Controversy-seeking fuels rumor-telling activity in polarized opinion networks Hugo P. Maia1Silvio C. Ferreira1 2and Marcelo L. Martins1 2 3 1Departamento de F sica Universidade Federal de Vi cosa 36570-900 Vi cosa Minas Gerais Brazil

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Controversy-seeking fuels rumor-telling activity in polarized opinion networks

Hugo P. Maia,1Silvio C. Ferreira,1, 2 and Marcelo L. Martins1, 2, 3

1Departamento de F´ısica, Universidade Federal de Vi¸cosa, 36570-900 Vi¸cosa, Minas Gerais, Brazil

2National Institute of Science and Technology for Complex Systems, 22290-180, Rio de Janeiro, Brazil

3Ibitipoca Institute of Physics - IbitiPhys, Concei¸c˜ao do Ibitipoca, 36140-000, MG, Brazil

(Dated: October 11, 2022)

Rumors have ignited revolutions, undermined the trust in political parties, or threatened the

stability of human societies. Such destructive potential has been signiﬁcantly enhanced by the

development of on-line social networks. Several theoretical and computational studies have been

devoted to understanding the dynamics and to control rumor spreading. In the present work, a

model of rumor-telling in opinion polarized networks was investigated through extensive computer

simulations. The key mechanism is the coupling between ones’ opinions and their leaning to spread

a given information, either by supporting or opposing its content. We report that a highly modular

topology of polarized networks strongly impairs rumor spreading, but the couplings between agent’s

opinions and their spreading/stiﬂing rates can either further inhibit or, conversely, foster informa-

tion propagation, depending on the nature of those couplings. In particular, a controversy-seeking

mechanism, in which agents are stimulated to postpone their transitions to the stiﬀer state upon in-

teractions with other agents of confronting opinions, enhances the rumor spreading. Therefore such

a mechanism is capable of overcoming the propagation bottlenecks imposed by loosely connected

modular structures.

I. INTRODUCTION

At the middle of 1789, from July 20 to August 6,

a rumor enigmatically spread like wildﬁre throughout

France. The news was that outlaw bands were sweep-

ing the prairies to cut the unripe wheat and destroy

the crops. Rapidly, a massive panic wave – the Grande

Peur (Great Fear) – raised, transforming a rural commo-

tion into an irreversible revolution [1] in France. Peas-

ants plundered and set ﬁre to landlords’ properties, in-

vaded registry oﬃces to burn property deeds, pillaged

churches and villages. Riots, attacks and ﬁres simultane-

ously erupted in many provincial towns (e.g., Marseille,

Lyon, Grenoble, Rennes, Le Havre and Dijon). Three

weeks after the fall of the Bastille, French feudal social

structure and its royal state machinery completely col-

lapsed.

These iconic events strongly highlight the centrality of

rumor spreading in human societies that were, to a lower

or higher degree, ever self-organized as informational net-

works [2]. They also provide evidence that rumors can

become or strategically be used to harm social stabil-

ity. Hence, understanding the mechanisms and design

means to regulate information dissemination are imper-

ative tasks for social sciences and even economics [3,4].

For the 21th century physics, in great measure focused

on the emergence and propagation of information in out-

of-equilibrium complex systems [5], the theoretical anal-

ysis of contact processes, epidemic spreading, and ru-

mor dissemination became central to understand phase

transitions, stochastic dynamics and irreversibility [6,7].

Regarding the dissemination of information, in 1964, in-

spired by the susceptible-infected-recovered (SIR) epi-

demic dynamics [8] for the spreading of an infectious

disease, Daley and Kendall reinterpreted and extended

this epidemic model aiming to describe rumor-telling [9].

This pioneer model was extended in many directions by

either adding traits to the original mechanism of rumor

propagation or varying the structural properties of the

underlying social networks [6]. Thus, several classes (e.g.,

asymptomatic, debunkers, exposed, hibernators, and la-

tent or skeptical) widen the classical ones – spreader,

ignorant, stiﬂer – present in the SIR model, leading to

diverse rumor spreading models [10–12]. These models

were designed to take into account hesitation, forget-

fulness, trust, refutation, forced silence, education, and

other human factors involved in realistic rumor propaga-

tion processes. Furthermore, the traditional approaches

based on ordinary (spatially implicit) and partial (spa-

tially explicit) diﬀerential equation models [10–13] were

joined to lattice and graph (homogeneous and heteroge-

neous) models [14–16], characterized by exponential and

power-law degree distributions, respectively [7]. These

studies revealed that topological properties of a com-

plex network substantially impact the dynamics of rumor

propagation, particularly the reach and speed of informa-

tion spreading.

