Trust, Opinion Diffusion, And Radicalization in Social Networks

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Trust, Opinion Diffusion and Radicalization inSocial Networks

Lin Li , Anna Scaglione , Ananthram Swami and Qing Zhao Department of Electrical and Computer Engineering

University of California, Davis, CA 95616Email: {llli, ascaglione, qzhao }@ucdavis.edu

Army Research Laboratory, Adelphi, MD 20783Email: [email protected]

Abstract Gossiping models have increasingly been applied tostudy social network phenomena. In this context, this paperis specically concerned with modeling how opinions of socialagents can be radicalized if the agents interact more stronglywith neighbors that share their beliefs. In our model, each agentsbelief is represented by a vector of probabilities that a givenstate is true. The agents average their opinions with that of theirneighbors over time, giving more weight to opinions that arecloser to their current beliefs. The increasing trust that mayexist among likeminded agents is modeled through a weight thatis a monotonically decreasing function of the distance in opinion.We consider a continuous (soft) and a discontinuous (hard) modelfor the weight and analyze the convergence properties.

I. INTRODUCTION

The concept of convergent social behavior or belief is awell-documented feature of social phenomenon in a numberof diverse elds, ranging from herding [1] in economics andnance to fad and trend [2] in social psychology and the Bandwagon effect [3] in political science. The techniquesused to elucidate this phenomenon can be generally divided

into two classes, i.e. , Bayesian models and non-Bayesianmodels. Bayesian models [1], [2] describe individuals asrational agents: opinions (or beliefs) of social agents areprobabilities of a given state, conditioned on all the availableinformation; individuals observe and update their beliefs usingBayes rule. In many scenarios, the relevant information isdispersed throughout a network; social agents only observea fraction of the total information, usually in the form of their past experiences. When the information available toan agent is not directly observable to others and agents donot know the structure of the social network, it is highlyimpractical to learn the state of nature in a Bayesian fashion. Incontrast, non-Bayesian models [4], [5], [6], [7], [8], [9] use asimple and heuristic local updating rule to capture the opiniondynamics over a complex network topology. For example,in the Hegselmann-Krause model [7], [8], opinions (whichare represented by a real number) are updated synchronously,as an average of all other opinions that differ from its ownby less than a condence level . Other studies that haveinvestigated the effects of using simple pair-wise interactionsbetween neighboring agents whose opinions differ by less thana threshold are [9] and [10]. Specically, Deffuant et al. in[9] model the network on a square grid: each agent can only

communicate with its four immediate neighbors. Weisbuch in[10] extends this topology to a scale free network model.

In our previous work [11], [12], we have introduced amodel slightly more general than the HK model, to studyrandom gossiping type of interactions among social agents inan arbitrary network. Interactions among pairs of these agents

occur at random. The opinion distance after the interactioncannot be larger than the distance prior to the interaction. Theamount of change in opinion is proportional to the degree of trust between agents and inversely proportional to the opiniondistance between agents. To characterize how different trustmodels will effect the evolution of opinions in a society,in [11], [12] we proposed two interaction models: a soft-interaction model and a hard-interaction model. In the former,trust always exists between two interacting agents, while inthe latter, trust only exists when the opinion distance betweenagents is below a threshold. In this paper, we extend thework in [11], [12] by analyzing the convergence propertiesof the two interaction models, nding their local rates of

convergence. We prove that, under the soft-interaction model,the average opinion distance and the average squared opiniondistance always converge at an exponential rate. Under thehard-interaction model, we show that there exists a phasetransition from a society of radicalized opinions to one witha consistent opinion at a critical opinion distance threshold.

