

⃗ u ⃗ w x y t 1 3 2000 ⋮ ⋮ 3 6 2002 6 10 2004 ⋮ ⋮ 7 15 2006 12 21 2008 a set of assignments s

Team semantics (Hodges 1997) Dependence logic (Väänänen 2007): FO+ =(t,y) Independence logic (Grädel, Väänänen 2013): FO + x ⊥ y

6/15

Team semantics & (In)dependence logic

⃗ u ⃗ w x y t 1 3 2000 ⋮ ⋮ 3 6 2002 6 10 2004 ⋮ ⋮ 7 15 2006 12 21 2008 a set of assignments s M ⊧s x ℓ y?

Team semantics (Hodges 1997) Dependence logic (Väänänen 2007): FO+ =(t,y) Independence logic (Grädel, Väänänen 2013): FO + x ⊥ y

6/15

Team semantics & (In)dependence logic

⃗ u ⃗ w x y t 1 3 2000 ⋮ ⋮ 3 6 2002 6 10 2004 ⋮ ⋮ 7 15 2006 12 21 2008 a set of assignments s

Team semantics (Hodges 1997) Dependence logic (Väänänen 2007): FO+ =(t,y) Independence logic (Grädel, Väänänen 2013): FO + x ⊥ y

6/15

Team semantics & (In)dependence logic

⃗ u ⃗ w x y t 1 3 2000 ⋮ ⋮ 3 6 2002 6 10 2004 ⋮ ⋮ 7 15 2006 12 21 2008 a set of assignments s a team D: M ⊧D x ℓ y

Team semantics (Hodges 1997) Dependence logic (Väänänen 2007): FO+ =(t,y) Independence logic (Grädel, Väänänen 2013): FO + x ⊥ y

6/15

Team semantics & (In)dependence logic

⃗ u ⃗ w x y t 1 3 2000 ⋮ ⋮ 3 6 2002 6 10 2004 ⋮ ⋮ 7 15 2006 12 21 2008 a set of assignments s a team D:

Team semantics (Hodges 1997) Dependence logic (Väänänen 2007): FO+ =(t,y) Independence logic (Grädel, Väänänen 2013): FO + x ⊥ y Inquisitive logic (Ciardelli, Groenendijk, Roelofsen 2011)

[stay tuned, stay for the next two talks...]

6/15

Team semantics & (In)dependence logic

⃗ u ⃗ w x y t 1 3 2000 ⋮ ⋮ 3 6 2002 6 10 2004 ⋮ ⋮ 7 15 2006 12 21 2008 a set of assignments s a team D:

Team semantics (Hodges 1997) Dependence logic (Väänänen 2007): FO+ =(t,y)

∃f

Independence logic (Grädel, Väänänen 2013): FO + x ⊥ y ×

6/15

Team semantics & (In)dependence logic

⃗ u ⃗ w x y t 1 3 2000 ⋮ ⋮ 3 6 2002 6 10 2004 ⋮ ⋮ 7 15 2006 12 21 2008 a set of assignments s a team D:

Team semantics (Hodges 1997) Dependence logic (Väänänen 2007): FO+ =(t,y)

∃f

Independence logic (Grädel, Väänänen 2013): FO + x ⊥ y × x ℓ y

y(t) = α0 + βx(t−ℓ) + α1(w1)(t−ℓ) + ⋅ ⋅ ⋅ + αn(wn)(t−ℓ) + ǫ

• Definition. M ⊧D x ℓ y iff ∃p(x, ⃗

w,y) as above with β > 0 s.t. M ⊧D x p

ℓ y,

6/15

Team semantics & (In)dependence logic

⃗ u ⃗ w x y t 1 3 2000 ⋮ ⋮ 3 6 2002 6 10 2004 ⋮ ⋮ 7 15 2006 12 21 2008 δ 2 2 2 2 2 a set of assignments s a team D:

Team semantics (Hodges 1997) Dependence logic (Väänänen 2007): FO+ =(t,y)

∃f

Independence logic (Grädel, Väänänen 2013): FO + x ⊥ y × x ℓ y

y(t) = α0 + βx(t−ℓ) + α1(w1)(t−ℓ) + ⋅ ⋅ ⋅ + αn(wn)(t−ℓ) + ǫ

term ℓ ∶∶= δ ∣ kδ

• Definition. M ⊧D x ℓ y iff ∃p(x, ⃗

w,y) as above with β > 0 s.t. M ⊧D x p

ℓ y,

6/15

Team semantics & (In)dependence logic

⃗ u ⃗ w x y t 1 3 2000 ⋮ ⋮ 3 6 2002 6 10 2004 ⋮ ⋮ 7 15 2006 12 21 2008 δ 2 2 2 2 2 s s′ a set of assignments s a team D:

Team semantics (Hodges 1997) Dependence logic (Väänänen 2007): FO+ =(t,y)

∃f

Independence logic (Grädel, Väänänen 2013): FO + x ⊥ y × x ℓ y

y(t) = α0 + βx(t−ℓ) + α1(w1)(t−ℓ) + ⋅ ⋅ ⋅ + αn(wn)(t−ℓ) + ǫ

term ℓ ∶∶= δ ∣ kδ

• Definition. M ⊧D x ℓ y iff ∃p(x, ⃗

w,y) as above with β > 0 s.t. M ⊧D x p

ℓ y, namely, for all s,s′ ∈ D,

s(t) = s′(t) + s′(ℓM) ⇒ s(y) ≈M βs′(x) + q(s′( ⃗ w)).

6/15

Dependence relation

M ⊧D x1,...,xn p

ℓ y iff for all s,s′ ∈ D,

s(t) = s′(t)+s′(ℓM) ⇒ s(y) ≈M β1s′(x1)+⋅⋅⋅+βns′(xn)+q(s′( ⃗ w)), where p is the above polynomial and β1,...,βn > 0. M ⊧D x1,...,xn ℓ y iff there exists p(⃗ x, ⃗ w,y) with β1,...,βn > 0 s.t. M ⊧D ⃗ x p

ℓ y

x1,...,xn ℓ y is defined similarly except that β1,...,βn are required to be < 0.

7/15

Dependence relation

M ⊧D x1,...,xn p

ℓ y iff for all s,s′ ∈ D,

s(t) = s′(t)+s′(ℓM) ⇒ s(y) ≈M β1s′(x1)+⋅⋅⋅+βns′(xn)+q(s′( ⃗ w)), where p is the above polynomial and β1,...,βn > 0. M ⊧D x1,...,xn ℓ y iff there exists p(⃗ x, ⃗ w,y) with β1,...,βn > 0 s.t. M ⊧D ⃗ x p

ℓ y

x1,...,xn ℓ y is defined similarly except that β1,...,βn are required to be < 0.

7/15

Matthew effects

+

y is subject to a (positive) direct ℓ-Matthew effect if y(t) = α0 + βy(t−ℓ) + α1(x1)(t−ℓ) + ⋅⋅⋅ + αn(xn)(t−ℓ) + ǫ, where β > 0. Define DMEℓy ∶= y ℓ y.

8/15

Matthew effects

yt−2ℓ yt−ℓ yt +

y is subject to a (positive) direct ℓ-Matthew effect if y(t) = α0 + βy(t−ℓ) + α1(x1)(t−ℓ) + ⋅⋅⋅ + αn(xn)(t−ℓ) + ǫ, where β > 0. Define DMEℓy ∶= y ℓ y.

