Polar Opinion Dynamics in Social...

Polar Opinion Dynamics in Social Networks

Victor Amelkin 〈[email protected]〉UC Santa Barbara

Co-authors: Francesco Bullo, Ambuj K. Singh

Special thanks to: Noah Friedkin

32nd Southern California Control Workshop

April 28, 2017

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mailto:[email protected]

Table of Contents

I Introduction

I Models for polar opinion dynamics

I Asymptotic behavior of the models

I Brief introduction into non-smooth convergence analysis

I Convergence analysis of the models for polar opinion dynamics

I Summary

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Motivation

• Social network consisting of agents

• Each agent holds an opinion

• Agents’ opinions change due to the agents’ interaction

• Opinions are polar (iOS vs. Android, Republicans vs. Democrats)

• Goal: model opinion formation and learn from the model

Figure: Zachary’s Karate Club network [Zac77]

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Problem statement

• Directed, strongly connected social network of n agents

• Row-stochastic adjacency matrix W ∈ [0, 1]n×n

• (W )ij = wij – how much agent i relatively “trusts” agent j

• x(t) ∈ [−1, 1]n – agents’ opinions at time t

• Goals:

Design a sociologically plausible model governing evolution of x(t)

x(t) = M(x(t), t)

The model must capture the competing nature of opinions.

Analyze the dynamical behavior of the model to understand

the dependency of x(∞) upon x(0) and W .

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Existing opinion dynamics models

DeGroot [DeG74] x(t + 1) = Wx(t) (x = −Lx, x = ∆x)

Time-varying DeGroot [Mor05] x(t + 1) = W (t)x(t)

Friedkin-Johnsen [FJ99] x(t + 1) = AWx(t) + (I −A)x(0)

A – diagonal matrix of susceptibilities

Bounded Confidence [HK02, Lor07] x(t + 1) = W (x(t))x(t)

wij(x) > 0⇔ |xi − xj | ≤ bound

0 0.05 0.1 0.15t

-1

-0.5

0

0.5

1

x(t

)

DeGroot Model

0 0.1 0.2 0.3t

-1

-0.5

0

0.5

1

x(t

)Friedkin-Johnsen Model

2 4 6 8 10 12 14 16 18t

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

x(t

)

Hegselmann-Krause BC Model (bound = 0.25)

Other models: SI/SIS/SIR-like, Independent Cascade, Linear Threshold, Voter,

Bayesian, . . .

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Table of Contents

I Introduction





I Summary

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General model for polar opinion dynamics

• Key idea – agents’ opinion formation behavior must changebased on the agents’ current beliefs

x(t+ 1) = A(x(t))Wx(t) + (I −A(x(t)))x(t)

x(t+ 1) = A Wx(t) + (I −A )x(0) (FJ)

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General model for polar opinion dynamics

• Key idea – agents’ opinion formation behavior must changebased on the agents’ current beliefs

x(t+ 1) = A(x(t))Wx(t) + (I −A(x(t)))x(t)

x(t+ 1) = A Wx(t) + (I −A )x(0) (FJ)

• General model for polar opinion dynamics [ABS17]:

x = −A(x)Lx

where

x = x(t) ∈ [−1, 1]n,

A(x) : [−1, 1]n → diag([0, 1]n),

L = (D out −W ) – out-degree Laplacian of the network.

[ABS17] Amelkin V., Bullo F., Singh A.K. “Polar Opinion Dynamics in Social

Networks”, IEEE Transactions on Automatic Control (2017)

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Three instances of the general model

• Model with stubborn positives:

A(x) = 1/2 (I − diag(x)), x = −1/2 (I − diag(x))Lx

• Model with stubborn neutrals:

A(x) = diag(x)2, x = −diag(x)2Lx

• Model with stubborn extremists:

A(x) = (I − diag(x)2), x = −(I − diag(x)2)Lx

-1 -0.5 0 0.5 1xi(t)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Aii(x

(t))

Susceptibility dependence on agent stateA(x) = 1=2(I ! diag(x))A(x) = diag(x)2

A(x) = (I ! diag(x)2)

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Table of Contents

I Introduction





I Summary

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Convergence: model with stubborn positives x = −1/2(I − diag(x))Lx

• If x(0) < 1, then limt→∞

x(t) = α1 for some α ∈ [0, 1).

