1 DeGroot-based opinion formation under a global steering mechanism

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DeGroot-based opinion formation

under a global steering mechanism

Ivan Conjeaud∗⋄ Philipp Lorenz-Spreen+Argyris Kalogeratos∗†

∗Centre Borelli, ENS Paris-Saclay, Gif-sur-Yvette, France

⋄Paris School of Economics, Paris, France

+Max Planck Institute for Human Development, Berlin, Germany

Abstract—This paper investigates how interacting agents arrive

to a consensus or a polarized state. We study the opinion

formation process under the effect of a global steering mechanism

(GSM), which aggregates the opinion-driven stochastic agent

states at the network level and feeds back to them a form of

global information. We also propose a new two-layer agent-based

opinion formation model, called GSM-DeGroot, that captures the

coupled dynamics between agent-to-agent local interactions and

the GSM’s steering effect. This way, agents are subject to the

effects of a DeGroot-like local opinion propagation, as well as

to a wide variety of possible aggregated information that can

affect their opinions, such as trending news feeds, press coverage,

polls, elections, etc. Contrary to the standard DeGroot model, our

model allows polarization to emerge by letting agents react to the

global information in a stubborn differential way. Moreover, the

introduced stochastic agent states produce event stream dynamics

that can ﬁt to real event data. We explore numerically the model

dynamics to ﬁnd regimes of qualitatively different behavior. We

also challenge our model by ﬁtting it to the dynamics of real

topics that attracted the public attention and were recorded on

Twitter. Our experiments show that the proposed model holds

explanatory power, as it evidently captures real opinion formation

dynamics via a relatively small set of interpretable parameters.

Keywords—Opinion formation dynamics, agent-based modeling,

DeGroot model, polarization, inﬂuence, global steering, public

opinion, public debate, information aggregation, media, social

networks, mass-movements, event data stream.

I. INTRODUCTION

The explosive development of new electronic communication

means is heavily impacting the self-organized social dynamics

of opinion formation and political participation, in ways that

are not fully-understood. Our limited view over the incurred

changes is partly due to the fact that we lack expressive

yet interpretable models that could account for the complex

multilevel information pathways that become available through

modern communication technology. To advance our under-

standing, what is mostly needed is rather simple models able

to highlight a meaningful prototypical agent-based mechanism

that drives opinion formation.

The landscape in which modern public debate takes place,

includes national and international broadcasting media, and

more recently online social networking platforms, which have

altered the way and the speed with which people exchange

†I.C. and A.K contributed equally to this work. Correspondence

to: A.K.; email: argyris.kalogeratos@ens-paris-saclay.fr.

Manuscript received December XX, XXXX; revised XXXX XX, XXXX.

information [1, 2]. Especially for the exchanges on online plat-

forms, these have substituted part of the physical interactions

between individuals, and have led to a reshaping of the social

network formed around each individual [3], e.g. by having

a wider set of contacts including weak-ties and contacts that

are geographically remote. The transition from one-to-many to

many-to-many communication that these platforms allow has

brought new attention to self-organized social behavior, like

the new ways of political participation through digital media

[4]. Among the interesting related phenomena, one can ﬁnd

some that are emergent, such as price formation, panic buying,

overnight formation of social movements, persistent rumors,

and self-organized fake news circulation [5, 6, 7, 8]. A recent

spur in modeling efforts for such phenomena from a complex

system perspective, largely concerns opinion dynamics [9].

Several recent analyses and models have either focused on

misinformation spreading or polarization dynamics [10, 11].

Only few modeling efforts have explicitly studied the interplay

between individual opinion dynamics, which are driven locally

by social inﬂuence, and the correlation of different debate

topics that co-evolve in a multidimensional space [12].

