Influence in Social Networks 2 1 4/23/18 Overview Influence - PDF document

4/23/18 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

4/23/18 Overview • Influence • Networks 3 2

4/23/18 Why model? — Understand how things work a complex system — Structure — Prediction — Interventions — Policy making Schelling’s residential segregation • Models based on “level Bayside of tolerance” (1971) Jamaica Thomas Schelling New York [Eric Fischer] Nobel Prize (2005) 6 3

4/23/18 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

4/23/18 Influence game 6 4 2 0 -2 No No -4 -6 +1 +2 Paul (R, KY) Johnson (R, WI) -1 -1 DeMint (R, SC) Yes Yes Schumer (D, NY) Sanders (I, VT) 9 Influence game 6 4 2 0 -2 No Yes -4 -6 +1 +2 Paul (R, KY) Johnson (R, WI) -1 -1 DeMint (R, SC) Yes Yes Schumer (D, NY) Sanders (I, VT) 10 5

4/23/18 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 LIG Values of Parameter variables values Model 11 Linear Influence Game (LIG) • Influence function (of the parameters and variables) • Best response of node i • Node i 's payoff function 12 6

4/23/18 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

4/23/18 Is it a Nash equilibrium? 6 4 2 0 -2 No Yes -4 -6 +1 +2 Paul (R, KY) Johnson (R, WI) -1 -1 DeMint (R, SC) Yes Yes Schumer (D, NY) Sanders (I, VT) 15 Meaning of Nash equilibrium Practical scenarios = Stable outcome = Nash equilibrium 16 8

4/23/18 Influence game 6 4 2 0 -2 -4 -6 +5 +2 +2 +3 17 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

4/23/18 Example 1 1 1 1 2 3 4 5 6 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 (d) (a) (b) (c) • 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 19 110 th Congress 2007-09 Who are the most influential senators? 1. Machine learning [Honorio & Ortiz, 2010] 10

4/23/18 110 th Congress 2007-09 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: 2 100 22 11

4/23/18 Hardness of computation Existence of (pure-strategy) Nash Eq. (even in bipartite graph) Existence of pure-strategy Nash eq. with a given set of players playing 1 NP-complete 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 Uniqueness of a PSNE Counting number of PSNE #P-complete (even in star graph) 23 Algorithms • Special case: Trees • Fast polynomial-time algorithm for trees • O ( n Δ ) vs. O ( n 2 Δ ) by TreeNash [Kearns et al. , 2001] • Δ is the maximum degree • General case • Effective computational scheme 24 12

4/23/18 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 1 -1 Not yet • Assign actions { − 1, 1} -1 1 1 selected -1 1 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

4/23/18 110 th Congress 2007-09 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 Given all Nash equilibria, this problem is Inapproximability ó Set-cover problem • Provable approximation algorithm for finding the most influential nodes 28 14

4/23/18 110 th Congress 2007-09 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 2011-13 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

4/23/18 Gang-of-six senators (2011) — How influential were they really? Chambliss Coburn Crapo Conrad Durbin Warner (R, GA) (R, OK) (R, ID) (D, ND) (D, IL) (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 Coburn Crapo Conrad Durbin Warner (R, GA) (R, OK) (R, ID) (D, ND) (D, IL) (D, VA) Bennet Johanns (D, CO) (R, NE) • Fiscal Cliff (January 1, 2013) • Consensus of Senate Majority Leader and Senate Minority Leader • 97% of outcomes – the majority are influenced (http://mtirfan.blogspot.com) 32 16

4/23/18 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

4/23/18 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 (1994–2004) Most Influential 18

4/23/18 Random LIG 19