I I . M ODEL

In our model, agents are treated as nodes V = {1, , n}in an undirected graph G = ( V , E ), where the edge set E connects agents who are able to interact. Without loss of generality, it is assumed that G is connected in the sense thatthere exists a path joining directly or indirectly any two nodesin V . The random interactions between agents are modeledthrough a time-invariant vector p , describing the probabilityof each agent initiating an interaction, and a stochastic matrixP = [P ij ], specifying the probability that an initiator interactswith one of its neighbors. Hence, the probability of the pair(i, j ) E interacting is P ij = p i P ij + pj P ij . Moreover, [P ij ]has the same sparsity structure as the graph G , that is, P ij > 0if (i, j ) E and P ij = 0 otherwise. This implies that eachedge in E is activated innitely often.


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14. ABSTRACT Gossiping models have increasingly been applied to study social network phenomena. In this context, thispaper is specifically concerned with modeling how opinions of social agents can be radicalized if the agentsinteract more strongly with neighbors that share their beliefs. In our model, each agent?s belief isrepresented by a vector of probabilities that a given state is true. The agents average their opinions withthat of their neighbors over time, giving more weight to opinions that are closer to their current beliefs.The increasing trust that may exist among likeminded agents is modeled through a weight that is amonotonically decreasing function of the distance in opinion. We consider a continuous (soft) and adiscontinuous (hard) model for the weight and analyze the convergence properties.

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Here we recall the modeling assumptions introduced in [11].Suppose that there are q possible outcomes of an experiment.The opinion or belief of the ith agent is expressed as a q -dimensional vector x i = [x i 1 , , xin ] in which xi is theprobability that the th outcome is believed to be true byagent i. Hence, the belief space of dimension q is X ={x |

q=1 x = 1 and x [0, 1]}. Since our interactionmodels depend on the opinion distance between agents, we

dene the distance metric d(x i , x j ) : X X R + to bea proper geometric distance. Thus, with respect to the normx := d(x , 0 ), the belief space X is bounded. Using thetriangular inequality, the distance d(x i , x j ) 2 sup i x i :=dmax . For notational convenience, let dij [k] denote the dis-

tance d(x i [k], x j [k]) after k network-wide interactions haveoccurred.

Recall that agents in the network interact at random. If trustexists between interacting agents, then their beliefs will movecloser to each other. Formally, the degree of trust betweenconnected agents is captured by the function (d) [0, 1sup k ]according to the following nonlinear model:

(a1 ) dij [k + 1] = (1 k (dij [k]))dij [k] , (1)where the trust function(a2 ) (d) is a non-increasing function of d;

And for technical reasons, the step-size k satises(a3 ) k : k =1 k , k =1 2k < .As the system in (1) evolves, we say that consensus is achieved

when all the agents attain the same belief vector as measuredby the distance metric. More precisely, dij [k] = 0 for somek and for i, j V . In contrast, the society is said to exhibitradicalized beliefs if the agents are divided into subgroups:agents attain consensus within the subgroup; no subgroup hasinuence on the other sub-groups beliefs.

Remark 1: The curious reader may be wondering how a pair of interacting agents update their beliefs to reect thesystem in (1). Indeed, we have many choices. One simplechoice is to have the beliefs move closer to each other throughtheir shortest path. More precisely, let x denote a agentsbelief after an update, then the displacement of the beliefs isd(x i , x j ) = d(x i , x j ) d(x i , x i ) d(x j , x j ).

III . A NALYSIS

This section studies in detail the asymptotic convergenceproperty of the system by rst deriving an ordinary differentialequation (ODE) of the stochastic approximation of (1) asfollows: under (a3),

dij (t) = (dij (t))dij (t) , (2)where dij denotes the derivative of dij with respect to acontinuous time variable t, replacing the discrete time variablek. For convenient, the time variable t is not explicitly shownin the rest of this section. Recall that P ij represents theprobability that the pair (i, j ) E interacts. We can dene anaveraged distance variable d over the edge set E as follows

d :=( i,j )E

P ij dij .

From (2), the ODE of the average belief distance d is

d = ( i,j )E P ij (dij )dij . (3)Note that the convergence property of the above expressiondepends on how the trust function (d) is modeled and isinvestigated in the following two sub-sections.

