# On the role of zealots in a best-of-n problem on a heterogeneous network

Thierry Njougouo  Namur Institute for Complex Systems (naXys), University of
Namur, Namur, Belgium  Andreagiovanni Reina  IRIDIA, Université Libre de
Bruxelles, Brussels, Belgium  Elio Tuci  Namur Institute for Complex Systems
(naXys), University of Namur, Namur, Belgium  Timoteo Carletti  Namur
Institute for Complex Systems (naXys), University of Namur, Namur, Belgium
Department of Mathematics, University of Namur, Namur, Belgium

###### Abstract

Both humans and social animals live in groups and are frequently faced to
choose between options with different qualities. When there are no leader
agents controlling the group decision, consensus can be achieved through
repeated interactions among group members. Various studies on collective
decision-making illustrate how the dynamics of the opinions are determined by
the structure of the social network and the methods that individuals use to
share and update their opinion upon a social interaction. In this paper, we
are interested in further exploring how cognitive, social, and environmental
factors interactively contribute to determining the outcome of a collective
best-of-$n$ decision process involving asymmetric options, i.e., different
costs and/or benefits for each option. We propose and study a novel model
capturing those different factors, i) the error in processing social
information, ii) the number of zealots (i.e., asocial agents who never change
their opinion), iii) the option qualities, iv) the social connectivity
structure, and v) the degree centrality of the asocial agents. By using the
heterogeneous mean-field approach, we study the impact of the above-mentioned
factors in the decision dynamics. Our findings indicate that when susceptible
agents, i.e., individuals who change their opinion to conform with others, use
the voter model as a mechanism to update their opinion, both the number and
the degree of connectivity of the zealots can lead the population to converge
towards the lowest quality option. Instead, when susceptible agents use
methods more cognitively demanding, the group is marginally impacted by the
presence of zealots. The results of the analytical model are complemented and
extended by agent-based simulations. Our analysis also shows that the network
topology can modulate the influence of zealots on group dynamics. In fact, in
homogeneous networks where all nodes have the same degree (numbers of
neighbours), any location of the zealots has similar impact on the group
dynamics. Instead, when the network is heterogeneous, our simulations confirm
the model predictions that show that placing the zealots in the network hubs
(nodes with several neighbours) has a much larger impact than placing them in
lower-degree nodes.

††preprint: APS/123-QED

##  I Introduction

Many decisions characterising the social life of humans as well as of other
social species are the consequence of information exchange among several
individuals and can be considered to be complex processes. In spite of the
fact that the outcomes of these decisions influence the life of all
individuals of a community, in some cases the decision process is governed by
a limited number of dominant individuals (e.g., political leaders in humans,
or socially dominant individuals in primates). In other cases, the decisions
are genuinely collective, meaning that they are the results of social
interactions between peers, and are characterised by the fact that the
decision made is not attributable to any individual of the group [10]. For
instance, primate groups like chimpanzees and baboons engage in a collective
decision-making process to determine their direction after a period of rest
[48, 50, 54]. Similarly, a flock of birds collectively decides when to leave a
foraging patch [13, 5] a swarm of honeybees, during reproductive swarming,
decides on the site where to build their new nest [4, 44, 45, 36].

Scenarios requiring collective decisions have been studied in different
scientific disciplines and with different methods, such as experimental
methods [10, 9, 15], computational modelling and simulations methods [19, 3,
21] and social network analysis [7, 51, 46]. Other study [53] provided a
descriptive framework for collective decision-making processes based on a
costs/benefits analysis for sampling and selecting the available options. In
particular, they distinguished between symmetric conditions (i.e., same costs
and/or same benefits for each option) and asymmetric conditions (i.e.,
different costs and/or different benefits for each option). A series of work
have shown that consensus for the best alternative can be achieved even when
individuals operate without a leader and make noisy individual estimates of
the option’s qualities (benefits/costs). Indeed, to make collective best-of-n
decisions, it is sufficient that the individuals use the estimated quality to
modulate the frequency (or persistence) with which they share their opinion.
This mechanisms has been observed in group-living animals, e.g., ants and
honeybees [44], and then employed to design artificial distributed systems,
such as robot swarms [29, 35]. There are multiple causal factors, interacting
in a complex way, that contribute to determining the outcome of collective
decision-making processes. Among these causal factors, a significant share of
research works has focused on the behaviours that each individual follows to
develop and update her opinion. It has been shown that decision-makers with
limited perceptual and cognitive resources can follow simple rules to make, as
a group, complex collective decisions [40, 41, 37]. Among these simple rules,
there are those based on social feedback, like the voter model [23, 27] where
each individual simply copies the opinion of a randomly selected individual
within her social network, and the majority model [14, 19] where each
individual selects the option held by the majority of the individuals in her
social network. Let us observe that the error in processing social information
in the former case of sampling a single randomly chosen opinion is larger than
in the latter case of aggregating all available opinions.

