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

Algorithmic Coalitional Game Theory Lecture 10: Game-Theoretic Network Centralities Oskar Skibski University of Warsaw 05.05.2020

Centrality Measures Centrality Measure 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) . In other words: what is the importance of a node in a graph? 𝑤 Notation: for a fixed set of vertices 𝑊 : • 𝐻 ! is the set of all graphs, and • 𝐷 ! is the set of all centralities. 𝑣 𝑥 𝑢

Centrality Measures 3 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Centrality Measures Standard Centralities • Degree Centrality: 𝐸 ' 𝐻 = 𝑣 ∈ 𝑊 ∶ 𝑤, 𝑣 ∈ 𝐹 ; • Closeness Centrality: 1 𝐷 ' 𝐻 = ∑ (∈!∖ ' 𝑒𝑗𝑡𝑢 𝑤, 𝑣 ; • Betweenness Centrality: 𝑞 ∈ Π 𝑡, 𝑢 ∶ 𝑤 ∈ 𝑞 𝐶 ' 𝐻 = : , Π 𝑡, 𝑢 +,-∈!∖ ' where Π 𝑡, 𝑢 = set of shortest paths between 𝑡 and 𝑢 ; and many more: Eigenvector, Katz, PageRank… 4 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Game-Theoretic Centralities Game-Theoretic Network 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. and set of all GTCs: 𝐻𝑈𝐷 ! . We will denote 𝜒 ∘ 𝑠 as 𝑠, 𝜒 Centrality Graph Coalitional Game 𝑠 𝜒 6 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Game-Theoretic Centralities / : Examples of representation functions 𝑠 𝐻 = 𝑔 . / 𝑇 = 1 if 𝐻[𝑇] is connected, 𝑔 / 𝑇 = 0 , otherwise • 𝑔 . . [Amer & Gimenez 2001] / 𝑇 = 𝐹 𝑇 / ∑ 0∈1[3] 𝜕(𝑓) if 𝐻[𝑇] is connected, 𝑔 / 𝑇 = • 𝑔 . . 0 , otherwise [Lindelauf et al. 2013] / 𝑇 = 2( 𝑇 − 𝐿 𝐻 𝑇 • 𝑔 ) [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 ! , 𝐽 5 = 𝑠, 𝜒 ∈ 𝐽 ∶ 𝜒 𝑗𝑡 𝑏 𝑞𝑝𝑡𝑗𝑢𝑗𝑤𝑓 𝑡𝑓𝑛𝑗𝑤𝑏𝑚𝑣𝑓 ). 7 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Game-Theoretic Centralities Semivalues A value is a semivalue if it is of the form: 𝜒 & 𝑂, 𝑤 = : 𝛾( 𝑇 ) 𝑤 𝑇 ∪ 𝑗 − 𝑤 𝑇 , 3⊆7∖{&} =>? 𝛾(𝑙) =>? for 𝛾: 0, … , 𝑜 − 1 → [0,1] such that ∑ :;< = 1 . : We will call a semivalue positive if 𝛾 𝑙 > 0 for every 𝑙 ∈ {0, … , 𝑜 − 1} . 8 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Game-Theoretic Centralities ? How to characterize GTCs? For 𝐽 ⊆ 𝐻𝑈𝐷 ! , we define 𝐽 = 𝑠, 𝜒 ∶ 𝑠, 𝜒 ∈ 𝐽 . 𝐷 ! 𝐻𝑈𝐷 ! (𝑠 1 , 𝜒 1 ) (𝑠 2 , 𝜒 2 ) 𝑑 1 [.] 𝐽 ! [𝐽 ! ] (𝑠 3 , 𝜒 3 ) 𝑑 2 (𝑠 4 , 𝜒 4 ) 𝑑 3 (𝑠 5 , 𝜒) 9 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

General result Characterization of GTCs [Skibski et al. 2018] We have 𝐻𝑈𝐷 ! = 𝐷 ! , i.e., for every centrality 𝐺 there exists (𝑠, 𝜒) such that 𝑠, 𝜒 = 𝐺 . Proof: On the blackboard. 10 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

