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

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CSCI 3210: Computational Game Theory

Mohammad T. Irfan

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

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Influence in Social Networks

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Overview

• Influence
• Networks

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Why model?

— Understand how things work a complex system

— Structure

— Prediction — Interventions

— Policy making

6

Schelling’s residential segregation

• Models based on “level
• f tolerance” (1971)

Thomas Schelling Nobel Prize (2005) New York [Eric Fischer]

Jamaica Bayside

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Modeling influence

• Threshold Models of Collective

Behavior (1978)

Mark Granovetter

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Modeling influence: “influence game”

• Players
• Actions
• Rule of the game (best response)

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Influence game

DeMint (R, SC) Schumer (D, NY) Paul (R, KY)

Yes No

Johnson (R, WI) Sanders (I, VT)

No Yes

2 4 6

• 6
• 2
• 1
• 1

+2 +1

• 4

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Influence game

DeMint (R, SC) Schumer (D, NY) Paul (R, KY)

Yes No

Johnson (R, WI) Sanders (I, VT)

Yes Yes

2 4 6

• 6
• 2
• 1
• 1

+2 +1

• 4

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Linear Influence Game (LIG)

• Variables
• Actions of node i, xi

{-1, 1}

• Parameters
• Influence factor from node j to i: wji
• Threhold of i: bi

∈

LIG Model

Parameter values Values of variables

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Linear Influence Game (LIG)

• Influence function (of the parameters and variables)
• Best response of node i
• Node i's payoff function

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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

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Practical scenarios are “stable outcomes”

• Nash Equilibrium
• Everyone chooses the best response to others
• We will work in the pure-strategy setting

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DeMint (R, SC) Paul (R, KY)

No

Johnson (R, WI) Sanders (I, VT)

Yes Yes

• 1
• 1

+2 +1

Is it a Nash equilibrium?

Schumer (D, NY)

Yes

2 4 6

• 6
• 2
• 4

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Meaning of Nash equilibrium

Practical scenarios = Stable outcome = Nash equilibrium

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Influence game

+2 +3 +2 +5

2 4 6

• 6
• 4
• 2

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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

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Example

• 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

1 2 3 4 5 6

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

(a) (b) (c) (d) Who are the

most influential senators?

110th Congress 2007-09

• 1. Machine learning

[Honorio & Ortiz, 2010]

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Who are the

most influential senators?

110th Congress 2007-09

• 1. Machine learning

[Honorio & Ortiz, 2010]

• 2. Compute stable
• utcome

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Challenge

• 100 senators
• Each has two actions
• Search space: 2100

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Hardness of computation

NP-complete co-NP-complete

Existence of (pure-strategy) Nash Eq. (even in bipartite graph) 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) Uniqueness of a PSNE

#P-complete

Counting number of PSNE (even in star graph)

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Algorithms

• Special case: Trees
• Fast polynomial-time algorithm for trees
• O(nΔ) vs. O(n2Δ) by TreeNash [Kearns et al., 2001]
• Δ is the maximum degree
• General case
• Effective computational scheme

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Computing all Nash equilibria

• Divide-and-conquer

(1, -1, 1, …, 1) (-1, 1, 1, …, -1) (1, 1, 1, …, -1) (-1, 1, 1, …, 1) (1, 1, 1, …, 1)

Merge

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Computing all Nash equilibria

• Backtracking search
• Select the next node
• Assign actions {−1, 1}

Not yet selected

1

• 1

1

• 1

1

• 1

1

É Question: Can (x1, x2, …, xi+1) possibly lead to a PSNE?

É No à Prune! É Otherwise à Propagation: Adapt NashProp [Ortiz & Kearns,

2002] to run in polynomial-time

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Who are the

most influential senators?

110th Congress 2007-09

• 1. Machine learning

[Honorio & Ortiz, 2010]

• 3. Find most

influential nodes

• 2. Compute stable
• utcome

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Finding the most influential nodes

• Provable approximation algorithm for finding the

most influential nodes

Inapproximability

Given all Nash equilibria, this problem is ó Set-cover problem

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Who are the

most influential senators?

110th Congress 2007-09

Kerry (D, MA) Enzi (R, WY) Inouye (D, HI) Bennett (R, UT) Sessions (R, AL) Lautenberg (D, NJ)

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) 112th Congress 2011-13

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Gang-of-six senators (2011) — How influential were they really?

• In 90% of the stable outcomes, not “powerful

enough”

• How to make this group “more powerful?”
• Add new senators! è Gang-of-eight (2012)

Chambliss (R, GA) Coburn (R, OK) Crapo (R, ID) Conrad (D, ND) Durbin (D, IL) Warner (D, VA)

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Gang-of-eight senators (2012) – How influential is this new group?

• 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)

Chambliss (R, GA) Coburn (R, OK) Crapo (R, ID) Conrad (D, ND) Durbin (D, IL) Warner (D, VA) Bennet (D, CO) Johanns (R, NE)

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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

— 110th Congress

Kerry (D, MA) Roberts (R, KS) Graham (R, SC)

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Filibusters – Who can “force” it?

• Coalition of senators that can block cloture by voting “no”

— 110th Congress

Kerry (D, MA) Nelson (D, FL) McConnell (R, KY)

Supreme court (1994–2004) Most Influential

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Random LIG