This study aims to unravel the mutual influence of psychological, social, and
environmental factors on the outcome of a collective decision-making process.
It focuses specifically on the best-of-$n$ scenario with $n=2$ options and an
asymmetric condition, i.e., the two options have different costs and/or
different benefits [11, 52, 42, 33]. The best-of-$n$ problem requires
selecting which option, out of $n$ available ones, is the best alternative
[53]. Considerable work has been dedicated to the study of the best-of-$n$
problem, in order to unveil opinion dynamics characterising a wide range of
phenomena, from political polarisation [6], to the spread of rumours [12]
misinformation [22]. We focus on an asymmetric best-of-$2$ scenario since we
intend to investigate which factors among those that we modelled hinder the
group from a consensus on the best option. In particular, we provide an
analysis of the combined effects of the following parameters (which we define
in detail in the next paragraph): i) the pooling error, ii) the number of
zealots present in the population, iii) the option qualities, iv) the social
network structure, and v) the degree centrality of zealot agents.

The pooling error reflects the mistakes introduced by individuals when
aggregating the opinions of others. In the given context, these errors may be
influenced by the amount of cognitive effort individuals invest in the pooling
process. In our model, the pooling error is correlated to the number of group
members that each agent samples to develop her own opinion [38]. One can
recast such error into a cognitive load, namely the lower the cognitive load
the larger the pooling error. For example, an agent can face limited cognitive
resources and update her opinion based on a peer randomly chosen within her
social network (i.e., the voter model). On the other hand, the cognitive load
increases when the agent samples a larger number of peers within her social
network (e.g., the majority model), and correspondingly the pooling error will
decrease. In our model, we consider a heterogeneous population, composed of
individuals that update their opinions using different rules. While in a
homogeneous population, all individuals use the same rule to select and update
their opinions, in a heterogeneous population there are sub-groups of
individuals that follow different rules. In particular, we consider a
heterogeneous population composed of two groups of individuals, susceptible
and zealots. Susceptible individuals at each time step update their opinion
based on the opinions of all (or a subset) of their peers in their social
network. Zealots are asocial individuals that never change their opinion [24,
18, 22, 39]. The option quality refers to combination the cost and benefit
that an agent receives from selecting an option. Our scenario is asymmetric
because the two options have different qualities. The social network structure
refers to the network that describes how decision-makers are connected and how
information flows between them [26]. Finally, the degree centrality of an
agent is the number of her neighbours, which are other agents in her social
network. Degree centrality has been defined in network science as one of the
measures to quantify the importance of a node within a network and it has
often used to understand the role and influence of decision-makers in social
networks [17, 25]. Nodes with a higher degree centrality are considered more
influential because they have a greater number of direct connections, enabling
them to spread information or exert their influence more efficiently within
the network.

The parameters mentioned above have been separately studied in previous
research works. For example, Moeinifar and Gündüç [25] investigates the effect
of the position and the number of zealots on the consensus formation and time
to reach consensus in both random and scale-free networks in a symmetric
opinions scenario. The results show that the degree centrality of the zealots
has a great impact on the time required for the group to achieve a consensus.
Other studies have shown that, in different quality options scenarios, the
asymmetry facilitates the formation of consensus towards the option with the
higher quality (or lower cost) [29, 36, 20]. In this paper, we conduct an
analysis of a model that we recently proposed [38] which allows generalising
popular previous models of collective decision-making, from models with high
cognitive requirements (such as the majority model) to models requiring low
cognitive load (e.g., the voter model). There are several works that
investigated the dynamics of the voter model [49, 47, 34, 43] or the majority
model [19, 14, 16, 28, 42] on networks. The model we presented in [38] allows
to interpolate between the two models by changing the (continuous) value of
the pooling error, showing the existence of a speed-accuracy trade-off. In
this work, we extend the model of [38] and its analysis by considering
heterogeneous populations that comprise susceptible and zealot agents. From
the methodological side, we make use of mathematical equations (hereafter, the
analytical model) to provide general insights and predictions on the decision
process outcomes, and of numerical simulations of an agent-based model (ABM),
to corroborate the predictions of the analytical model. By looking at
populations with different numbers of zealots holding the lowest quality
option, we show interesting correlations between the cognitive load, the
number of zealots in the population and their degree centrality, in
differently connected populations. In particular, we identify the conditions
under which zealots favouring the lowest-quality option manage to
counterbalance the quality difference and drive the population toward a
consensus on the lowest-quality option. We also demonstrate the effect of
network heterogeneity, in particular its sparsity, on the conditions for
achieving consensus towards the low-quality option.

The rest of this work is organised as follows. In Section II, we describe the
agent-based model with social interactions occurring on a scale-free network
and in Section III we present the results of the numerical simulation of this
agent-based model. Section IV presents the mathematical model defined by an
ordinary differential equation (ODE) allowing us to study the evolution of
group opinion taking into account multiple parameters such as the pooling
error (cognitive load), the heterogeneity of the population, the option
qualities, the connectivity structure, and the degree centrality of the zealot
agents. In Section V, we show the numerical results of this analytical model
showing the combined effects of the previous parameters. Finally, we present
our conclusions in Section VII.

