Influence in Social Networks

Transcription

1 CSCI 3210: Computational Game Theory Influence Games Ref: Irfan & Ortiz, AI (2014) Reading: Sections 1 3(up to pg. 86), Sections 4.5, 5 (no proof), 6 bowdoin.edu/~mirfan/papers/irfan_ortiz_influence_games_ai2014.pdf Mohammad T. Irfan Influence in Social Networks 2 1

2 Overview Influence Networks 3 2

3 Why model? Understand how things work a complex system Structure Prediction Interventions Policy making Schelling s residential segregation Models based on level of tolerance (1971) Bayside Jamaica Thomas Schelling Nobel Prize (2005) New York [Eric Fischer] 6 3

4 Modeling influence Threshold Models of Collective Behavior (1978) Mark Granovetter 7 Modeling influence: influence game Players Actions Rule of the game (best response) 8 4

5 Influence game No Paul (R, KY) Johnson (R, WI) No Yes -1 DeMint (R, SC) -1 Yes Schumer (D, NY) Sanders (I, VT) 9 Influence game No Paul (R, KY) Johnson (R, WI) Yes Yes -1 DeMint (R, SC) -1 Yes Schumer (D, NY) Sanders (I, VT) 10 5

6 Linear Influence Game (LIG) Variables Actions of node i, x i {-1, 1} Parameters Influence factor from node j to i: w ji Threhold of i: b i Parameter values LIG Model Values of variables 11 Linear Influence Game (LIG) Influence function (of the parameters and variables) Best response of node i Node i's payoff function 12 6

7 Representation size Graph One number (threshold) for each node One number (influence factor) for each edge Size is linear in the size of the graph Yardstick for time-complexity of algorithms Size of the graph 13 Practical scenarios are stable outcomes Nash Equilibrium Everyone chooses the best response to others We will work in the pure-strategy setting 14 7

8 Is it a Nash equilibrium? No Paul (R, KY) Johnson (R, WI) Yes Yes -1 DeMint (R, SC) -1 Yes Schumer (D, NY) Sanders (I, VT) 15 Meaning of Nash equilibrium Practical scenarios = Stable outcome = Nash equilibrium 16 8

9 Influence game Most influential individuals Inputs Influence Game (player, action, rule of the game) A desirable outcome Definition (Most influential individuals) They can influence everyone to strictly follow the desirable outcome 18 9

10 Example (a) (b) (c) (d) Each node wants to behave like majority of neighbors Desirable outcome: every node choosing black {1, 2, 3} is NOT a most influential set of nodes {3, 4} is a most influential set th Congress Who are the most influential senators? 1. Machine learning [Honorio & Ortiz, 2010] 10

11 110 th Congress Who are the most influential senators? 1. Machine learning [Honorio & Ortiz, 2010] 2. Compute stable outcome Challenge 100 senators Each has two actions Search space:

12 Hardness of computation Existence of (pure-strategy) Nash Eq. (even in bipartite graph) NP-complete Existence of pure-strategy Nash eq. with a given set of players playing 1 Existence of PSNE with at least k players playing 1 Existence of k most influential nodes (all PSNE and desired state given) co-np-complete #P-complete Uniqueness of a PSNE Counting number of PSNE (even in star graph) 23 Algorithms Special case: Trees Fast polynomial-time algorithm for trees O(nΔ) vs. O(n2 Δ ) by TreeNash [Kearns et al., 2001] Δ is the maximum degree General case Effective computational scheme 24 12

13 Computing all Nash equilibria Divide-and-conquer (1, -1, 1,, 1) (-1, 1, 1,, -1) (1, 1, 1,, -1) Merge (-1, 1, 1,, 1) (1, 1, 1,, 1) 25 Computing all Nash equilibria Backtracking search Select the next node Assign actions { 1, 1} Not yet selected É Question: Can (x 1, x 2,, x i+1 ) possibly lead to a PSNE? É No à Prune! É Otherwise à Propagation: Adapt NashProp [Ortiz & Kearns, 2002] to run in polynomial-time 26 13

14 110 th Congress Who are the most influential senators? 1. Machine learning [Honorio & Ortiz, 2010] 2. Compute stable outcome 3. Find most influential nodes Finding the most influential nodes Inapproximability Given all Nash equilibria, this problem is ó Set-cover problem Provable approximation algorithm for finding the most influential nodes 28 14

15 110 th Congress Who are the most influential senators? Kerry (D, MA) Enzi (R, WY) Inouye (D, HI) Bennett (R, UT) Sessions (R, AL) Lautenberg (D, NJ) 112 th Congress Who are the most influential senators? Reid (D, NV) Enzi (R, WY) Sanders (I, VT) Crapo (R, ID) Inouye (D, HI) Johnson (R, WI) Reed (D, RI) DeMint (R, SC) Hagan (D, NC) Collins (R, ME) 15

16 Gang-of-six senators (2011) How influential were they really? Chambliss (R, GA) Coburn (R, OK) Crapo (R, ID) Conrad (D, ND) Durbin (D, IL) Warner (D, VA) In 90% of the stable outcomes, not powerful enough How to make this group more powerful? Add new senators! è Gang-of-eight (2012) 31 Gang-of-eight senators (2012) How influential is this new group? Chambliss (R, GA) Coburn (R, OK) Crapo (R, ID) Conrad (D, ND) Durbin (D, IL) Warner (D, VA) Bennet (D, CO) Johanns (R, NE) Fiscal Cliff (January 1, 2013) Consensus of Senate Majority Leader and Senate Minority Leader 97% of outcomes the majority are influenced (

17 Filibusters Does there exist a small set of senators who can prevent filibusters? Filibusters Who can prevent it? Small coalition of senators that can break filibusters Kerry (D, MA) Roberts (R, KS) Graham (R, SC) 110 th Congress 17

18 Filibusters Who can force it? Coalition of senators that can block cloture by voting no Kerry (D, MA) Nelson (D, FL) McConnell (R, KY) 110 th Congress Supreme court ( ) Most Influential 18

19 Random LIG 19