A. Soft-Interaction Model

As mentioned earlier, trust always exists between agentsunder the soft-interaction model. Formally, we impose:

(a4 ) (d) is C 2 -differentiable for d (0, dmax );(a5 ) limdd max (d) = min > 0;(a6 ) (d)d is concave for d [0, dmax ].

The following lemma as proved in [11] implies asymptoticbelief convergence.

Lemma 1: Under (a1) (a6), (0, 12 ] such that thedynamics of d is upper and lower bounded by

(d)d d (d)d.Because of the strictly positive condition imposed on (d),both the upper bound and lower bound systems have one andonly one stable equilibrium at d = 0 and thus (3) will alwaysconverge to d = 0 , implying that dij = 0 for (i, j ) E.Since the network G is assumed to be connected, then for anytwo agents , m V , their opinion distance equals d ,m = 0 ,as the opinion distances along any path joining the two nodes

and m are zero. Therefore, the soft-interaction model alwayslead to belief consensus.

Notice that since (d) is differentiable, the rate of conver-gence can be computed locally. In fact, the system locallyresembles the form of the logistic equation and the local rateof convergence is exponential (See Corollary 1). Recall that

a system is said to converge exponentially to d if C, r > 0such that |d(t) d| Cert |d(0) d|. Note that d = 0 inour case.Corollary 1: Let r(d) be the local rate of convergence in

the neighborhood of d. Under (a1) (a6), (0, 12 ] suchthat the local rate of converge is exponential and is boundedby

[(d) d (d)] r (d) (d) d (d).Proof: See Appendix.

Lemma 1 and Corollary 1 imply that the average belief distance between agents asymptotically approaches zero witha rate that is locally exponential. We will prove next that theaverage squared distance as expressed below also approacheszero.

Dene the average squared distance as

d2 :=( i,j )E

P ij d2ij .

Property 1: Under (a2) and (a6), the function g(a) =( a)a is concave for a [0, d2max ].Proof: See Appendix.Using Property 1, the following lemma indicates that theaverage squared distance in belief converges to zero.


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Lemma 2: Under (a1) (a6), (0, 12 ] such that thedynamics of d2 is upper and lower bounded by

2 d2 d2 d2 d2 d2 .Proof: See Appendix.

Since ( d2) is strictly positive, both the upper and lowerbound systems have only one equilibrium at d

2

= 0 . There-fore, the average squared distance in belief will asymptoticallyapproach zero. Following a procedure similar to that in theproof of Corollary 1, the rate of convergence can be computedas stated below.

Corollary 2: Under (a1) (a6), the local rate of conver-gence in the neighborhood of d2 is exponential and is boundedby

( d2) d2 ( d2) r (d2) 2 ( d2) d2 ( d2) .In summary, as long as trust exists between any pair of

connected agents, as in the case of the soft-interaction model,

agents will keep updating their beliefs until consensus is at-tained. On the other hand, the next sub-section investigates theconvergence property of the system under the hard-interactionmodel.

B. Hard-Interaction Model

Recall that hard-interaction model describes the situationin which agents only decide to trust their neighbors whoseopinions differ by less than a threshold. To be precise,

(a7 ) : d (d) = 0;In addition, the following assumptions replace (a4) (a6) of the soft-interaction model.

(a8 ) (d) is C 2 -differentiable for d (0, );(a9 ) (0) / ( ) < ;(a10 ) (d)d is concave for d [0, ].Intuitively, if a society has a large threshold , i.e. is more

open-minded, it is possible that it will asymptotically attainconsensus. In contrast, if a society is close-minded with a smallvalue of , it will fail to converge and several opinion clusterswill emerge. Hence, the asymptotic belief prole varies with .As the value of increases, a society may transition from onewith radicalized beliefs to a society with a consistent belief.This suggests the existence of a phase transition. Indeed, thefollowing lemma as proved in [11] provides some insights onhow large needs to be for a society to reach consensus.