##  II The Agent-Based Model

The aim of this section is to introduce the agent-based model, rooted on the
parameters previously introduced and to discuss their impact on the system
outcome. We consider the $N$ agents to be the nodes of a scale-free network
composed of $L$ undirected edges, each one representing a possible social
interaction between two agents and thus a channel to acquire/share
information. We assume the number of neighbours of a generic node $i$ (i.e.,
its degree) follows a power-law distribution [1, 30], $p_{k}\sim
1/k^{\gamma}$, where $\gamma>2$. Note that the closer $\gamma$ to $2$ the more
heterogeneous the degree distribution. Nodes with a very large degree can be
observed since $\langle k^{2}\rangle$ becomes unbounded as $\gamma\rightarrow
2$. When $\gamma\gg 3$ the network becomes sparse, very high degree nodes are
very rare and the degree spread is well described by finite variance of the
degree distribution. The networks we consider are simple (i.e., at most one
edge can connect any two nodes) and connected (i.e., starting from any node it
is possible to reach any other node by traversing the network using the
available links).

We classify nodes into susceptible (i.e., representing agents capable of
changing their opinions) and zealots (i.e., representing agents that never
change their opinion). We model a best-of-$n$ problem with $n=2$ options where
each node can hold either opinion $A$ or opinion $B$. We also associate to
each option a quality, $Q_{A}>0$ for opinion $A$, and $Q_{B}>0$ for opinion
$B$. The quality defines the strength or the probability with which the option
is communicated to the neighbours. Without lack of generality, in the rest of
the work, we assume $Q_{A}=1$, $Q_{B}\leq Q_{A}$ and hence the quality ratio
$Q:=Q_{B}/Q_{A}\leq 1$. In this work, zealots hold the opinion with the lowest
quality option (i.e., opinion $B$), since we are interested in investigating
under which circumstances they can hinder the population from reaching a
consensus (i.e., a general agreement) on the option with the best quality.

The system evolves asynchronously: at each time step one susceptible
individual $i$ is randomly selected with uniform probability from the
population. This agent has a probability $P_{\alpha}(x_{i})$ as defined in Eq.
(1), of changing her opinion.

$P_{\alpha}(x_{i})=\begin{dcases}\frac{1}{2}-\frac{1}{2}\left(1-2x_{i}\right)^{\alpha}&\text{if
$0\leq x_{i}\leq\frac{1}{2}$}\\\
\frac{1}{2}+\frac{1}{2}\left(2x_{i}-1\right)^{\alpha}&\text{if
$\frac{1}{2}<x_{i}\leq 1$}\end{dcases}\,,$ (1)

where, $x_{i}$ is the weighted fraction of agents connected to agent $i$
holding opinion $A$ or $B$, where the weight is given by the quality ratio
$Q$. The parameter $\alpha\geq 0$ is the pooling error (and thus is negatively
correlated with the cognitive load); for increasing $\alpha$, agents make
larger errors because they use less resources to sample the opinions of their
neighbouring agents. More specifically, when $\alpha=0$, from Eq. (1) it
follows that agents do not make any sampling error because they change their
opinions based on the weighted average of all their neighbours, corresponding
thus to the majority model. This action requires a higher cognitive load
compared to the case where $\alpha>0$. For instance, for $\alpha=1$, agents
change their opinions by copying the opinion of a randomly selected neighbour,
replicating thus a (weighted) voter model. Therefore, by varying $\alpha$,
$P_{\alpha}$ is capable of generalising the two opinion selection mechanisms
existing in the literature, i.e., the voter model (see Fig. 1, case
$\alpha=1.0$) and the majority model (see Fig. 1, case $\alpha=0.0$). The
function of Eq. (1) interpolates the cognitive load level in the form of
pooling error between the two models in a continuous way (e.g., $\alpha=0.5$).
Additionally, it can also model the case of high pooling errors where the
changes are made approximately (e.g., Fig. 1, case $\alpha=1.5$). Using Eq.
(1), we investigate the effect of the pooling error (cognitive load) on the
best-of-$2$ problem.

Figure 1: Probability function $P_{\alpha}(x_{i})$ for the agent $i$ to change
opinion according to the weighted fraction $x_{i}$ of neighbours holding
opinion $A$ for several values of the pooling error parameter, $\alpha$. For
$\alpha=0$, our model aligns with the majority model model and for $\alpha=1$,
our model aligns with the voter model. In the range $0<\alpha<1$, our model
interpolates between the two models, while for $\alpha>1$, agents make large
pooling errors and change their opinion with little regard for others’
opinions.

