Algorithmic Coalitional Game Theory Lecture 10: Game-Theoretic - - PowerPoint PPT Presentation

Oskar Skibski

University of Warsaw

Algorithmic Coalitional Game Theory

Lecture 10: Game-Theoretic Network Centralities

05.05.2020

Centrality Measures

In other words: what is the importance of a node in a graph? Notation: for a fixed set of vertices 𝑊:

• 𝐻! is the set of all graphs, and
• 𝐷! is the set of all centralities.

𝑤 𝑣 𝑥 𝑢

Centrality (measure) is a function 𝐺: 𝐻! → ℝ! that assigns to each node, 𝑗, in a graph 𝐻 = (𝑊, 𝐹) a real value 𝐺&(𝐻). Centrality Measure

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Centrality Measures

Centrality Measures

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• Degree Centrality:

𝐸' 𝐻 = 𝑣 ∈ 𝑊 ∶ 𝑤, 𝑣 ∈ 𝐹 ;

• Closeness Centrality:

𝐷' 𝐻 = 1 ∑(∈!∖ ' 𝑒𝑗𝑡𝑢 𝑤, 𝑣 ;

• Betweenness Centrality:

𝐶' 𝐻 = :

+,-∈!∖ '

𝑞 ∈ Π 𝑡, 𝑢 ∶ 𝑤 ∈ 𝑞 Π 𝑡, 𝑢 , where Π 𝑡, 𝑢 = set of shortest paths between 𝑡 and 𝑢; and many more: Eigenvector, Katz, PageRank… Standard Centralities

Game-Theoretic Centralities

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory 𝑠

Graph

𝜒

Coalitional Game Centrality

Game-theoretic network centrality is a pair, (𝑠, 𝜒), where:

• 𝑠: 𝐻! → 2! → ℝ is a representation function,
• 𝜒: 2! → ℝ → ℝ! is a solution concept.

We will denote 𝜒 ∘ 𝑠 as 𝑠, 𝜒 and set of all GTCs: 𝐻𝑈𝐷!. Game-Theoretic Network Centrality

Game-Theoretic Centralities

Examples of representation functions 𝑠 𝐻 = 𝑔

. /:

• 𝑔

. / 𝑇 = 1 if 𝐻[𝑇] is connected,𝑔 . / 𝑇 = 0, otherwise

[Amer & Gimenez 2001]

• 𝑔

. / 𝑇 = 𝐹 𝑇 / ∑0∈1[3] 𝜕(𝑓) if 𝐻[𝑇] is connected, 𝑔 . / 𝑇 =

0, otherwise [Lindelauf et al. 2013]

• 𝑔

. / 𝑇 = 2( 𝑇 − 𝐿 𝐻 𝑇

) [Skibski et al. 2019b] The Shapley value is mostly used as the solution concept. We will discuss only GTCs where the solution concept is a positive semivalue (marked as + in the subscript – formally, for 𝐽 ⊆ 𝐻𝑈𝐷!, 𝐽5 = 𝑠, 𝜒 ∈ 𝐽 ∶ 𝜒 𝑗𝑡 𝑏 𝑞𝑝𝑡𝑗𝑢𝑗𝑤𝑓 𝑡𝑓𝑛𝑗𝑤𝑏𝑚𝑣𝑓 ). 7

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Game-Theoretic Centralities

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

A value is a semivalue if it is of the form: 𝜒& 𝑂, 𝑤 = :

3⊆7∖{&}

𝛾( 𝑇 ) 𝑤 𝑇 ∪ 𝑗 − 𝑤 𝑇 , for 𝛾: 0, … , 𝑜 − 1 → [0,1] such that ∑:;<

=>? 𝛾(𝑙) =>? :

= 1. Semivalues We will call a semivalue positive if 𝛾 𝑙 > 0 for every 𝑙 ∈ {0, … , 𝑜 − 1}.