• If ∃i ∈ 1, . . . , n : xi = 1, then limt→∞

x(t) = 1.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1x1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x2

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Convergence: model with stubborn neutrals x = −diag(x)2Lx

• If x < 0 (negative semi-open orthant) or

x > 0 (positive semi-open orthant),

then limt→∞

x(t) = α1 for some

α ∈ [−1, 0) or

α ∈ (0, 1], respectively.

• Otherwise, limt→∞

x(t) = 0.

-0.9

-0.5

-0.10 0.1

0.5

0.9

x1

-0.9

-0.5

-0.10

0.1

0.5

0.9

x2

x3

x1

x2

-0.9

0.9

0

0

0.9

-0.90

-0.9

0.9

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Convergence: model with stubborn extremists x = −(I − diag(x)2)Lx

• If |x(0)| < 1, then x(∞) = α1 for some α ∈ (−1, 1).

• If all closed agents in x(0) have the same state, −1 or 1,then x(∞) is either −1 or 1, respectively.

• If there are some closed agents having different opinions x1(0), and theremay be some open agents x2(0), then

x(∞) = PT[x1(0)T, (I −W 22)−1W 21x1(0)T]T

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Table of Contents

I Introduction





I Summary

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General idea behind convergence analysis

Lyapunov Direct Method [Lya92]

Find V (x) : D → R ∈ C1(D) with a unique minimum at x ∗, whose directionalderivative Lf (V (x)) = 〈∇V (x), f(x)〉 along the trajectories of the systemx = f(x) is non-positive and zero at x ∗. Then, x ∗ is asymptotically stable.

LaSalle Invariance Principle [LL61]

Given a compact S ⊆ D forward-invariant w.r.t. system x = f(x), findV (x) : D → R ∈ C1(D) with Lf (V (x)) ≤ 0. Then, trajectories starting in Sconverge to the largest invariant subset of the 0-level set of Lf (V (x)).

Quadratic Lyapunov function

(The image is due to Matt Kawski @ ASU)

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When smooth analysis does not work

• Quadratic Lyapunov functions generally1 do not work well for directednetworks.

Formally, for a negative Laplacian flow x = −Lx = f(x), ifV (x) = ‖x− x ∗‖22, then LfV (x) ≤ 0⇔ G(L) is weight-balanced.

Informally, while such V (x) are often non-negative and 0 only at x ∗, theirLie derivative occasionally happens to be positive.

0 0.05 0.1 0.15 0.2 0.25 0.3t

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x(t

)

V (x) = kx ! x$k22

V (x)_V (x)

• Thus, it may worth time to look for a non-smooth Lyapunov function,even for a system x = f(x) with a continuous vector field f .

1Quadratic Lyapunov functions may work for directed networks; for example, paper [KBG14]shows and ad hoc design of one such function for a specific non-linear system – individual-levelnetwork SIS.

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Tools from non-smooth stability analysis – Generalized Gradient

Let V : Rd → R be locally Lipschitz2, and ΩV ⊂ Rd be the set of points whereV fails to be differentiable.

Generalized Gradient [Cla90]:

∂V (x) = co

limi→∞

∇V (xi) | xi → x,xi /∈ N0 ∪ ΩV

,

co – convex hull,

N0 – any set of Lebesgue measure zero.

In English:

• compute ∇V (x) around each point where V is non-differentiable,

• and take the convex hull of all discovered ∇V (“around x”) as ∂V (x).

