Opinion Dynamics over Signed Social Networks Guodong Shi Research - PowerPoint PPT Presentation

Opinion Dynamics over Signed Social Networks Guodong Shi Research School of Engineering The Australian Na@onal University, Canberra, Australia Ins@tute for Systems Research The University of Maryland, College Park May 2, 2016 1

Joint work with Alexandre Prou-ere, Mikael Johansson, Karl Henrik Johansson KTH Royal Ins@tute of Technology, Sweden John S. Baras , University of Maryland, US Claudio Altafini Linkoping University, Sweden 2

Social Networks 3

Opinions iPhone, Blackberry, or Samsung? Republican or Democrat? Sell or buy AAPL? The rate of economic growth this year? Social cost of carbon? French 1956; Benerjee 1992; Galam 1996 4

Dynamics of Opinions ⇣ ⌘ x i ( k ); x j ( k ) , j ∈ N i x i ( k + 1) = f i 5

De Groot Social Interac@ons x ( k + 1) = Px ( k ) • is a stochas@c matrix P DeGroot 1974 6

De Groot Social Interac@ons Before: x i ( k + 1) = x i ( k ) + α ( x j ( k ) − x i ( k )) After: 0 < α < 1 DeGroot 1974 7

De Groot Social Interac@ons x ( k + 1) = Px ( k ) An agreement is achieved if is ergodic in the sense that P k !1 x i ( k ) = v > x (0) lim Trust and coopera@on lead to social consensus! 8

Wisdom of Crowds (with Trust) Golub and Jackson 2010 9

Disagreement Models — Memory of ini@al values Friedkin and Johnsen (1999) — Bounded confidence Krause (1997); Hegselmann-Krause (2002); Blondel et al. (2011); Li et al. (2013) — Stubborn agents Acemoglu et al. (2013) — Homophily Dandekar et al. (2013) 10

This Talk A model and theory for opinion dynamics over social networks with friendly and adversarial interpersonal rela-ons coexis-ng. 11

Friends and Adversaries _ + + + + + + + _ _ + + + _ + _ + _ + + 12

Structural Balance Theory • S trongly balanced if the node set can be divided into two disjoint subsets such that nega@ve links can only exist between them; • Weakly balanced if such a par@@on contains maybe more than two subsets. Heider (1947), Harary (1953), Cartwright and Harary (1956), Davis (1963) 13

Structural Balance _ _ _ + _ _ + + + + _ _ _ + 14

When do nodes interact? 15

Underlying World G = G + ∪ G − _ • Fixed + • Undirected + • Determinis@c + + + + + • Connected _ _ + + + N i = N + _ i ∪ N − + _ + _ i + + 16

Gossip Model Neighbors 17

Gossip Model Independent with other time and node states, at time k , (i) A node i is drawn with probability 1 /N ; (ii) Node i selects one of its neighbor j with probability 1 / |N i | . 18

How a pair of nodes interacts with each other when they meet? 19

Posi@ve and Nega@ve Interac@ons A pair (i,j) is randomly selected. The two selected nodes update. + � � x i ( k + 1) = x i ( k ) + α x j ( k ) − x i ( k ) 0 < α < 1 _ � � x i ( k + 1) = x i ( k ) + β x i ( k ) − x j ( k ) β > 0 Altafini 2013 20

Posi@ve and Nega@ve Interac@ons Before: x i ( k + 1) = x i ( k ) + α ( x j ( k ) − x i ( k )) After: � � x i ( k + 1) = x i ( k ) − β x j ( k ) − x i ( k ) After: 21

Mean/Mean-Square Evolu@on 22

Rela@ve-State-Flipping Model x ( k + 1) = W ( k ) x ( k ) • It’s an eigenvalue perturba@on problem! 23

Phase Transi@on Theorem. Suppose G + is connected and G − is non-empty. Then there exists β ∗ such that (i) lim k →∞ E { x i ( k ) } = P N i =1 x i (0) /N for all i = 1 , . . . , N if β < β ∗ ; (ii) lim k →∞ max i,j k E { x i ( k ) } � E { x j ( k ) } k = 1 if β > β ∗ . • It’s possible to prove that the expecta@on of the state transi@on matrix is eventually posi.ve . 24

Phase Transi@on Let G − be the Erdos-Renyi random graph with link Let G be the complete graph. appearance probability p . α (i) If p < α + β , then ⇣ ⌘ Consensus in expectation → 1 P as the number of nodes N tends to infinity; α (ii) If p > α + β , then ⇣ ⌘ Divergence in expectation → 1 P as the number of nodes N tends to infinity. 25