Online communications networks neatly exhibit ho-

mophily which leads to a natural polarization in groups

sharing distinct perspectives [17]. The mutual interac-

tions within these groups create echo chambers, in which

beliefs are reinforced due to repeated interactions with

individuals sharing the same points of view, as observed,

for instance, in the impeachment of the Brazilian pres-

ident Dilma Roussef in 2016 [18] and French elections

of 2017 [19]. Moreover, these communities form a new

topological structure of the communication network as

interconnected modules. So, superimposed to the hetero-

geneous degree distribution, there is an additional level

of heterogeneity associated with community sizes in mod-

ular networks. Hence, the goal of the present paper is to

investigate rumor spreading onto networks generated by

arXiv:2210.04103v1 [physics.soc-ph] 8 Oct 2022

adaptive opinion formation processes that lead to loosely

connected modular networks forming echo chambers. A

new content is released within a community of the polar-

ized network and follows a rumor spreading process [20]

coupled with the opinion of the interacting individuals ac-

cording to diﬀerent rules raging from beliefs’ alignment to

controversy-seeking where contrasting opinions hampers

lost of interest on an issue. In the current paper, we show

that, as would be expected, the highly modular structure

of opinion polarized networks strongly impairs rumor

spreading. However, the introduction of couplings be-

tween agent’s opinions and their spreading/stiﬂing rates

has a striking eﬀect on rumor-telling. Indeed, depending

on the nature of those couplings, information propaga-

tion can be either further inhibited or enhanced up to the

level observed in unpolarized networks, thus suppressing

the modularity bottleneck.

The rest of the paper is organized as follows. In Sec-

tion II the model is presented. Initially, it is described

how the polarized opinion networks considered in our

analysis are generated. The mechanism for rumor spread-

ing onto such networks is proposed. The simulation re-

sults obtained are reported in Section III and discussed

in Section IV. Finally, our major ﬁndings are summarized

as well as some perspectives addressed in Section V. Ad-

ditional methodological details are presented at A.

II. MODEL

The fundamental issue addressed in the present work

is how a rumor spreads in a polarized opinion’s networks

formed previously in a possibly diﬀerent context. For

example, how an anti-vaccine rumor spreads onto an on-

line network formed during political or ideological de-

bates. As a concrete contemporary example, consider

the on-line network formed by supporters or opponents

of Brazilian president Jair Bolsonaro, who deliberately

preached against vaccine safety, especially for kids. So,

how strongly an anti-vaccine rumor, started within the

bubble of supporters, would reach the opponents depend-

ing on how the individuals react to this content?

A. Creating Polarized Networks

The major trait of political polarization is the split-

ting of a social group in diverse subgroups sharing simi-

lar opinions and committed to a common ideology. Such

communities self-organize in densely connected modules

loosely interconnected among them. In order to gener-

ate networks with polarized opinions and modular topol-

ogy, we used an adaptive network model with bounded

conﬁdence opinions, inspired in the classical Deﬀuant

model [21], proposed in Ref. [22]. A quick rundown of

the model follows:

•Initially, Nindividuals supporting uniformly dis-

tributed opinions oi∈[0,1] are organized in a

random network described by a power-law degree

distribution P(k)∼k−γ. This network is gen-

erated according the uncorrelated-conﬁguration-

model (UCM) [23]. An upper cutoﬀ of kmax =√N

and a degree exponent of γ= 2.7 were used.

•In each time step, each individual com-

putes its neighbor’s average opinion hoii=

ki(t)Pj∈νi(t)oj(t) and how much it diﬀers from

his or her own opinion, ∆i=oi− hoii. If the

diﬀerence is below the threshold i, he or she

changes their opinions in order to resemble those

of their neighbors. The opinion of each agent is

updated as:

oi(t+ 1) = oi(t) + µ∆i(t) if ∆i(t)≤i

oi(t) if ∆i(t)> i,

where µis the opinion’s convergence rate. The pa-

rameter iis the tolerance threshold of agent ito

diﬀerent opinions and is constant along the time.

Low imeans low tolerance and increasing polar-

ization.

•Every agent reconsiders their connections after

opinion update. If any pair of connected individ-

uals iand jsustains opinions diﬀering by ∆ij =

|oi−oj|>min{i, j}, they may break their con-

nection with probability p= 1 −e−κ∆ij .

•Broken connections can be reconnected at a fu-

ture time step with probability q=e−dik /d0, if

∆ik <min{i, k}. This probability is a function

of the distance dik between the individuals iand

k. Here, κis the link rupture tendency, and d0is

the characteristic rewiring distance, assumed to be

uniform for all agents.

Here, we assume 5% of the individuals has a high tol-

erance of = 0.5 while the remaining 95% has a lower

tolerance = 0.04, distributed at random. We ﬁxed

µ= 0.8, κ= 3.5, d0= 4 and an iteration time of

T= 120. This parameter set produces networks where

a small number of individuals act as “bridges” between

well connected communities of diﬀerent opinions. Also,

a balanced opinion distribution, i. e., hoii= 0.5 that

does not favor any side is obtained. The initial size of

N= 18000 produces a largest connected component of

sizes around N= 10000 individuals, in which the rumor

spreading is investigated. Figure 1shows a typical mod-

ular network with ﬁxed opinions generated according to

this protocol. Indeed, although in the original adaptive

model [22] both opinions and connections change in time,

here we use static networks with ﬁxed opinions assum-

ing that rumor spreading proceeds at a time scale much

smaller than opinion formation processes.

FIG. 1. Typical network obtained with the opinion formation

model of Ref·[22]. Colors represent individual’s opinions.