General result " / ) 𝑠 𝐻 = 𝑔 𝜒(𝑔 𝐻 ! . ' (∅) $ ({𝑤, 𝑣}) $ ({𝑤}) $ ({𝑣}) 𝑔 𝑔 𝑔 𝑔 & ! # " # " # " 𝑤 𝑤 ' ) 𝜒 𝑤 (𝑔 & ! 𝑠 𝜒 𝑣 𝑣 ' ) 𝜒 𝑣 (𝑔 𝐻 1 & ! ' ({𝑤, 𝑣}) ' (∅) $ ({𝑤}) $ ({𝑣}) 𝑔 𝑔 𝑔 𝑔 # ! # ! & " & " 𝑤 𝑤 ' ) 𝜒 ( (𝑔 & " 𝑠 𝜒 𝑣 𝑣 ' ) 𝜒 𝑣 (𝑔 𝐻 2 & " 11 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

General result Sketch of the proof: Take 𝐺 ∈ 𝐷 ! and define 𝑠 𝐻 = 𝑔 / as follows: . / 𝑇 = : 𝑔 𝐺 & 𝐻 . . &∈3 / is additive (non-essential) where every player is a Game 𝑔 . 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 Consider the following restrictions on the representation function. General Separable Induced 𝑤 𝑣 𝑤 𝑣 𝑤 𝑣 c c c c c c 13 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Restrictions on GTCs … 𝐻 1 𝐻 2 𝐻 3 𝐻 4 𝑤 𝑤 𝑤 𝑤 General … Separable 𝑤 𝑤 𝑤 𝑤 … 𝑤 𝑤 𝑤 𝑤 Induced … 14 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Restrictions on GTCs … 𝐻 6 𝐻 7 𝐻 8 𝐻 5 𝑤 𝑤 𝑤 𝑤 General … Separable 𝑤 𝑤 𝑤 𝑤 … 𝑤 𝑤 𝑤 𝑤 Induced … 15 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Separable GTCs Representation function 𝑠 is separable if for every coalition 𝑇 and every two graphs 𝐻, 𝐻 E : 𝐻 𝑇 = 𝐻 E 𝑇 ∧ 𝐻 𝑊 ∖ 𝑇 = 𝐻 E 𝑊 ∖ 𝑇 𝑇 = 𝑠 𝐻 E → 𝑠 𝐻 𝑇 . All GTCs with separable 𝑠 : 𝑇𝐻𝑈𝐷 ! ! ? ? What is 𝑇𝐻𝑈𝐷 5 16 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

Separable GTCs Characterization of Separable GTCs [Skibski et al. 2018] We have: ! = 𝐺 ∈ 𝐷 ! ∶ 𝐺 𝑡𝑏𝑢𝑗𝑡𝑔𝑗𝑓𝑡 𝐺𝑏𝑗𝑠𝑜𝑓𝑡𝑡 . 𝑇𝐻𝑈𝐷 5 In other words: • ⊆: every separable game-theoretic centrality satisfies Fairness • ⊇ : every centrality that satisfies Fairness can be obtained as a separable game-theoretic centrality Proof: On the blackboard. 17 Oskar Skibski (UW) Algorithmic Coalitional Game Theory

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 𝑊, 𝐹 = a1 if 𝑤 = 𝑣 and 𝑁 = 𝐹, 𝑓 ' 0 otherwise. 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: {(},F 𝑊, 𝐹 = a1 if 𝑤 = 𝑣 and 𝑁 ⊆ 𝐹, 𝑑 ' 0 otherwise. 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: G,F 𝑊, 𝐹 = a1 if 𝑤 ∈ 𝑉 and 𝑁 ⊆ 𝐹, 𝑑 ' 0 otherwise. 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, 𝐺 E satisfy Fairness, then 𝐺 + 𝐺 + and 𝑑 ⋅ 𝐺 also satisfy Fairness. 21 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: 𝐶 IJ&/ = 𝑑 G,F ∶ 𝑊, 𝑁 ∈ 𝐻 ! , 𝑉 ∈ 𝐿 𝑊, 𝑁 . We need to prove that: • elements from 𝐶 IJ&/ satisfy Fairness • elements from 𝐶 IJ&/ are linearly independent • 𝐶 IJ&/ equals dimension of vector space 23 Oskar Skibski (UW) Algorithmic Coalitional Game Theory