2∀x ∈ Rd∃R,C(x) ∈ R+∀y ∈ B(x,R) : |V (x)− V (y)| ≤ C(x)‖x− y‖16 / 29

Tools from non-smooth stability analysis – Set-valued Lie Derivative

Set-valued Lie Derivative [Cor08, BC99]: For a locally Lipschitz V : Rd → Rand system x = f(x), the set-valued Lie derivative LfV (x) of V w.r.t.system 〈Rd, f〉 is defined as

LfV (x) = a ∈ R | ∀ξ ∈ ∂V (x) : 〈ξ, f(x)〉 = a

LfV (x) = 〈ξ, f(x)〉, ξ=∇V (x) – compare

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Tools from non-smooth stability analysis – Invariance Principle

Generalized Invariance Principle ∼ [Cor08, BC99]If the following holds

(i) V : Rd → R is locally Lipschitz and regular, (was V ∈ C1(D))

(ii) S ⊂ Rd is compact and invariant for system x = f(x), and

(iii) max LfV (x) ≤ 0 for each x ∈ S, (was LfV (x) ≤ 0)

then all solutions x : [0,∞)→ Rd starting in S converge to the largest invariantsubset M of

S ∩ clx ∈ Rd | 0 ∈ LfV (x) (was S ∩ x ∈ Rd | LfV (x) = 0)

If M is finite, then the limit of each solution x(0) ∈ S exists and is an element of M .

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Table of Contents

I Introduction





I Summary

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Convergence analysis of the model with stubborn positives

• Model

x = −A(x)Lx, – general model,

A(x) = (I − diag(x))/2,

x = −1/2(I − diag(x))Lx – model with stubborn positives.

• State space partition:

[−1, 1]n = limε→+0

S0(ε) ∪ S1,

S0(ε) = [−1, 1− ε]n, N0(ε) = α1 | α ∈ [−1, 1− ε],

S1 = PTx | x ∈ ∪nk=11k × [−1, 1)n−k], N1 = 1.

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Convergence analysis of the model with stubborn positives: S0(ε) N0(ε)

• Convergence from S0(ε) = [−1, 1− ε]n to N0(ε) = α1 | α ∈ [−1, 1− ε].

• Computing ∂V (x):

∂Vmax−min(x) = PT[αT,−βT,0T]T

• Computing LfVmax−min(x):

• Invariance of compact S0(ε): comes at no cost from max LfVmax(x) ≤ 0.

• From Invariance Principle, all solutions starting in S0(ε) convergeasymptotically to N0(ε).

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• Lyapunov function candidate [Mor04, Hen08, Bul16]

Vmax−min(x) = max (x)−min (x).

0 0.05 0.1 0.15 0.2 0.25t-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

x(t

)

Lyapunov function's behavior along a trajectory

V (x)_V (x)






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∂(c1g1(x) + c2g2(x)) = c1∂g1(x) + c2∂g2(x) (∗)

∂max g1(x), . . . , gk(x) = co∪∂gi | gi attains max at x (∗∗)

gi(x) = xi, i ∈ 1, . . . , n – locally Lipschitz, regular

∂Vmax(x) = ∂(max (x)) = PT[αT,0T]T, α – “convex”

∂V−min(x) = ∂(−min (x)) = PT[−βT, 0T]T, β – “convex”


• Computing LfVmax−min(x):• Invariance of compact S0(ε): comes at no cost from max LfVmax(x) ≤ 0.


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• Computing LfVmax−min(x): w.l.o.g., assume P = I• Invariance of compact S0(ε): comes at no cost from max LfVmax(x) ≤ 0.


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Vmax−min(x) = max (x)−min (x).• Computing ∂V (x):



LfVmax−min(x) = a ∈ R | ∀ξ ∈ ∂V (x) : 〈ξ, f(x)〉 = a

ξTf(x) = −

α−β

0

Amax(x) 0 00 Amin(x) 00 0 Amid(x)

××

(I −W 11) −W 12 −W 13−W 21 (I −W 22) −W 23−W 31 −W 32 (I −W 33)

xmaxxminxmid

=

= −(α

TAmax(x)(xmax − [W 11W 12W 13]x)+

βTAmin(x)([W 21W 22W 23]x− xmin)

)≤ 0.