Game-Theoretic Centralities

For 𝐽 ⊆ 𝐻𝑈𝐷!, we define 𝐽 = 𝑠, 𝜒 ∶ 𝑠, 𝜒 ∈ 𝐽 . 9

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

How to characterize GTCs?

?

(𝑠3, 𝜒3) (𝑠4, 𝜒4) (𝑠1, 𝜒1) (𝑠2, 𝜒2) 𝑑1 𝑑2 𝑑3

𝐻𝑈𝐷!

[.]

𝐽! 𝐷! [𝐽!]

(𝑠5, 𝜒)

General result

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We have 𝐻𝑈𝐷! = 𝐷!, i.e., for every centrality 𝐺 there exists (𝑠, 𝜒) such that 𝑠, 𝜒 = 𝐺. Characterization of GTCs [Skibski et al. 2018] Proof: On the blackboard.

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

𝑤 𝑣 𝑤 𝑣 𝐻 𝜒

𝜒(𝑔

. /)

𝑤 𝑣 𝑤 𝑣 𝑠 𝑠 𝜒

𝜒𝑤(𝑔

&! ' )

𝜒𝑣(𝑔

&! ' )

𝜒((𝑔

&" ' )

𝜒𝑣(𝑔

&" ' )

𝑔

&" ' ({𝑤, 𝑣})

𝑔

#! $ ({𝑣})

𝑔

#! $ ({𝑤})

𝑔

&" ' (∅)

𝐻1 𝐻2

𝑔

#" $ ({𝑤, 𝑣})

𝑔

#" $ ({𝑣})

𝑔

#" $ ({𝑤})

𝑔

&! ' (∅)

𝑠 𝐻 = 𝑔

! "

General result

General result

Sketch of the proof: Take 𝐺 ∈ 𝐷! and define 𝑠 𝐻 = 𝑔

. / as follows:

𝑔

. / 𝑇 = : &∈3

𝐺& 𝐻 . Game 𝑔

. / is additive (non-essential) where every player is a

dummy player – the marginal contribution of 𝑗 to every coalition 𝑇 ⊆ 𝑊 ∖ 𝑗 equals 𝑔

. /

𝑗 = 𝐺&(𝐻). Hence, for every semivalue 𝜒 we have: [ 𝑠, 𝜒 ] = 𝐺. 12

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Restrictions on GTCs

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

General 𝑤

c c

𝑣 Separable 𝑤

c c

𝑣 Induced 𝑤

c c

𝑣 Consider the following restrictions on the representation function.

Restrictions on GTCs

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

𝑤 𝑤 𝑤 𝑤 𝑤 𝑤 𝑤 𝑤 𝑤 𝑤 𝑤 𝑤 Separable Induced General

𝐻1 𝐻2 𝐻3 𝐻4

… … … …

Restrictions on GTCs

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

𝑤 𝑤 𝑤 𝑤 𝑤 𝑤 𝑤 𝑤 𝑤 𝑤 𝑤 𝑤 Separable Induced General

𝐻5 𝐻6 𝐻7 𝐻8

… … … …

Separable GTCs

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Representation function 𝑠 is separable if for every coalition 𝑇 and every two graphs 𝐻, 𝐻E: 𝐻 𝑇 = 𝐻E 𝑇 ∧ 𝐻 𝑊 ∖ 𝑇 = 𝐻E 𝑊 ∖ 𝑇 → 𝑠 𝐻 𝑇 = 𝑠 𝐻E 𝑇 . All GTCs with separable 𝑠: 𝑇𝐻𝑈𝐷! What is 𝑇𝐻𝑈𝐷5

! ?

?

Separable GTCs

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

In other words:

• ⊆: every separable game-theoretic centrality satisfies

Fairness

• ⊇: every centrality that satisfies Fairness can be obtained

as a separable game-theoretic centrality We have: 𝑇𝐻𝑈𝐷5

! = 𝐺 ∈ 𝐷! ∶ 𝐺 𝑡𝑏𝑢𝑗𝑡𝑔𝑗𝑓𝑡 𝐺𝑏𝑗𝑠𝑜𝑓𝑡𝑡 .