For the randomly selected susceptible agent $i$, the quality-weighted
proportion of neighbours holding opinion $A$ (i.e., $n_{i,A}^{\\#}$), and the
quality-weighted proportion of neighbours holding opinion $B$ (i.e.,
$n_{i,B}^{\\#}$) are computed as follows:

$n_{i,A}^{\\#}=\frac{Q_{A}n_{i,A}}{Q_{A}n_{i,A}+Q_{B}n_{i,B}}\text{ and
}n_{i,B}^{\\#}=\frac{Q_{B}n_{i,B}}{Q_{A}n_{i,A}+Q_{B}n_{i,B}}\,,$ (2)

where $n_{i,A}$ and $n_{i,B}$ are the number of neighbours of agent $i$
holding opinion $A$ and $B$, respectively. $n_{i,A}^{\\#}+n_{i,B}^{\\#}=1$,
$\forall i$. Let us denote by $k_{i}=n_{i,A}+n_{i,B}$ the degree of agent $i$.
We can rewrite Eq. (2) as follows:

$n_{i,A}^{\\#}=\frac{n_{i,A}/k_{i}}{(1-Q)n_{i,A}/k_{i}+Q}\quad\text{and}\quad
n_{i,B}^{\\#}=1-n_{i,A}^{\\#}\,.$ (3)

To distinguish between susceptible voters with opinion $B$ and zealots, let us
introduce $z_{k}$ to represent the number of zealots with opinion $B$ and with
a degree $k$. Therefore, the total number of neighbours of agent $i$ holding
opinion $B$ can be written as $n_{i,B}=s_{i,B}+z_{i,B}$ and then,
$k_{i}=n_{i,A}+s_{i,B}+z_{i,B}$.

In the process of changing opinions, if the selected agent $i$ is susceptible
and has opinion $A$, then with probability $P_{\alpha}(n_{i,B}^{\\#})$ the
agent changes her mind and commits to opinion $B$, and with probability
$1-P_{\alpha}(n_{i,B}^{\\#})$ she remains committed to opinion $A$. If instead
the agent has opinion $B$, then with probability $P_{\alpha}(n_{i,A}^{\\#})$
the agent changes her mind and commits to opinion $A$, and with probability
$1-P_{\alpha}(n_{i,A}^{\\#})$ she remains committed to opinion $B$. Let us
observe that because of the functional form of Eq. (1) and because
$n_{i,A}^{\\#}+n_{i,B}^{\\#}=1$, we can conclude that
$P_{\alpha}(n_{i,A}^{\\#})+P_{\alpha}(n_{i,B}^{\\#})=1$.

##  III Numerical results of the agent-based-model

The objective of this section is to present the results of the numerical
simulation of the agent-based model with social interactions occurring on a
scale-free network. In this section, we limit our analysis to studying the
combined effects of three factors: i) the number of zealots ($z$), ii) their
location in the network, and iii) the pooling error ($\alpha$), while we keep
constant the network structure and the option’s quality ratio
$Q=Q_{B}/Q_{A}=0.9$.

We consider a system composed of $N=1000$ agents whose social interactions are
described by a scale-free network obtained by using the Barabási-Albert
algorithm [1] with parameter $m=8$ (i.e., a network in which the minimum
degree is $k_{min}=8$). The exponent of the power law of such a network is
$\gamma=3$. In our experimental plan, we have $20$ different parameters
conditions obtained by considering $4$ different values of
$\alpha\in\\{0.0,0.5,1.0,1.5\\}$, and $5$ different quantities of zealots,
$z\in\\{0,32,64,128,150\\}$. We consider cases where the zealots, committed to
opinion $B$, have similar locations in the social network. That is, they can
be peripheral agents (i.e., associated with nodes with the smallest degree
$k_{\text{min}}=8$), or agents associated to nodes with an intermediate
degree, or a large degree. The opinion qualities have been fixed to $Q_{A}=1$
and $Q_{B}=0.9$. The initial opinions are randomly distributed with half of
the agents holding opinion $A$ ($n_{A}(0)=N/2$) and half opinion $B$
($n_{B}(0)=N/2$), and the maximal simulation time has been set to
$T_{max}=400\,000$. Each experimental plan is repeated $30$ times by changing
each time the seed of the pseudo-random number generator.

Figure 2: We report the average final fraction of agents with opinion $A$ in
the population of $N=1000$, interacting on a Barabási-Albert network with
$m=8$, and the standard deviation (yellow line), resulting from $30$
independent simulation runs. Different values of the pooling error $\alpha$
are considered (on the x-axis) together with a varying number of zealots
(different colours). zealots are committed to opinion $B$ are set into leaves
nodes, i.e., $k_{i}=8$. $Q_{A}=1$ and $Q_{B}=0.9$ for options $A$ and $B$,
respectively.