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max LfVmax−min(x) ≤ 0,

N0(ε) – 0-level set of LfVmax−min(x).



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Convergence analysis for other models

• For other part(s) of the state space and other models, use sametechniques.

• Lyapunov functions based on a combination of Vmax, V−min usually work.

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Table of Contents

I Introduction





I Summary

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Summary

• Polar opinion dynamics can be modeled with non-linear dynamical systemwhose behavior can be formally analyzed.

• Generally, non-smooth Lyapunov functions can be applied to the stabilityanalysis of continuous non-linear systems over directed networks.

• Our convergence proofs did not rely on a particular shape of A(x) (onlywhen its diagonal elements are zero / non-zero) and, thus, can be carriedover to other models of similar form.

• Similar techniques can be used to design Lyapunov functions out of convexcomponents.

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References I

I Victor Amelkin, Francesco Bullo, and Ambuj K. Singh, Polar opiniondynamics in social networks, IEEE Transactions on Automatic Control(2017).

I Andrea Bacciotti and Francesca Ceragioli, Stability and stabilization ofdiscontinuous systems and nonsmooth Lyapunov functions, ESAIM: Control,Optimisation & Calculus of Variations 4 (1999), 361–376.

I Francesco Bullo, Lectures on network systems, March 2016.

I Frank H Clarke, Optimization and Nonsmooth Analysis, Classics in AppliedMathematics, vol. 5, SIAM, 1990.

I Jorge Cortes, Discontinuous dynamical systems – A tutorial on solutions,nonsmooth analysis, and stability, IEEE Control Systems Magazine 28(2008), no. 3, 36–73.

I Morris H DeGroot, Reaching a consensus, Journal of the AmericanStatistical Association 69 (1974), no. 345, 118–121.

I Noah E Friedkin and Eugene C Johnsen, Social influence networks andopinion change, Advances in Group Processes 16 (1999), no. 1, 1–29.

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References II

I Julien M Hendrickx, Graphs and networks for the analysis of autonomousagent systems, Ph.D. thesis, Universite Catholique de Louvain, Belgium,February 2008.

I Rainer Hegselmann and Ulrich Krause, Opinion dynamics and boundedconfidence models, analysis, and simulation, Journal of Artificial Societiesand Social Simulation 5 (2002), no. 3.

I Ali Khanafer, Tamer Basar, and Bahman Gharesifard, Stability of epidemicmodels over directed graphs: A positive systems approach, arXiv preprintarXiv:1407.6076 (2014).

I Joseph P LaSalle and Solomon Lefschetz, Stability by liapunov’s directmethod: with applications, vol. 4, Academic Press New York, 1961.

I Jan Lorenz, Continuous opinion dynamics under bounded confidence: Asurvey, International Journal of Modern Physics C 18 (2007), no. 12,1819–1838.

I Aleksandr Lyapunov, General problem of the stability of motion, 1892.

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References III

I Luc Moreau, Stability of continuous-time distributed consensus algorithms,IEEE Conf. on Decision and Control (Nassau, Bahamas), vol. 4, 2004,pp. 3998–4003.

I , Stability of multiagent systems with time-dependent communicationlinks, IEEE Transactions on Automatic Control 50 (2005), no. 2, 169–182.

I Wayne Zachary, An information flow model for conflict and fission in smallgroups, Journal of Anthropological Research 33 (1977), no. 4, 452–473.

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Questions

∼ Thanks ∼

Questions?

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[Backup] Convergence: general model x = −A(x)Lx

Let S ⊆ [−1, 1]n be a non-empty compact forward-invariant set, and

N = S ∩ α1 | α ∈ [−1, 1]

be its non-empty subset of consensus states. Further, assume that in S, theagents’ susceptibility functions Aii(x) agree upon their zeros in that

∀x ∈ S ∀i, j ∈ 1, . . . , n : Aii(x) = Ajj(x) = 0→ xi = xj .

Then, all trajectories x(t) of x = −A(x)Lx starting in S converge to N ast→∞.

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