The theoretical literature on modeling opinion dynamics com-

prises mainly two streams, one with models considering

opinions as continuous variables, and another one considering

them as binary or discrete variables. The ﬁrst one contains

models based on the DeGroot model [13], which is itself

a generalization of French’s seminal work [14]. A great

variety of generalizations and variations of this model have

been proposed, mainly by relaxing the assumption that the

inﬂuence between any two agents is ﬁxed, and allowing instead

to vary as functions of time or the opinion of the nodes

[15, 16, 17]. Continuous modeling is not restricted to use

DeGroot-Friedkin models [13, 18, 19] as a basis, but rather

includes a variety of other models [9, 10, 20, 21]. The other

stream of research, initiated by Granovetter [22], considers

opinions as binary (or discrete) variables and frequently adopts

a game theoretical approach, in which opinions are considered

as strategies that give each time the best response to the

state of the local environment [23, 24], or a physics-like

approach in which opinions are states, with models adapted

from physics to social sciences [25, 26]. Often, these models

can be summed up to threshold models, where an opinion

state is adopted when a sufﬁciently large proportion of a

node’s neighborhood has done so [27]. Both these research

streams have boosted the interest in understanding the opinion

arXiv:2210.12274v2 [cs.SI] 2 Nov 2023

formation process, consensus formation [13, 28], maintenance

of diversity despite increasing local resemblance [29], with

some attempts to model global interaction on top of the one at

the local level [30, 31]. Such models are limited as they deﬁne

global interactions to be also peer-to-peer, whereas with other

arbitrarily distant agents.

DeGroot-based modeling. At the core of many of the opinion

formation studies is the DeGroot-based modeling [13, 18],

which is also central in this work. The classic DeGroot

model considers only local interactions between neighboring

agents, and brings their opinions closer and closer. An agent

can still be inﬂuenced by any other if there exists a path

connecting them, but only through step-by-step bilateral in-

teractions involving intermediaries. Essentially, this simulates

the primordial idea that an agent’s opinion is driven mainly

by locally inﬂuential individuals [32] and her tendency to

conform with her social environment. The DeGroot model is

prototypical and insightful as a mechanism, but, it comes with

a number of notable limitations, most of which have occupied

the literature.

First, the local smoothing of opinions, under weak assump-

tions, leads always to global consensus. Consequently, it is

unable to generate opinion diversity or polarization (i.e. mul-

timodal consensus) on its own. To ﬁll the gap, there have

been conjectures and speculations about mechanisms that

could allow such phenomena to emerge. One idea is that

polarization can come from stubborn agents that are not eager

to change their positions regardless the changes in their social

surrounding, and therefore act as diverse attractors [21, 33, 34].

Another one, also at the local level, stipulates that signed

networks, which model local attraction-repulsion, can also lead

to polarization [35]. We discuss in technical terms that these

approaches lead to limited polarization, speciﬁcally upper-

bounded by the initial conditions (see Sec. III-C). One may

point out that the attempts to explain opinion divergence

introduce pre-inscribed features to the system, either at the

connectivity level, or at the agents’ opinion update level. This

implies that divergence is not really generated by the process

itself, but is due to the pre-inscribed features that push the

system to polarized states. The pre-inscribed features can be

the result of deeper beliefs or psychological factors that do not

change during a short-term debate, such as those taking place

in social media. Few works have tried to include psychological

factors that can cause an agent’s behavior to change during the

opinion formation, e.g. the notion of tolerance that makes an

agent’s opinion to saturate the more agreement there is in her

neighborhood [36].

Second, by conceptualizing the opinion formation as taking

place strictly through peer-to-peer interactions, it lacks any

mechanism of broadcasting or aggregation of agents’ opinions,

or ways for agents to get feedback from the global state

of the debate over the network; hence it leaves mass-media

effects completely out of its scope. In reality, such mechanisms

become more and more relevant due to the fact that it is natural

for agents who operate under cognitive and time constraints

to seek for summarized or ﬁltered information sources. In

the modern landscape there are new interacting entities and

information pathways [37, 38, 39, 40], as well as the increased

coupling of local and global information ﬂows (e.g. mass

media picking up on social media trends), which are usually

in place simultaneously [41].

Third, a point of our criticism that is somewhat related to the

previous one, the DeGroot-based modeling rarely considers

political participation as an important aspect of the opinion

formation. However, political participation has been found

to be reliably associated with media usage, and especially

social media [42, 43]. We accordingly argue that for an agent,

public expression beyond her narrow social environment and

political participation are intertwined with her opinion, which

is a mostly overlooked feature in the literature. In this work,

we regard agents as being in conversation with both their

local environments and the global state-of-things represented

by information aggregation. Furthermore, and related to the

ﬁrst point of criticism about polarization, we argue that

the attraction or repulsion to information aggregation can

be more important as a factor producing opinion diversity,

compared to similar local level reactions, for several reasons.