In Fig. 2, we report the results of the numerical simulations performed by
adopting the strategy of introducing zealots with low degree centrality. More
precisely we illustrate the average final fraction of susceptible agents
holding opinion $A$ computed over $30$ independent replicas together with the
standard deviation (vertical yellow lines). In the absence of zealots (blue
bars, $z=0$, in Fig. 2), the system always converges to a consensus toward
opinion $A$ for $\alpha\in\\{0.0,0.5,1.0\\}$. Indeed the standard deviations
associated to those values of $\alpha$ are zero, thus, the yellow lines reduce
to points. On the other hand, for $\alpha=1.5$, and still $z=0$, the group
faces a decision deadlock where the population splits into two coexisting
groups of agents, one having opinion $A$ and the other opinion $B$, the
average fraction of agents committed to $A$ is $0.5$ and the standard
deviation is relatively small. In other words, due to high pooling error
($\alpha>1$), agents fail to coordinate with each other and the population
remains polarised and unable to reach a consensus for any alternative,
especially when the quality difference between the two options is small. With
the introduction of zealots in the population, for
$\alpha\in\\{0.5,1.0,1.5\\}$, we notice that the increase in the number of
zealots lowers the final proportion of susceptible agents with opinion $A$. In
the cases $\alpha\in\\{0.5,1.0\\}$, a sufficiently large number of zealots
($z=150$) is even able to drive consistently the system toward a full
consensus to opinion $B$ (observe that in Fig. 2 the red bars for $\alpha=0.5$
and $\alpha=1.0$ vanish, and the standard deviation is reduced to zero). For
$\alpha=1.5$, agents with opinion $A$ persist in the population despite the
presence of (many) zealots, this is due to the large pooling error that keeps
the population polarised and deadlocked at indecision, unable to choose one
opinion over the other. This last observation holds true even for a large
number of zealots (see the cases $z=128$ and $z=150$), the average fraction of
$A$ agents decreases and the standard deviation increases, indicating a larger
variability among the different runs. For $\alpha=0.0$, indicating the absence
of pooling error (majority model), the existence of zealots does not change
the outcomes observed without the presence of zealots (see all the non-blue
bars in Fig. 2), again we can observe a null standard deviation.

To study the combined impact of the number of zealots and their location in
the social network, we have performed a further series of simulations in order
to verify whether the position of the zealots within a scale-free network of
$N=1000$ nodes, generates a similar effect to the one observed with respect to
their number. We considered $60$ different experimental conditions, given by
$5$ different amount of zealots, $z\in\\{0,1,4,8,16\\}$, $3$ values of
$\alpha\in\\{0.5,1.0,1.5\\}$, and $4$ different strategies to locate zealots
in the network according to agent’s degree $k$. With the first strategy,
zealots correspond to nodes with degree $k_{min}=8$. With the second strategy,
zealots correspond to nodes with a degree $15\leq k\leq 25$. With the third
strategy, zealots correspond to nodes with degree $35\leq k\leq 45$. Finally,
with the fourth strategy, zealots correspond to nodes with a degree $k\geq
60$. Note that, in the last three strategies, we first associate zealots with
nodes with highest degree and we progressively select nodes with lower degree
only once there are no more available nodes with the larger degree in the
network. As in the previous set of simulations, the quality of option $A$ is
set to $Q_{A}=1$, and that of opinion $B$ to $Q_{B}=0.9$; the initial opinions
are randomly distributed with half of the agents holding opinion $A$
($n_{A}(0)=N/2$) and half opinion $B$ ($n_{B}(0)=N/2$). We repeated each
experiment $30$ times by using different random seeds, and each simulation
lasts for $T_{max}=400\,000$ time steps.

The results of these simulations are shown in Fig. 3, where we report the
average fraction of agents holding opinion $B$ at the end of the simulations,
and the associated standard deviations (vertical lines) for each considered
value of $z$. For $\alpha=0.5$ (see Fig. 3a) and for $\alpha=1.0$ (see Fig.
3b), we notice two clear trends: first, with the exception of $z=1$, for any
given number of zealots within the population, the higher their degree the
higher the final proportion of susceptible agents terminating with opinion
$B$. Second, for zealots with a given degree centrality, the higher the number
of zealots, the higher the final proportion of susceptible agents terminating
with opinion $B$. The increase in the proportion of susceptible agents
committed to opinion $B$ at the end of the simulations appears to be non-
linearly correlated with the number of zealots. These trends indicate that, in
an asymmetric best-of-$2$ problem, given a fixed number of zealots committed
to the lowest quality option, their capability to influence the opinion
dynamics of susceptible agents using opinion selection strategies with either
small or large pooling error, is determined by both their quantity within the
population and their degree centrality. Note that, for the values of the
considered parameters, when zealots correspond to small degree nodes,
approximately $z=150$ zealots are required to converge to a consensus on the
lowest quality opinion $B$, while for the same values, at least $z=128$
zealots are needed in the case of $\alpha=1$ (see Fig. 2). In the same vein,
in the case of $\alpha=1.5$ (see Fig. 3c) for any number of zealots, and
regardless of their degree, the population always splits into two groups of
(almost) equal sizes with different opinions. As previously stated, the
standard deviation increases with the number of zealots and with the pooling
error.

