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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

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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

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Joint work with

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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

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Social Networks

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Opinions

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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

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Dynamics of Opinions

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xi(k + 1) = fi ⇣ xi(k); xj(k), j ∈ Ni ⌘

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  • is a stochas@c matrix

De Groot Social Interac@ons

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x(k + 1) = Px(k)

DeGroot 1974

P

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DeGroot 1974

After: Before:

xi(k + 1) = xi(k) + α(xj(k) − xi(k))

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De Groot Social Interac@ons

0 < α < 1

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De Groot Social Interac@ons

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x(k + 1) = Px(k)

An agreement is achieved if is ergodic in the sense that

P

lim

k!1 xi(k) = v>x(0)

Trust and coopera@on lead to social consensus!

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Wisdom of Crowds (with Trust)

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Golub and Jackson 2010

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Disagreement Models

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— 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)

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

This Talk

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Friends and Adversaries

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+ + + + + + + + + + + + + +

_ _ _ _ _ _

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  • Strongly 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)

Structural Balance Theory

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Structural Balance

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+ + + +

_ _ _

+ +

_ _ _ _ _

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When do nodes interact?

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Underlying World

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+ + + + + + + + + + + + + +

_ _ _ _ _ _

  • Fixed
  • Undirected
  • Determinis@c
  • Connected

Ni = N +

i ∪ N − i

G = G+ ∪ G−

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Gossip Model

Neighbors

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Gossip Model

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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/|Ni|.

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

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Posi@ve and Nega@ve Interac@ons

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_

+

xi(k + 1) = xi(k) + α

  • xj(k) − xi(k)
  • A pair (i,j) is randomly selected.

The two selected nodes update.

xi(k + 1) = xi(k) + β

  • xi(k) − xj(k)
  • 0 < α < 1

β > 0

Altafini 2013

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Posi@ve and Nega@ve Interac@ons

After: Before:

xi(k + 1) = xi(k) + α(xj(k) − xi(k))

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After:

xi(k + 1) = xi(k) − β

  • xj(k) − xi(k)
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Mean/Mean-Square Evolu@on

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Rela@ve-State-Flipping Model

x(k + 1) = W(k)x(k)

  • It’s an eigenvalue perturba@on problem!
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Phase Transi@on

  • It’s possible to prove that the expecta@on of the state

transi@on matrix is eventually posi.ve.

  • Theorem. Suppose G+ is connected and G− is non-empty. Then there exists β∗ such

that (i) limk→∞ E{xi(k)} = PN

i=1 xi(0)/N for all i = 1, . . . , N if β < β∗;

(ii) limk→∞ maxi,j kE{xi(k)} E{xj(k)}k = 1 if β > β∗.

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Let G be the complete graph. Let G− be the Erdos-Renyi random graph with link appearance probability p. (i) If p <

α α+β, then

P ⇣ Consensus in expectation ⌘ → 1 as the number of nodes N tends to infinity; (ii) If p >

α α+β, then

P ⇣ Divergence in expectation ⌘ → 1 as the number of nodes N tends to infinity.

Phase Transi@on

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Sample Path Behavior

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Live-or-Die Lemma

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Lemma.

Suppose G+ is connected. Then (i) P(Cx0) + P(Dx0) = 1; (ii) P(C ∗

x0) + P(D∗ x0) = 1.

As a consequence, almost surely, one of the following events happens:

  • lim

k→∞ max i,j |xi(k) − xj(k)| = 0

;

  • lim

k→∞ max i,j |xi(k) − xj(k)| = ∞

;

  • lim inf

k→∞ max i,j |xi(k) − xj(k)| = 0; lim sup k→∞

max

i,j |xi(k) − xj(k)| = ∞

. Introduce Cx0 . =

  • lim sup

k→∞

max

i,j |xi(k) − xj(k)| = 0

, Dx0 . =

  • lim sup

k→∞

max

i,j |xi(k) − xj(k)| = ∞

C ∗

x0 .

=

  • lim inf

k→∞ max i,j |xi(k) − xj(k)| = 0

, D∗

x0 .

=

  • lim inf

k→∞ max i,j |xi(k) − xj(k)| = ∞

,

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Zero-One Law

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C . =

  • lim sup

k→∞

max

i,j |xi(k) − xj(k)| = 0 for all x0 ∈ Rn

, D . =

  • ∃ (deterministic) x0 ∈ Rn,

s.t. lim sup

k→∞

max

i,j |xi(k) − xj(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.

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No-Survivor Theorem

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  • Theorem. There always holds

P ⇣ lim inf

k→∞

  • xi(k) xj(k)
  • = 1
  • lim inf

k→∞ max i,j

  • xi(k) xj(k)
  • = 1

⌘ = 1 for all i 6= j.

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Phase Transi@on

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Theorem.

(i) Suppose G+ is connected. Then there is a β∗ > 0 such that P ⇣ lim

k→∞ xi(k) = N

X

i=1

xi(0)/N ⌘ = 1 for all i if β < β∗. (ii) There is β∗ > 0 such that P ⇣ lim inf

k→∞ max i,j

  • xi(k) − xj(k)
  • = ∞

⌘ = 1 for all i if β > β∗.

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Bounded State Model

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Bounded States

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  • Let A > 0 be a constant and define PA(·) by PA(z) = −A, z < −A, PA(z) =

z, z ∈ [−A, A], and PA(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: xs(t + 1) = PA

  • (1 − θ)xs(t) + θx−s(t)
  • , s ∈ {i, j}.
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Clustering of Opinions

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  • Theorem. Let α ∈ (0, 1/2). Assume that G is a weakly structurally balanced complete

graph under the partition V = V1 ∪V2 · · ·∪Vm with m ≥ 2. Let α ∈ (0, 1/2). When β is sufficiently 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, l1(x(0)), . . . , lm(x(0)), each of which taking values in {−A, A}, such that: P ⇣ lim

t→∞ xi(t) = lj(x(0)), i ∈ Vj, j = 1, . . . , m

⌘ = 1.

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Separa@on Events

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  • The power of minority groups.
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Numerical Example

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Oscilla@on of Opinions

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  • Theorem. Let α ∈ (0, 1/2). Assume that G is a complete graph and the positive graph

G+ is connected. When β is sufficiently large, for almost all initial value x(0) w.r.t. Lebesgue measure, there holds for all i ∈ V that P ⇣ lim inf

t→∞ xi(t) = −A, lim sup t→∞

xi(t) = A ⌘ = 1.

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Numerical Example

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Related Publica@ons

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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.

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Thank you!

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