To mention a few: i) local reactions going against an agent’s

social surrounding is likely to be frictional and costly; ii) the

effects of this kind of local disagreement can be negligible

compared to the -usually more frequent- interactions with

global information that is supposed to be more representative

for the state of the debate at the whole network level; iii) for

the same reason, information aggregation is likely to generate

structured reactions, while local disagreement is not.

Fourth, a point of general criticism to all the stream of classical

opinion formation modeling is that it idealizes the process

(e.g. by assuming that opinions are visible and subject to direct

exchange between agents, by considering simplistic opinion

propagation and update rules, or by ignoring psychological

aspects in agents’ reactions) and does not offer in the end

sufﬁcient tools for addressing problems involving real data

[44, 45, 46, 47, 48].

Contribution. In this paper, we present the GSM-DeGroot

model that aims at capturing the intertwined relationship

between each agent’s opinion (a continuous variable) and

the publicly visible political expression or participation

(e.g. protest participation, posting on social networking plat-

forms, etc.), which is represented by an opinion-dependent

stochastic state. It is thereby a hybrid model that combines

elements from different literature streams.

The proposed model consists of three mechanisms, where the

last two represent distinct but potentially contradicting forces:

i) an event generation mechanism (EGM) that introduces an

opinion-based stochastic state (binary) for each agent corre-

sponding to events of public manifestation or participation;

ii) a typical local opinion propagation mechanism (OPM) that

is a converging force making agents more and more alike; and

iii) a global steering mechanism (GSM) that is a polarizing

force acting at the global level, and can make agents moving

apart from each other. More speciﬁcally, the GSM computes a

summary of the agents’ states and feeds it back to the agents,

who are allowed to have stubborn differential reactions to

it, hence contrasting opinion updates. Note that, the deﬁned

process can also be seen as a point-process over a graph, with

the difference to existing processes (e.g. like Hawkes process

[48]) that here the agents do not interact directly through their

states (i.e. the occurring events) but through their opinions

(latent variables) that drive the states.

The originality of our approach is that it goes beyond the stan-

dard DeGroot-based modeling: on the top of a DeGroot-like

idealization, GSM-DeGroot accounts for information aggrega-

tion phenomena that can lead to structured agent reactions and

polarization, while also builds a stochastic process that can ﬁt

to real event data and offer quantitative insight.

By both extensive numerical simulations and deriving math-

ematical properties, we show how the interaction of these

mechanisms allow richer and more complex dynamics, such

as disagreement, polarization, and radicalization. We show

that there are areas of distinct behavior in different regions

of the model’s parameter space, and that the model offers

interpretable descriptions of the associated dynamics. We also

show that our model is capable of ﬁtting to the approximate

dynamics of several phenomena of recent collective movement

or action recorded on Twitter. The model parameters allow

the interpretation and comparison of different public events,

or the same event across different linguistic areas, and this

way to get insight about their characteristics. An improved

ﬁtting is achieved when combining our approach with (fully)

stubborn agents. Contrary, when removing the proposed GSM,

the remaining model equipped only with stubbornness cannot

ﬁt well to the event data.

The organization of the rest is as follows: Sec. II presents the

proposed model. Sec. III investigates some of its mathemat-

ical properties. In Sec. IV we study empirically the model

dynamics in synthetic scenarios. In Sec. V, we ﬁt our model

to real event data and we highlight its interpretability. We give

our conclusions in Sec. VI. The Appendix provides technical

proofs and additional material.

II. THE ENHANCED GSM-DEGROOT MODEL

A. Model statement

Nagents are represented as nodes in a ﬁxed, strongly con-

nected, weighted digraph G= (V, W ), where V={1,...,N }

is the set of node indexes. W={wji}i,j∈Vis a matrix with

normalized incoming edge weights, i.e. ∀i∈V, PN

j=1 wji = 1,

where wji indicates the inﬂuence level of agent jto i.