Sample Path Behavior 26

Live-or-Die Lemma Introduce C x 0 . D x 0 . � � = lim sup max i,j | x i ( k ) − x j ( k ) | = 0 = lim sup max i,j | x i ( k ) − x j ( k ) | = ∞ , k →∞ k →∞ x 0 . x 0 . � � C ∗ = lim inf k →∞ max i,j | x i ( k ) − x j ( k ) | = 0 D ∗ = lim inf k →∞ max i,j | x i ( k ) − x j ( k ) | = ∞ , , Lemma. Suppose G + is connected. Then (i) P ( C x 0 ) + P ( D x 0 ) = 1; (ii) P ( C ∗ x 0 ) + P ( D ∗ x 0 ) = 1. As a consequence, almost surely, one of the following events happens: � k →∞ max lim i,j | x i ( k ) − x j ( k ) | = 0 ; � k →∞ max lim i,j | x i ( k ) − x j ( k ) | = ∞ ; � lim inf k →∞ max i,j | x i ( k ) − x j ( k ) | = 0; lim sup max i,j | x i ( k ) − x j ( k ) | = ∞ . k →∞ 27

Zero-One Law C . i,j | x i ( k ) − x j ( k ) | = 0 for all x 0 ∈ R n � = lim sup max , k →∞ D . ∃ (deterministic) x 0 ∈ R n , � = s.t. lim sup max i,j | x i ( k ) − x j ( k ) | = ∞ k →∞ Theorem. Both C and D are trivial events (i.e., each of them occurs with probability equal to either 1 or 0) and P ( C ) + P ( D ) = 1. 28

No-Survivor Theorem Theorem. There always holds � ⇣ ⌘ � = 1 � = 1 � � � � lim inf � x i ( k ) � x j ( k ) � lim inf k →∞ max � x i ( k ) � x j ( k ) = 1 P � k →∞ i,j for all i 6 = j . 29

Phase Transi@on Theorem. (i) Suppose G + is connected. Then there is a β ∗ > 0 such that N ⇣ ⌘ X k →∞ x i ( k ) = lim x i (0) /N = 1 P i =1 for all i if β < β ∗ . (ii) There is β ∗ > 0 such that ⇣ ⌘ � = ∞ � � lim inf k →∞ max � x i ( k ) − x j ( k ) = 1 P i,j for all i if β > β ∗ . 30

Bounded State Model 31

Bounded States • Let A > 0 be a constant and define P A ( · ) by P A ( z ) = − A, z < − A , P A ( z ) = z, z ∈ [ − A, A ], and P A ( z ) = A, z > A . • Define the function θ : E → R so that θ ( { i, j } ) = α if { i, j } ∈ E + and θ ( { i, j } ) = − β if { i, j } ∈ E − . Consider the following node interaction under relative-state flipping rule: � � x s ( t + 1) = P A (1 − θ ) x s ( t ) + θ x − s ( t ) , s ∈ { i, j } . 32

Clustering of Opinions Theorem. Let α ∈ (0 , 1 / 2). Assume that G is a weakly structurally balanced complete graph under the partition V = V 1 ∪ V 2 · · · ∪ V m with m ≥ 2. Let α ∈ (0 , 1 / 2). When β is su ffi ciently large, almost sure boundary clustering is achieved in the sense that for almost all initial value x (0) w.r.t. Lebesgue measure, there are there are m random variables, l 1 ( x (0)) , . . . , l m ( x (0)), each of which taking values in { − A, A } , such that: ⇣ ⌘ t →∞ x i ( t ) = l j ( x (0)) , i ∈ V j , j = 1 , . . . , m lim = 1 . P 33

Separa@on Events • The power of minority groups. 34

Numerical Example 35

Oscilla@on of Opinions Theorem. Let α ∈ (0 , 1 / 2). Assume that G is a complete graph and the positive graph G + is connected. When β is su ffi ciently large, for almost all initial value x (0) w.r.t. Lebesgue measure, there holds for all i ∈ V that ⇣ ⌘ lim inf t →∞ x i ( t ) = − A, lim sup x i ( t ) = A = 1 . P t →∞ 36

Numerical Example 37

Related Publica@ons Shi et al. 2013 IEEE Journal on Selected Areas in Communica.ons; Shi et al. 2015, 2016 IEEE Transac.ons on Control of Network Systems; Shi et al. 2016 Opera.ons Research. 38

Thank you! 39