---
(a)
(b)
(c)
Figure 3: We report the average final fraction of agents with opinion $B$ in
the population of $N=1000$, interacting on a Barabási-Albert network with
$m=8$, and the standard deviation (yellow line), resulting from $30$
independent simulation runs. Panel (a) corresponds to pooling error
$\alpha=0.5$, panel (b) to $\alpha=1.0$, and panel (c) to $\alpha=1.5$. The
colours of the bars refer to zealots with different degrees of centrality (see
text). $Q_{A}=1$ and $Q_{B}=0.9$ for options $A$ and $B$, respectively.

The results shown in this section indicate that adding zealots with the lower
quality option can drive the system to a consensus toward the latter. More
importantly, this outcome is generated by a smaller number of zealots if they
have a higher degree. The analysis has been performed on a scale-free network
obtained by using the Barabási-Albert algorithm. The goal of the next section
is to move a step forward in confirming such conclusions in the whole
generality in the framework of scale-free networks by using the Heterogeneous
Mean-Field theory.

##  IV Mathematical model

We build a mathematical model defined by an ordinary differential equation
(ODE) to look at the combined effects of the pooling error, the ratio of the
opinion qualities $Q:=Q_{B}/Q_{A}$, the fraction of zealots, and the network
structure $\gamma$, on the evolution of opinion dynamics in the best-of-$2$
problem.

To make some analytical progress we rely on the Heterogeneous Mean Field
assumption [32, 31, 8] (HMF), namely we hypothesise that nodes with the same
degree are dynamically equivalent. Therefore, nodes are grouped into degree
classes, more precisely we define $A_{k}$ (resp. $B_{k}$), as the number of
nodes with degree $k$ and opinion $A$ (resp. opinion $B$). To distinguish
between susceptible agents with opinion $B$ and zealots, we introduce $Z_{k}$
to denote the number of zealots with opinion $B$ and degree $k$. Therefore, by
letting $N_{k}$ to denote the total number of nodes with degree $k$, we have:

$A_{k}+Z_{k}+S_{k}=N_{k}\,,$ (4)

where $S_{k}$ denotes the number of susceptible agents having opinion $B$ and
degree $k$. $a_{k}=A_{k}/N_{k}$ is the fraction of agents having opinion $A$
and degree $k$; $b_{k}=S_{k}/N_{k}$ is the fraction of susceptible agents
having opinion $B$ with degree $k$, and $\zeta_{k}=Z_{k}/N_{k}$ the fraction
of zealots with degree $k$. Therefore, for all $k$,

$a_{k}+b_{k}+\zeta_{k}=1\,.$ (5)

The goal of the HMF is to derive an ODE ruling the evolution of $a_{k}$ and
$b_{k}$. In particular, we extend the model recently developed in [38] by
adding zealots; in this way, we obtain the analytical model described by

$\displaystyle\frac{d\langle a\rangle}{dt}$ $\displaystyle=$
$\displaystyle-\langle
a\rangle+\sum_{k}q_{k}(1-\zeta_{k+1})\sum_{\ell=0}^{k+1}\binom{k+1}{\ell}\langle
a\rangle^{k+1-\ell}\times$ (6) $\displaystyle\left(1-\langle
a\rangle\right)^{\ell}P_{\alpha}\left(\frac{k+1-\ell}{k+1-\ell+\ell
Q}\right)\,,$

where we define $\langle a\rangle=\sum_{k}q_{k}a_{k+1}$, being $q_{k}$ the
probability for a node to have an excess degree $k$, namely

$q_{k}=\frac{\left(k+1\right)p_{k+1}}{\langle k\rangle}\quad\forall k\geq
0\,,$

with $\langle k\rangle=\sum_{k}kp_{k}$ the average node degree and $p_{k}$ the
probability a generic node has degree $k$. Eq. (6) contains the relevant
parameters of the model, the zealots ($\zeta_{k}$), the model of opinion
dynamics ($P_{\alpha}$) incorporating the pooling error $\alpha$, the opinion
quality ratio ($Q$), and the network structure ($q_{k}$). The aim of the next
subsection is to determine the equilibria of the HMF equation and their
stability, and thus the system fate. Let us observe that, similarly to what we
did in [38], by knowing $\langle a\rangle(t)$ from Eq. (6) we can obtain the
evolution of $a_{k}$ for all $k$ by using the following equation:

$\displaystyle\frac{da_{k}}{dt}$ $\displaystyle=$ $\displaystyle-
a_{k}+\left(1-\zeta_{k}\right)\sum_{\ell=0}^{k-1}\left(\begin{array}[]{c}k\\\
\ell\\\ \end{array}\right)\langle a\rangle^{k-\ell}\left(1-\langle
a\rangle\right)^{\ell}\times$ $\displaystyle
P_{\alpha}\left(\frac{k-\ell}{k-\ell+\ell Q}\right)\,.$

###  IV.1 Equilibria of the analytical model and their stability

The equilibria of the system are obtained by setting the right-hand side of
Eq. (6) equal to zero. Let us thus define the function $f_{\alpha}(a)$