Characterization of Separable GTCs [Skibski et al. 2018] Proof: On the blackboard.

Separable GTCs

Sketch of the proof: Before we start… What is the basis of all centralities? For every node 𝑤 ∈ 𝑊 and graph 𝑊, 𝑁 ∈ 𝐻! we define elementary centrality 𝑓 {(},F as follows: 𝑓'

{(},F 𝑊, 𝐹 = a1

if 𝑤 = 𝑣 and 𝑁 = 𝐹,

• therwise.

Clearly, 𝑓 {(},F ∶ 𝑊, 𝑁 ∈ 𝐻!, 𝑣 ∈ 𝑊 forms a basis. 18

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Separable GTCs

Sketch of the proof (continued): We can also express the basis using unanimity centralities. For every node 𝑤 ∈ 𝑊 and graph 𝑊, 𝑁 ∈ 𝐻! we define unanimity centrality 𝑑 {(},F as follows: 𝑑'

{(},F 𝑊, 𝐹 = a1

if 𝑤 = 𝑣 and 𝑁 ⊆ 𝐹,

• therwise.

Clearly, 𝑑 {(},F ∶ 𝑊, 𝑁 ∈ 𝐻!, 𝑣 ∈ 𝑊 forms a basis. 19

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Separable GTCs

Sketch of the proof (continued): We will also use unanimity centralities generalized to sets: for 𝑉 ⊆ 𝑊 and 𝑊, 𝑁 ∈ 𝐻! we define: 𝑑'

G,F 𝑊, 𝐹 = a1

if 𝑤 ∈ 𝑉 and 𝑁 ⊆ 𝐹,

• therwise.

Now, for example, for 𝑤∗ ∈ 𝑊 the basis of all centralities is: 𝑑 G,F ∶ 𝑊, 𝑁 ∈ 𝐻!, 𝑉 ∈ { 𝑣 ∶ 𝑣 ∈ 𝑊 ∖ 𝑤∗} ∪ {𝑊} 20

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Separable GTCs

Sketch of the proof (continued):

• 1. Every Separable GTC satisfies Fairness

If representation function is separable, then edge {𝑤, 𝑣} affects only values of coalitions that contain either both nodes 𝑤, 𝑣 or none of them. Hence, from Symmetry and Additivity of the Shapley value we get Fairness.

• 2. Centralities that satisfy Fairness form a vector space

Clearly, if 𝐺, 𝐺E satisfy Fairness, then 𝐺 + 𝐺+ and 𝑑 ⋅ 𝐺 also

satisfy Fairness.

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Separable GTCs

Sketch of the proof (continued):

• 3. Dimension of this vector space is not greater than the

number of components in all graphs For every function 𝑕: 2!× 𝐻! → ℝ there exists at most one centrality index 𝑑 ∈ 𝒟! satisfying Fairness and ∑'∈3 𝑑' 𝑇 = 𝑕 𝑇, 𝐻 for every 𝐻 ∈ 𝐻! and 𝑇 ∈ 𝐿 𝐻 . Myerson (1977) proved a version for 𝑕: 2! → ℝ, but his proof can be easily translated to get this more general result. 22

Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Separable GTCs

Sketch of the proof (continued):

• 4. The basis of centralities that satisfy Fairness is:

𝐶IJ&/ = 𝑑 G,F ∶ 𝑊, 𝑁 ∈ 𝐻!, 𝑉 ∈ 𝐿 𝑊, 𝑁 . We need to prove that:

• elements from 𝐶IJ&/ satisfy Fairness
• elements from 𝐶IJ&/ are linearly independent
• 𝐶IJ&/ equals dimension of vector space

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Separable GTCs

Sketch of the proof (continued):

• 5. For every 𝑑 G,F ∈ 𝐶IJ&/ and every positive semivalue 𝜒,

there exists 𝑠 such that 𝑠, 𝜒 = 𝑑 G,F . We define 𝑠 𝐻 = 𝑔

. / as follows:

𝑔 !,#

$

𝑇 = 1 𝛾 𝑉 + 𝛾( 𝑉 − 1) . 𝑗𝑔 𝑇 = 𝑉, 𝑁 ⊆ 𝐹, 𝛾( 𝑉 ) 𝛾 𝑉 + 𝛾 𝑉 − 1 𝛾( 𝑊 − 1) . 𝑗𝑔 𝑇 = 𝑊, 𝑁 ⊆ 𝐹, 0. 𝑝𝑢ℎ𝑓𝑠𝑥𝑗𝑡𝑓.

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Induced GTCs

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Representation function 𝑠 is induced if for every coalition 𝑇 and every two graphs 𝐻, 𝐻E: 𝐻 𝑇 = 𝐻E 𝑇 → 𝑠 𝐻 𝑇 = 𝑠 𝐻E 𝑇 . All GTCs with induced 𝑠: 𝐽𝐻𝑈𝐷! What is 𝐽𝐻𝑈𝐷5

! ?

?

Induced GTCs

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

In other words:

• ⊆: every separable game-theoretic centrality satisfies Edge

Balanced Contributions

• ⊇: every centrality that satisfies Edge Balanced

Contributions can be obtained as a separable game- theoretic centrality We have: 𝐽𝐻𝑈𝐷5

! = 𝐺 ∈ 𝐷! ∶ 𝐺 𝑡𝑏𝑢𝑗𝑡𝑔𝑗𝑓𝑡 𝐹𝑒𝑕𝑓 𝐶𝐷 .

Characterization of Induced GTCs [Skibski et al. 2018]

Induced GTCs

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Oskar Skibski (UW) Algorithmic Coalitional Game Theory

For a graph 𝐻 = (𝑊, 𝐹), edge 𝑓 ∉ 𝐹, node 𝑗 ∈ 𝑊 and centrality 𝐺 define: 𝛦&

I 𝑓, 𝐻 = 𝐺& 𝐻 + 𝑓 − 𝐺& 𝐻 .

In particular, Fairness states: 𝛦&

I {𝑗, 𝑘}, 𝐻 = 𝛦K I {𝑗, 𝑘}, 𝐻 for

every 𝑗, 𝑘 ∈ 𝑊. Edge Balanced Contribution: for every 𝐻 = (𝑊, 𝐹), every 𝑓 = {𝑗, 𝑗E}, 𝑓E = 𝑘, 𝑘E , 𝑓, 𝑓E ∉ 𝐹 we have: 𝛦&

I 𝑓, 𝐻 + {𝑓E} − 𝛦& I 𝑓, 𝐻 = 𝛦K I 𝑓E, 𝐻 + 𝑓

− 𝛦K

I 𝑓E, 𝐻 .

Conclusions

We discussed game-theoretic centralities.

• We showed that every centrality can be represented as a

game-theoretic centrality.

• We showed that reasonable game-theoretic centralities are

characterized by Fairness. 28

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References

• [Amer & Gimenez 2001] R. Amer, J.M. Giménez.

A connectivity game for graphs. Mathematical Methods of Operations Research 60, pp. 453-470, 2004.

• [Lindelauf et al. 2013] R. Lindelauf, H. Hamers, B. Husslage.

Cooperative game theoretic centrality analysis of terrorist networks: The cases of Jemaah Islamiyah and Al Qaeda. European Journal of Operational Research, 229, pp. 230-238, 2013.

• [Skibski et al. 2018] O. Skibski, T.P. Michalak, T. Rahwan.

Axiomatic Characterization of Game-Theoretic Centrality. Journal of Artificial Intelligence Research 62, pp. 33-68, 2018.

• [Skibski et al. 2019b] O. Skibski, T. Rahwan, T.P. Michalak, M. Yokoo.

Attachment centrality: Measure for connectivity in networks. Artificial Intelligence 274, pp. 151-179, 2019.

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