Lemma 3: Under (a1)(a3) and (a7)(a10), a necessarycondition for the system in (3) to converge almost surely is > d (0) .

However, the above condition is not sufcient. To show itsinsufciency, consider a society H: there are two groups H1and H2 existing in H, i.e. H1H2 = H; dij = 0 if i and j arein the same group; dij = + if i and j are in different groups.When H1 H2 P ij d (0) is not a sufcient condition.

IV. S IMULATIONS

A. Soft-Interaction Model

Fig. 1 illustrates the evolution of a belief prole over timeunder the soft-interaction model. Specically, the underlyingnetwork graph G consists of n = 100 agents randomlydistributed over an unit diameter disk and is generated usinga random geometric graph (RGG), i.e.

G =

G(n, r ), with

radius of communication r = 0 .6. Note that r = 1 implies afully connected RGG and r = 0 corresponds to a completelydisconnected network. The choice of RGG is arbitrary sinceour analysis applies to any type of network topology. Theinitial belief prole is uniformly distributed in X of dimension3. The distance dij between agents beliefs is dened usingthe L2 norm and thus dmax = 2sup i x i 2 = 2 . Agentsupdate their beliefs through the shortest path connecting themwith a constant step-size k = 1 and a trust function (d) =0.5 0.2d. This means that if agent i initiates an interactionwith agent j , then they will adjust their beliefs to d(x i , x i ) =(dij )dij and d(x j , x j ) = (dij )dij , respectively, where(dij ) + (dij ) = (dij ). We use the uniform communicationscheme to model the rate of interaction between agents. Let

N i be the set of neighbors of agent i. Uniform communicationcorresponds to homogeneous rates pi = 1/n for i Vwith P ij = 1 / |N i | uniform across neighbors. As the systemevolves, the belief prole asymptotically converges a singlebelief as seen from Fig. 1. Fig. 2 shows the convergencerate of d (i.e. , solid line) averaged over 300 trials. It is thencompared with its local upper bound and lower bound ( i.e. ,dotted lines) around various values of d as derived in Corollary1. Observe that the actual rate of convergence always lies inbetween its analytical bounds. Fig. 3 shows two histograms of 300 belief proles at time zero (left) and after the dynamicshave stabilized (right). Each belief prole is associated with arandomly generated RGG.

Fig. 1. The evolution of a belief prole with time: each line segmentcorresponds to a node and a segment terminates when a node interacts and

changes its belief. B. Hard-Interaction Model

We ran a suite of 300 trials; each trial starts with anuniformly distributed random initial belief prole in X of dimension 3, over a randomly generated RGG similar to thatin the soft-interaction model. The trust function (d) equals1 if d < and 0 otherwise. The underlying communicationgraph and updating rule are the same as in the soft-interactionmodel. Networks of three sizes n = 50 , 100, 200 were simu-lated. We dene the effective graph as Geff = (V , E eff ), with


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0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.1

0.2

0.3

0.4

0.5

A v e r a g e

D i s t a n c e

k

Actual rate of convergenceUpper bound on the rateLower bound on the rate

Fig. 2. The rate of convergence (solid line) is locally bounded from aboveand from below by two exponential rate functions (dotted lines).

00.5

1

0

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100

150

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2000

3000

4000

xavg

[0,1,0]

[0,0,1][0,0,1][0,1,0]

[1,0,0] [1,0,0]

Fig. 3. A histogram of 300 belief proles at time zero (left) and afterthe dynamics have stabilized (right). Each initial belief prole is uniformlydistributed in X of dimension 3 .

E eff := {(i, j ) E|dij [k] < } where k is sufciently large.Figure 3 shows the algebraic connectivity of the effectivegraph; it is clear that the network converges with a probabilityone if is sufciently above d(0) .