8/15

Matthew effects

xt−2ℓ xt−ℓ yt−3ℓ yt−2ℓ yt−ℓ yt reviews sales + +

y is subject to a (positive) x-mediated ℓ-Matthew effect if y(t) = α0 + β(x)(t−ℓ) + α1(x1)(t−ℓ) + ⋅⋅⋅ + βn(xn)(t−ℓ) + ǫ, x(t) = γ0 + δ(y)(t−ℓ) + γ1(x1)(t−ℓ) + ⋅⋅⋅ + γn(xn)(t−ℓ) + ǫ, where β,δ > 0. Define MMEℓ(y,x) ∶= (x ℓ y) ∧ (y ℓ x) DMEℓy ⊧ MMEℓ(y,y)

9/15

Matthew effects

xt−2ℓ xt−ℓ yt−3ℓ yt−2ℓ yt−ℓ yt reviews sales + +

y is subject to a (positive) x-mediated ℓ-Matthew effect if y(t) = α0 + β(x)(t−ℓ) + α1(x1)(t−ℓ) + ⋅⋅⋅ + βn(xn)(t−ℓ) + ǫ, x(t) = γ0 + δ(y)(t−ℓ) + γ1(x1)(t−ℓ) + ⋅⋅⋅ + γn(xn)(t−ℓ) + ǫ, where β,δ > 0. Define MMEℓ(y,x) ∶= (x ℓ y) ∧ (y ℓ x) DMEℓy ⊧ MMEℓ(y,y)

9/15

Matthew effects

• x
• y

reviews sales

y is subject to a (positive) comple ℓ-Matthew effect w.r.t. x: CMEℓy(x) ∶∶= MMEℓ(y,x) ∧ DMEℓy x and y are subject to a (positive) double comple ℓ-Matthew effect: CMEℓ(x,y) ∶∶= MMEℓ(y,x) ∧ DMEℓx ∧ DMEℓy

10/15

Matthew effects

• x
• y

reviews sales

y is subject to a (positive) comple ℓ-Matthew effect w.r.t. x: CMEℓy(x) ∶∶= MMEℓ(y,x) ∧ DMEℓy x and y are subject to a (positive) double comple ℓ-Matthew effect: CMEℓ(x,y) ∶∶= MMEℓ(y,x) ∧ DMEℓx ∧ DMEℓy

10/15

Properties of dependence relation

(Commutativity) x1,⋯,xn ℓ y ⊧ xi1,⋯,xin ℓ y (Duplication) x1,...,xn ℓ y ⊧ xi,x1,...,xn ℓ y (Projection) x1,...,xn ℓ y ⊧ xi1,...,xik ℓ y, where k ≤ n (Regrouping) (⃗ x p

ℓ y),(⃗

z p

ℓ y) ⊧ ⃗

x, ⃗ z p

ℓ y

(Transitivity) (⃗ x ℓ y), (y ℓ′ z) ⊧ ⃗ x ℓ+ℓ′ z (Enhancing) x ℓ x ⊧ x kℓ x (Reflexivity) ⊧ x 0 x

11/15

Properties of Matthew effects

MMEℓ(x,y),MMEℓ(y,z) ⊧ MME2ℓ(x,z), i.e., (x ℓ y),(y ℓ x),(y ℓ z),(z ℓ y) ⊧ (x 2ℓ z) ∧ (z 2ℓ x) MMEℓ(y,x) ⊧ DME2ℓx ∧ DME2ℓy, i.e., (x ℓ y),(y ℓ x) ⊧ (x 2ℓ x) ∧ (y 2ℓ y)

12/15

Properties of Matthew effects

xt−2ℓ xt−ℓ yt−3ℓ yt−2ℓ yt−ℓ yt

MMEℓ(x,y),MMEℓ(y,z) ⊧ MME2ℓ(x,z), i.e., (x ℓ y),(y ℓ x),(y ℓ z),(z ℓ y) ⊧ (x 2ℓ z) ∧ (z 2ℓ x) MMEℓ(y,x) ⊧ DME2ℓx ∧ DME2ℓy, i.e., (x ℓ y),(y ℓ x) ⊧ (x 2ℓ x) ∧ (y 2ℓ y)