Each agent iis characterized by: an opinion-dependent

stochastic state Si,t ∈ {0,1}, produced by the event genera-

tion mechanism (EGM), indicating whether or not the agent

generates an event to manifest her views beyond her local

environment; a time-dependent opinion Xi,t ∈R, which is

exchanged locally with neighboring agents through an opinion

propagation mechanism (OPM); and a ﬁxed inherent (i.e. stub-

born) way βi∈ B ⊆ R, in a value range Baround 0, in

which she responds to received global information. Moreover,

S = 0

Global aggregation

Local interactions

X4X5X6

S = 1

Legend

Node with negative

reaction to GSM

Node with positive

reaction to GSM

Node with

negative opinion

Node with

positive opinion

Edge weights

Stochasic states

Fig. 1: Scheme of the proposed two-layer GSM-DeGroot model.

At the bottom there is the local interaction layer, and at the top the

global information aggregation layer. We are at time t(here omitted

in the notations). The model assumes that the opinions X1,..., X8

(their value scale is shown as red or blue areas inside the nodes), are

exchanged at the local level between connected agents through the

opinion propagation mechanism (OPM). Then, according to the event

generation mechanism (EGM), each agent ienters stochastically a

state Si={0,1}depending on her opinion Xi. Next, the global

steering mechanism (GSM) aggregates the states at a global level, and

ﬁnally feeds back a view over this information to the agents. Each

agent reacts to global information in a different but ﬁxed way βi,

positive or negative (shown as dashed green or red node boundaries).

we consider g(St)to be a function representing the global

steering mechanism (GSM) that aggregates information from

the network at a global level and feeds it back to the agents.

Given agent i’s current opinion Xi,t, the discrete-time evolu-

tion of her state and opinion for time t+1 is given by:

State update: Si,t ∼Bernoulli

| {z }

event generation

1

1+exp(−λXi,t)(1)

Opinion update: Xi,t+1 =βi

|{z}

agent’s

reaction

g(St)

|{z}

global

steering

j=1

wjiXj,t

| {z }

local opinion

propagation

(2)

According to the EGM (Eq. 1), P(Si,t = 1) = 1

1+exp(−λXi,t)

and P(Si,t = 0) = 1 −P(Si,t = 1), with λbeing a sensitivity

parameter. In the rest, we consider g(St) = γ˜

St, where

St=1

NPN

i=1 1{Si,t = 1}, and γ≥0is a parameter expressing

the GSM’s scaling effect. We call the value of g(St)as

the GSM’s steering strength at time t. Note that, by setting

γ= 0, the GSM is neutralized and leaves only the OPM in

effect, thus this model becomes equivalent to the classical

DeGroot model. Fig. 1 shows schematically the elements of

the proposed model.

B. Model interpretation

GSM-DeGroot introduces, for of each agent i, a stochastic

state Si,t and a ﬁxed predisposition βiover the received global

information. These two additions to the classical DeGroot

model [13] are explained next.

Opinion-dependent states.. At time t, the EGM generates

stochastically the state of agent ias a function that is in-

creasing with her current opinion value, and independently of

her previous state. The GSM-DeGroot model is a particular

discrete-time stochastic process generating one-sided opinion-

driven events (i.e. agents getting in state 1) with variable

probability intensity over time (i.e. non-iid events). This is

totally different to typical state-based models as there is

no notion of agent’s transition from one state to the other.

The model could be seen as a discrete-time point-process

over a graph, however the difference to existing processes

(e.g. Hawkes process [48]) is that the agents in our model

do not interact directly through their states (i.e. the events),

but only through their opinions, which are latent variables

driving the states; then, states affect the process only through

the global aggregation of the GSM.

State 1might be regarded as any kind of behavior or action

induced by the agent’s opinion. For instance, an opinion on a

governmental policy can lead to protesting against it. Here, the

one-sidedness of the process means that an agent remaining in

state 0does not imply she protests in support of the policy. For

cases in which protests are two-sided, an agent in state 1can

be interpreted as protesting for one of the side and her opinion

as measuring how important the matter is to her. In this case,

the number of agents in state 1should be interpreted as a

measure of how controversial a topic is. A non-deterministic

state means that the decision is taken considering additional

factors that are external to the model, which are here assumed

to be randomly distributed. E.g. deciding an agent whether to

participate in a protest can be a function of her view on the

seriousness of a situation, psychological factors (e.g. social

pressure, fatigue), her whereabouts or time availability, which

are not explicitly modeled. Instead, such factors are captured

by the global parameter λthat controls the opinion-driven

actions (see Sec. II-C).