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

threshold

N o r m a l i z e

d

A l g e b

r a i c C o n n e c t i v

i t y

n=50n=100n=200E[d

c[0]]

0

0.5

1

0

0.5

1

0

1000

2000

3000

4000

0

0.5

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2000

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= 0.8 = 0.6

[0,0,1][0,1,0]

[1,0,0][1,0,0]

= 0.4

[0,0,1]

[1,0,0]

[0,0,1][0,1,0][0,1,0]

Fig. 4. Phase transition (top) of a society from radicalized beliefs to a

consistent belief. Histogram (bottom) of 300 asymptotic opinion proles withn = 100 at = 0 . 4 , 0 . 6 , 0 . 8 .

C. Heterogeneous Model

It is also of interest to learn how a belief prole evolvesin a heterogeneous model, that is, one in which a fraction of the population uses the hard-interaction model and the restuses the soft-interaction model. We want to investigate howthe critical threshold shifts as the number of agents using thehard-interaction model decreases. Fig. 5 shows two simple

heterogeneous interaction models compared with the hard-interaction model in a network of n = 50 agents. Specically,h represents the number of agents using the hard-interactionmodel and s is the number of agents using the soft-interactionmodel. One observes that, on average, as the society becomesmore open-minded ( i.e. s increases and h decreases), thecritical threshold shifts down.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

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1

threshold

N o r m a

l i z e

d

A l g e

b r a

i c C o n n e c

t i v

i t y

h=50,s = 0

h=35,s=10

h=25,s=25

Fig. 5. Heterogeneous Model

V. C ONCLUSION

In this paper, we have extended the analysis of the conver-gence properties of both the soft-interaction model and hard-interaction model, which were originally proposed in [11].Under the soft-interaction model, we have proved that theaverage belief distance converges to zero; moreover, the aver-age squared distance also converges. The rate of convergenceis locally exponential. Under the hard-interaction, we haveshowed the existence of a phase transition from radicalizedbeliefs to a consistent belief at a critical threshold , which isclosely related to the averaged initial belief distance.

VI . A PPENDIX : PROOFS

Corollary 1: Suppose that d(t + s) is in a neighborhood of

d(t) provided that s is small. Hence, Taylors expansion gives d(t + s) d(t) + d(t + s) d(t) d(t) , where(d) = d/dd . Since (d(t)) 0 given (a2), the upper boundsystem d(t + s) (d(t + s))d(t + s) becomes

d(t + s) (d(t))d(t + s) if d(t) = 0;

f (s; t) if d(t) < 0.where f (s; t) = d(t) d(t) d(t) 1

d( t + s )d ( t ) d(t +

s). In the rst case, (d(t)) is locally constant and hence, thelocal rate of convergence around d(t) is exponential and islower bounded by r (d(t)) (b(t)) . In the second case,dene (s; t) := d(t + s)/d (t). The dynamics of become

(s; t) = d(t + s)d(t) R (t) (s; t) 1

(s; t)K (t)

(4)

where R(t) = d(t) d(t) d(t) and K (t) = 1 +(d ( t ))d ( t ) (d ( t ))

> 1. Note that the dynamics on resemble thelogistic equation. Provided that (0; t) = 1 and R(t) = 0 , thesolution to (4) is

(s; t) K (t)eR ( t )s

K (t) 1 + eR ( t ) ss

K (t)K (t) 1

eR ( t ) s ,


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where R (t) is the rate of convergence of the upper bound sys-tem. Combining the results from both cases yields r d(t) R (t). Similarly, using the lower bound system d(t) (d(t))d(t), one can also nd an upper bound on the rateof convergence around d(t), that is, r d(t) R(t).

Property 1: g(a) is concave if its second derivative isnegative. Taking the rst derivative of g(a) yields g(a) = ( a) + a2 ( a). Taking the second derivative yields

g(a) = ( a) a + 2 ( a)

4 a + ( a)

4 a . (5)(a6) implies that the second derivative of (d)d is negative,i.e. , (d)d + 2 (d) < 0 for d [0, dmax ]. Let a = d2 , thenfor a [0, d2max ], the rst term in (5) is negative. Moreover, ( a) 0 from (a2). Hence, g(a) < 0 and g(a) is concave.