The opinion distribution p(o) is presented besides the color

bar; the latter indicate the opinion scale in range [0,1]. The

ﬁnal number of individuals is around 10000 and a special care

was taken to pick only networks that maintain a balanced

opinion distribution, i. e., hoii= 0.5 that does not favor any

side.

B. Opinion Driven Rumor Modeling

Usually, rumor-spreading models divide the social

group into three compartments in analogy to SIR epi-

demic dynamics [7,9,20]: ignorants (presented by S

in analogy with the susceptible individuals) that are

completely unaware of the spreading information/rumor;

spreaders (represented by I in analogy with infectious in-

dividuals) that actively disseminate the information, and

stiﬂers (represented by R in analogy with the removed

individuals) that is aware of the information but are no

longer interested in spreading it around. Here we focus

on the version introduced by [24] in which, upon pairwise

social interactions, individuals can change their compart-

ments according three diﬀerent events:

I+S λ

−→ 2I,I+R α

−→ 2R,I+I α

−→ R+I.

Ignorant individuals become aware with rate λwhen in-

teracting with a spreader, while spreaders lose interest in

the information with rate αwhen interacting with agents

aware of the rumor. Diﬀerently from the SIR epidemic

model, individuals do not recover spontaneously, but only

if they perceive that spreading rumor is no longer novel

nor interesting.

In realistic contexts, the members of a social group

are very heterogeneous not only in their connectivity but

also in their behaviors. So, individual’s opinion is an im-

portant factor since one can choose either to spread the

information or keep it depending on what he or she thinks

about the rumor. For this reason, we altered the tradi-

tional rumor-spreading model in order to replicate such

phenomena. The rate in which spreaders disseminate

the rumor is now dependent on their alignment/opinion

oithrough a coupling function f(oi) . Speciﬁcally, each

social agent has his own spreading rate per contact given

λi=λ∗f(oi),(1)

in which the parameter λ∗is the average spreading rate.

In turn, the rate in which an individual will grow bored

of a rumor depends on both his own opinion oiand that

of his interacting neighbor ojthrough another coupling

function g(oi, oj). Speciﬁcally, each link in the network

has their own stiﬂing rate given by

αij =α∗g(oi, oj),(2)

where α∗is the average stiﬂing rate. The exact forms of

the coupling functions f(oi) and g(oi, oj) will naturally

depend on the rumor in question. So, we propose few dis-

tinct functional forms for the coupling between opinions

and the spreading/stiﬂing rates. The decoupled case,

corresponding to f(o) = 1 or g(o) = 1 depending whether

one considers spreading or stiﬂing processes, respectively,

is also investigated for sake of comparison. Three cou-

pling functions are considered: linear and unimodal cou-

pling for spreading rate and controversy-seeking (CS) for

stiﬂing rate.

Linear coupling. In this model, the impetus to fur-

ther propagate a new rumor is directly correlated with

the individual’s opinion and given by

f(oi)=2ηλ(oi−1

2)+1,(3)

where ηλ∈[−1,1]. The case ηλ= 0 corresponds to

the decoupled case while ηλ>0 or <0 favors opinions

closest to 1 or 0, respectively. For instance, a rumor

about vaccines causing side-eﬀects would spread much

more eﬃciently in anti-vaccination groups (o∼+1) than

in pro-vaccination ones (o∼0) implying ηλ>0. So, the

linear coupling results in a better spreading rate when

there is an ideological alignment between the individual’s

opinion and the rumor.

Unimodal coupling. The function f(oi) has a non-

monotonic symmetric form centered at o= 0.5 given by

f(oi) = 1 + 2ηλ|2oi−1| − 1

2.(4)

Again, ηλ∈[−1.+ 1] and ηλ= 0 means no coupling

at all. But now ηλ= 1 favors opinions in either ex-

tremes (o= 0 or o= 1), whereas ηλ=−1 favors mod-

erated opinions (o= 0.5). Particularly, in the present

work we are interested ηλ>0 for which a rumor will be

more eﬀectively spread by both extremes. Indeed, while

one extreme side agrees with the rumor and promotes its

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Controversy-seekingfuelsrumor-tellingactivityinpolarizedopinionnetworksHugoP.Maia,1SilvioC.Ferreira,1,2andMarceloL.Martins1,2,31DepartamentodeFsica,UniversidadeFederaldeVicosa,36570-900Vicosa,MinasGerais,Brazil2NationalInstituteofScienceandTechnologyforComplexSystems,22290-180,RiodeJaneiro,Brazi...

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Controversy-seeking fuels rumor-telling activity in polarized opinion networks Hugo P. Maia1Silvio C. Ferreira1 2and Marcelo L. Martins1 2 3 1Departamento de F sica Universidade Federal de Vi cosa 36570-900 Vi cosa Minas Gerais Brazil.pdf

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Controversy-seeking fuels rumor-telling activity in polarized opinion networks Hugo P. Maia1Silvio C. Ferreira1 2and Marcelo L. Martins1 2 3 1Departamento de F sica Universidade Federal de Vi cosa 36570-900 Vi cosa Minas Gerais Brazil

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