12/15

Properties of Matthew effects

xt−2ℓ xt−ℓ yt−4ℓ

• yt−2ℓ
• yt

MMEℓ(x,y),MMEℓ(y,z) ⊧ MME2ℓ(x,z), i.e., (x ℓ y),(y ℓ x),(y ℓ z),(z ℓ y) ⊧ (x 2ℓ z) ∧ (z 2ℓ x) MMEℓ(y,x) ⊧ DME2ℓx ∧ DME2ℓy, i.e., (x ℓ y),(y ℓ x) ⊧ (x 2ℓ x) ∧ (y 2ℓ y)

12/15

A logic of Matthew effects (ML)

Syntax φ ∶∶= α ∣ ¬α ∣ ⃗ x ℓ y ∣ ⃗ x ℓ y ∣ φ ∧ φ ∣ φ ∨ φ ∣ φ ⊗ φ ∣ ∃xφ ∣ ∀xφ Team semantics M ⊧D α iff for all s ∈ D, M ⊧s α M ⊧D ¬α iff for all s ∈ D, M / ⊧s α M ⊧D φ ∧ ψ iff M ⊧D φ and M ⊧D ψ M ⊧D φ ∨ ψ iff M ⊧D φ or M ⊧D ψ M ⊧D φ ⊗ ψ iff there exist D0,D1 ⊆ D with D = D0 ∪ D1 s.t. M ⊧D0 φ and M ⊧D1 ψ

13/15

A logic of Matthew effects (ML)

Syntax φ ∶∶= α ∣ ¬α ∣ ⃗ x ℓ y ∣ ⃗ x ℓ y ∣ φ ∧ φ ∣ φ ∨ φ ∣ φ ⊗ φ ∣ ∃xφ ∣ ∀xφ Team semantics M ⊧D α iff for all s ∈ D, M ⊧s α M ⊧D ¬α iff for all s ∈ D, M / ⊧s α M ⊧D φ ∧ ψ iff M ⊧D φ and M ⊧D ψ M ⊧D φ ∨ ψ iff M ⊧D φ or M ⊧D ψ M ⊧D φ ⊗ ψ iff there exist D0,D1 ⊆ D with D = D0 ∪ D1 s.t. M ⊧D0 φ and M ⊧D1 ψ

... x y t 2 3 1 5 2 2 ... 10 4 3 12 15 4 16 21 5 (x < 10)

13/15

A logic of Matthew effects (ML)

Syntax φ ∶∶= α ∣ ¬α ∣ ⃗ x ℓ y ∣ ⃗ x ℓ y ∣ φ ∧ φ ∣ φ ∨ φ ∣ φ ⊗ φ ∣ ∃xφ ∣ ∀xφ Team semantics M ⊧D α iff for all s ∈ D, M ⊧s α M ⊧D ¬α iff for all s ∈ D, M / ⊧s α M ⊧D φ ∧ ψ iff M ⊧D φ and M ⊧D ψ M ⊧D φ ∨ ψ iff M ⊧D φ or M ⊧D ψ M ⊧D φ ⊗ ψ iff there exist D0,D1 ⊆ D with D = D0 ∪ D1 s.t. M ⊧D0 φ and M ⊧D1 ψ

... x y t 2 3 1 5 2 2 ... 10 4 3 12 15 4 16 21 5 (x < 10)

13/15

A logic of Matthew effects (ML)

Syntax φ ∶∶= α ∣ ¬α ∣ ⃗ x ℓ y ∣ ⃗ x ℓ y ∣ φ ∧ φ ∣ φ ∨ φ ∣ φ ⊗ φ ∣ ∃xφ ∣ ∀xφ Team semantics M ⊧D α iff for all s ∈ D, M ⊧s α M ⊧D ¬α iff for all s ∈ D, M / ⊧s α M ⊧D φ ∧ ψ iff M ⊧D φ and M ⊧D ψ M ⊧D φ ∨ ψ iff M ⊧D φ or M ⊧D ψ M ⊧D φ ⊗ ψ iff there exist D0,D1 ⊆ D with D = D0 ∪ D1 s.t. M ⊧D0 φ and M ⊧D1 ψ