Lemma 2: From (a1), we have

d2ij [k +1] d2ij [k] = 2 k (dij [k])d2ij [k]+ 2k 2(dij [k])d2ij [k].Since (d) 1sup k 1k for d [0, dmax ], then

k (dij [k]) 1 and thus2 k (dij [k])d2ij [k] d2ij [k+1]d2ij [k] k (dij [k])d2ij [k].Stochastic approximation implies that

2( i,j )E

P ij (dij )d2ij d2 ( i,j )E

P ij (dij )d2ij . (6)

Let aij = d2ij and a := ( i,j )E P ij a ij . Hence, a = d2 .Using Property 1 and Jensens inequality, we get( i,j )E

P ij ( a ij )a ij a a = d2 d2 . (7)Hence, d2 is lower bounded by d2

2

d2 d2 .

To prove the upper bound, let S = {( , m) E c |d2m d2}, S c = {(i, j ) E c |d2ij > d 2} s.t. S S c =E c . From (6), d2 f c (d2) := ( i,j )S c P ij (dij )d2and d f (d2) := ( ,m )S P m ( d2)d2m = d2 ( i,j )S c P ij d2ij ( d2). A weighted sum yields

d f ( d2)+(1 )f c ( d2) = ( d2)d2g(d2), (8)where g(d2) = ( i,j )S c P ij (1 )(dij )d2 ( d2)d2ij .The term g(d2) will vanish if = 11+ , where =

( i,j ) S c P ij d2ij ( d 2 )

( i,j ) S c P ij (d ij )d 2 1 because for (i, j ) S c ,d2ij > d 2 and ( d2) (dij ). Moreover, by (a5),

( d 2 )( i,j ) S c P ij (d ij )

< . Therefore, 0, 12 suchthat d2 is upper bounded by d2 ( d2)d2 .

ACKNOWLEDGMENT

This work of the rst two authors was supported by NSFgrant CCF-1011811.

R EFERENCES

[1] C.P. Chamley, Rational Herds: Economic Models of Social Learning ,Cambridge University Press, 2004.

[2] S. Bikhchandani, D. Hirshleifer, and I. Welch, A theory of fads, fashion,custom, and cultural change as informational cascades, in J. Pol. Econ. ,1992, vol. 100, pp. 9921026.

[3] H. Leibenstein, Bandwagon, snob, and Veblen effects in the theoryof consumers demand, in Quarterly Jounrnal of Economics , 1950,vol. 64, pp. 183207.

[4] M. H. DeGroot, Reaching a consensus, in Journal of AmericanStatistical Association , 1974, vol. 69, pp. 118121.[5] G. Deffuant, D. Neau, F. Amblard, and G. Weisbuch, Mixing beliefs

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[7] R. Hegselmann and U. Krause, Opinion dynamics and boundedcondence models, analysis and simulations, in Journal of ArticialSocieties and Social Simulation , 2002, vol. 5.

[8] V.D. Blondel, J.M. Hendrickx, and J.N. Tsitsiklis, On Krauses multi-agent consensus model with state-dependent connectivity, in IEEE Trans. Auto. Control , 2009, vol. 54, pp. 25862597.

[9] G. Deffuant, F. Amblard, and G Weisbuch, How can extremism prevail?a study based on the relative agreement interaction model, Journal of Articial Societies and Social Simulation , vol. 5, 2002.

[10] G. Weisbuch, Bounded condence and social networks, The European

Physical Journal B - Condensed Matter and Complex Systems , vol. 38,no. 2, pp. 339343, 2004-03-25.[11] L. Li, A. Scaglione, A. Swami, and Q. Zhao, A study on opinion

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Trust, Opinion Diffusion, And Radicalization in Social Networks