$\displaystyle f_{\alpha}(a):$ $\displaystyle=$ $\displaystyle-\langle
a\rangle+\sum_{k}q_{k}(1-\zeta_{k+1})\sum_{\ell=0}^{k+1}\binom{k+1}{\ell}\langle
a\rangle^{k+1-\ell}\times$ (8) $\displaystyle\left(1-\langle
a\rangle\right)^{\ell}P_{\alpha}\left(\frac{k+1-\ell}{k+1-\ell+\ell
Q}\right)\,,$

hence by denoting $\langle a^{*}\rangle$ a system equilibrium, we have by
definition

$f_{\alpha}(\langle a^{*}\rangle)=0\,.$

A direct inspection of Eq. (8) allows to demonstrate that $f_{\alpha}(0)=0$,
hence $\langle a^{*}\rangle=0$, i.e., the absence of agents holding opinion
$A$ represents an equilibrium of the system. Conversely,
$f_{\alpha}(1)=-\sum_{k}q_{k}\zeta_{k+1}\neq 0$, signifying that the presence
of zealots (with opinion $B$) prevents the system from converging to a
situation where all susceptible agents hold opinion $A$. Finally, the
existence of a nontrivial solution $0<\langle a^{*}\rangle<1$ to the equation
$f(\langle a^{*}\rangle)=0$ indicates an equilibrium for the coexistence of
opinions $A$ and $B$ in the network.

The stability of those equilibria can be determined by evaluating the
derivative of the function $f_{\alpha}$ at those points. This analysis will be
detailed in the following section, where we will also explore the effects of
the key model parameters.

##  V Results of the HMF model

In this section, we present the results obtained for the analytical model
described in the previous section. As previously noted, our focus lies on the
influence of the pooling error $\alpha$, the network structure encapsulated
into the exponent $\gamma$ of the power law, and the ratio of zealots present
in the population and their degree centrality, more precisely if they sit onto
hubs or leaves nodes. To place zealots in hubs, we set $\zeta_{k}=1$ for all
$k\geq k_{M}$, for some sufficiently large $k_{M}>0$; this accounts to add
into the model an average number of zealots equal to $Z_{tot}=\sum_{k\geq
k_{M}}N_{k}\sim\sum_{k\geq k_{M}}Nc_{\gamma}/k^{\gamma}$, where $c_{\gamma}$
is a normalisation constant such that $\sum_{k}p_{k}=1$ and $N$ is the total
number of nodes in the network. In the scenario where zealots are assumed to
be on leaf nodes, for a fair comparison with the prior condition, we maintain
the same number of zealots as positioned in the hubs. This is achieved by
assuming $\zeta_{k_{min}}=Z_{tot}/N_{k_{min}}$, where $k_{min}>0$ represents a
sufficiently small degree. Specifically:

$\displaystyle\zeta_{k_{min}}$ $\displaystyle=$
$\displaystyle\frac{Z_{tot}}{N_{k_{min}}}\sim\frac{Z_{tot}}{Np_{k_{min}}}$
$\displaystyle=$ $\displaystyle k_{min}^{\gamma}\sum_{k\geq
k_{M}}\frac{1}{k^{\gamma}}\sim\frac{k_{M}}{\gamma-1}\left(\frac{k_{min}}{k_{M}}\right)^{\gamma}\,.$

Let us observe that the above strategy implies that $\zeta_{k}$ can be
interpreted as the probability to find a zealot into the class of nodes with
degree $k$; the final number of added zealots will be always finite, indeed in
any network realisation, e.g., by using the configuration model, there is a
finite number of nodes with degree larger than $k_{M}$ and thus $Z_{tot}$ is
also a finite quantity.

Fig. 4 summarises our main results. The quality of options $A$ and $B$ are set
as in the previous section, i.e., $Q_{A}=1$ and $Q_{B}=0.9$, resulting in
$Q=Q_{B}/Q_{A}=0.9$. We then proceed to vary the power law exponent $\gamma$
and the location of zealots in the network. Subsequently, we (numerically)
determine the zeros of the function $f_{\alpha}$ for varying values of
$\alpha$ within the range of $[0,2]$, enabling us to derive the system
equilibria. Once the latter have been found, we evaluate the derivative of
$f_{\alpha}$ and we determine its sign, if $f_{\alpha}(\langle
a^{*}\rangle)>0$ then the equilibrium $\langle a^{*}\rangle$ is unstable and
marked with red points in Fig. 4. On the other hand, if $f_{\alpha}(\langle
a^{*}\rangle)<0$ then the equilibrium $\langle a^{*}\rangle$ is stable and we
represent it in green. The two top panels (see Fig. 4(a) and (b)) refer to the
strategy consisting of setting the zealots in the leaves (here $k_{min}=8$),
and the two bottom panels (see Fig. 4(c) and (d)) refer to the opposite
strategy with the zealots in the hubs, $k_{M}=60$; this strategy is thus the
same of the one used for the ABM in Section III. Moving from left to right we
increase $\gamma$, passing from $\gamma=2.5$ (Fig. 4(a) and (c)), to
$\gamma=3.5$ (Fig. 4(b) and (d)).