... x y t 2 3 1 5 2 2 ... 10 4 3 12 15 4 16 21 5 (x < 10) ⊗ MMEℓ(x,y)

13/15

Comparison with dependence and independence logic

(Väänänen, Kontinen 2009) Dependence Logic (D ∶= FO+=(⃗ x,y)) can express all existential second-order downward closed properties. (Galliani 2012) Independence Logic (I ∶= FO +⃗ x ⊥ ⃗ y) can express all existential second-order properties. ML ≤ D < I, i.e., for every ML-formula φ, there is D-formula τ(φ) s.t. M ⊧D φ ⇐ ⇒ M ⊧D τ(φ). There is a deduction system (via translation into independence logic) such that Γ ⊧ φ ⇐ ⇒ Γ ⊢ φ, where φ is ⊗-free and has no quantification over ⃗ x ℓ y. (follows from (Hannula 2013)&(Y. 2016))

14/15

Comparison with dependence and independence logic

(Väänänen, Kontinen 2009) Dependence Logic (D ∶= FO+=(⃗ x,y)) can express all existential second-order downward closed properties. (Galliani 2012) Independence Logic (I ∶= FO +⃗ x ⊥ ⃗ y) can express all existential second-order properties. ML ≤ D < I, i.e., for every ML-formula φ, there is D-formula τ(φ) s.t. M ⊧D φ ⇐ ⇒ M ⊧D τ(φ). There is a deduction system (via translation into independence logic) such that Γ ⊧ φ ⇐ ⇒ Γ ⊢ φ, where φ is ⊗-free and has no quantification over ⃗ x ℓ y. (follows from (Hannula 2013)&(Y. 2016))

14/15

Future work

(Full) axiomatization of ML without going through the translation. Comparison with other dependency notions. To study the notion of ⃗ x r

ℓ y, where r represents a regression

that has been (actually) performed on the dataset in question. There is (indeed) a difference between ⃗ x p

ℓ y and ⃗

x r

ℓ y, even if

r generates the same regression function for y as p. Different levels of abstraction: ⃗ x ℓ y, ⃗ x p

ℓ y and ⃗

x r

ℓ y

To consider other parameters in a Matthew effect. E.g., the strength of a Matthew effect (which roughly corresponds to the β in y(t) = β(y)(t−ℓ) + q( ⃗ w)).

15/15

Future work

(Full) axiomatization of ML without going through the translation. Comparison with other dependency notions. To study the notion of ⃗ x r

ℓ y, where r represents a regression

that has been (actually) performed on the dataset in question. There is (indeed) a difference between ⃗ x p

ℓ y and ⃗

x r

ℓ y, even if

r generates the same regression function for y as p. Different levels of abstraction: ⃗ x ℓ y, ⃗ x p

ℓ y and ⃗

x r

ℓ y

To consider other parameters in a Matthew effect. E.g., the strength of a Matthew effect (which roughly corresponds to the β in y(t) = β(y)(t−ℓ) + q( ⃗ w)).

15/15

Future work

(Full) axiomatization of ML without going through the translation. Comparison with other dependency notions. To study the notion of ⃗ x r

ℓ y, where r represents a regression

that has been (actually) performed on the dataset in question. There is (indeed) a difference between ⃗ x p

ℓ y and ⃗

x r

ℓ y, even if

r generates the same regression function for y as p. Different levels of abstraction: ⃗ x ℓ y, ⃗ x p

ℓ y and ⃗

x r

ℓ y

To consider other parameters in a Matthew effect. E.g., the strength of a Matthew effect (which roughly corresponds to the β in y(t) = β(y)(t−ℓ) + q( ⃗ w)).

15/15