Steering mechanism and agents’ reaction. Beyond the

assumption of most variations of the DeGroot model that an

agent’s opinion is only affected by her local social interactions,

our model formalizes the idea that the global network state

has also an important role in the opinion formation. The

GSM represents any form of information aggregation that

may modify agents’ opinions over a topic of public debate.

The underlying idea for g(St)summarizing the agents’ states

is that the steering mechanism relies on aggregated coarse

information from the whole network, contrary to peer-to-

peer interaction that is based on repeated social exchanges

and allows for more nuance. The GSM is characterized by

the proportion of positively-reacting agents in the population,

β=1

NPN

i=1 1{βi>0}, and the function g(·)described by the

parameter γ. Essentially, our model assumes that the agents

have already formed their views, biases or predispositions,

prior to the debate, and those determine the stubborn way they

react to global information. This reaction can be due to either

a sense of alignment, or as a reaction in opposition to what

the agent perceives as the opponent ‘other’ (e.g. believing that

the media have an agenda or corrupted and distort reality).

C. Model extensions

Fully stubborn agents with ﬁxed opinions can be easily incor-

porated to the GSM-DeGroot model by allowing them to skip

the opinion update step at each iteration. We use this approach

in our experiments. More generally, as in the Friedkin-Johnsen

(FJ) model [18], we could express by (1 −ξi),ξi∈[0,1],

the extent to which each agent is stubborn about her initial

opinion: Xi,t+1 =ξi(“opinion update Eq. 2”)+ (1−ξi)Xi,0.

Another direction to extend the model is to introduce time-

dependency to some of its elements. For instance, the βi’s and

the graph structure may evolve with time, however, we suppose

this takes place in a much longer time-scale compared to

shorter-term opinion formation dynamics (e.g. those observed

in social media), and therefore can be ignored. Besides, an

individual λi,t for each agent icould represent effects such

as her engagement in the debate over time and saturation

of interest. That would be an attractive feature, yet in this

work we choose to keep the model simpler by assuming

homogeneity across the population and no temporal variation,

thus ∀i,t, λi,t =λ. Since events are proportional to the agent’s

opinion value (deﬁned to be around 0), a notable implication

of the chosen setting is that all opinions need to get very

negative for no events to be generated (e.g. at the beginning or

the end of an information spread). In that sense, the opinion

value Xi,t should be perceived as a combination of agent’s

opinion and her interest to participate to the associated debate.

Contrary, a time-dependent λi,t would make ﬁtting to real

data more complex, but it would also allow agents to seize

generating events while remaining with non-negative opinions.

It would be interesting to also combine local features, such as

psychological factors that vary individually an agent’s behavior

throughout the process (e.g. the tolerance proposed in [36]),

with the GSM-DeGroot that emphasizes large-scale effects.

III. TECHNICAL RESULTS

A. Distinct properties of the opinion propagation and the

global steering mechanisms

GSM-DeGroot’s opinion update rule (see Eq. 2), incorporates

formally two mechanisms. The second term corresponds to

the OPM’s effect on agent ithrough direct social inﬂuence.

The ﬁrst term corresponds to the GSM’s effect, subject to the

agent’s βireaction to it. Next, we discuss the distinct prop-

erties of these two mechanisms when considered separately,

along with the default EGM. We show that each of them

exhibits stereotypical behavior with a clear role: the OPM acts

as a converging force, whereas the GSM acts as a polarizing

force. All technical proofs are provided in Appendix A.

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1DeGroot-basedopinionformationunderaglobalsteeringmechanismIvanConjeaud∗⋄PhilippLorenz-Spreen+ArgyrisKalogeratos∗†∗CentreBorelli,ENSParis-Saclay,Gif-sur-Yvette,France⋄ParisSchoolofEconomics,Paris,France+MaxPlanckInstituteforHumanDevelopment,Berlin,GermanyAbstract—Thispaperinvestigateshowinteractinga...

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