Figure 4: Bifurcation diagrams of the HMF. We report the equilibria $\langle
a^{*}\rangle$ of Eq. (6) as a function of the pooling error $\alpha$ for
$Q=0.9$. Stable equilibria, i.e., associated to $f^{\prime}_{\alpha}(\langle
a^{*}\rangle)<0$, are coloured in green while unstable ones, i.e., associated
to $f^{\prime}_{\alpha}(\langle a^{*}\rangle)>0$, are coloured in red. Top
panels (a) and (b), correspond to zealots set into leaves nodes, i.e.,
$k_{i}=8$, while bottom panels (c) and (d) correspond to zealots set into the
large degree nodes, more precisely into nodes with $k_{i}\geq 60$. The
underlying support is a scale–free network, with $\gamma=2.5$ (panels (a) and
(c)) and $\gamma=3.5$ (panels (b) and (d)).

Several conclusions can be drawn from those results. For large enough values
of $\alpha$, the system always sets into a state where opinions $A$ and $B$
coexist, leading to a state of decision deadlock where the population is
unable to choose one opinion over the other. The larger is $\alpha$ the closer
is the stable equilibrium to $0.5$; this behaviour is independent of the
strategy used to place the zealots or the network structure determined by the
exponent $\gamma$. Hence, a too-large pooling error $\alpha$, and thus a small
cognitive load, prevents the agents from reaching a consensus for their
alternative regardless of the position of zealots.

For intermediate values of the pooling error $\alpha$, e.g., close to
$\alpha=1$, placing the zealots into the low-degree nodes allows the system to
reach a consensus for the best option (see panels 4(a) and 4(b)) irrespective
from the value of $\gamma$, indeed there is a stable equilibrium $\langle
a^{*}\rangle$ very close to $1$. This behaviour completely changes once
zealots are set into hubs nodes (see panels 4(c) and 4(d)), indeed for small
$\gamma$, i.e., a network with a pronounced degree heterogeneity, there exists
an interval of $\alpha\in[0.41,1.10]$ (see panel 4(c)) values for which the
unique stable equilibrium is $\langle a^{*}\rangle=0$, the HMF predicts thus
the system to converge to the opinion with lower quality, $B$. By increasing
$\gamma$, i.e., dealing with a more homogeneous network, we recover once again
convergence to a large fraction of agents committed to opinion $A$.

Finally, for very low values of the pooling error $\alpha$, i.e., $\alpha\sim
0$, placing zealots in low-degree nodes allows the system to reach a consensus
for the best option. Therefore, a stable equilibrium $\langle a^{*}\rangle\sim
1$ is achieved for any value of $\gamma$ (see panels 4(a) and 4(b)). However,
when these zealots are placed on hubs in a network with much greater
heterogeneity, the stable equilibrium representing a majority of agents with
opinion $A$ reduces to values lower than 1. Thus, while most agents adopt the
higher-quality opinion, there is also a significant proportion of agents with
opinion $B$. This is shown in Fig. 4(c) by the stable equilibrium point
$0.5<\langle a^{*}\rangle<1$. Furthermore, when the network structure becomes
more homogeneous (e.g., $\gamma=3.5$), the presence of zealots does not
prevent the system from reaching a consensus towards the higher-quality
option, thereby reaching the stable equilibrium point $\langle a^{*}\rangle=1$
(see panel 4(d)).

To achieve a broader understanding of the complex relationship between the
parameters, we studied the equilibrium $\langle a^{*}\rangle$ as a function of
the pooling error $\alpha$ and the exponent $\gamma$, while maintaining
constant the ratio of the opinion qualities $Q=0.9$ (we report the results in
Fig. 5). Furthermore, for each examined scenario, we assessed the influence of
the strategy involving the placement of zealots on leaf nodes (see Fig. 5(a)
and (b)) or on hubs nodes (see Fig. 5(c) and (d)). In Fig. 5(a) and (c) we
represented by a colour code (yellow high values of $\langle a^{*}\rangle$
close to $1$ and blue $\langle a^{*}\rangle\sim 0$) the equilibrium reached by
the system starting from an initial population with half agents holding
opinion $A$ and half opinion $B$ (note that half of agents holding opinion $A$
are all susceptible agents while the other half, holding opinion $B$, also
includes the zealots). One can observe a striking difference between Fig. 5(a)
corresponding to zealots placed into leaves nodes, here $k_{min}=8$, with
respect to the Fig. 5(c), where the zealots have been set into hubs nodes,
here $k_{M}\geq 60$. In the former case, the equilibrium $\langle
a^{*}\rangle$ is almost independent from $\gamma$ and the system exhibits two
main behaviours: for $\alpha\lesssim 1$ the whole group converges to a
consensus to $A$, while for $\alpha\gtrsim 1$ the population faces a deadlock
where agents with opinion $A$ and $B$ coexist. On the other hand, once zealots
are placed into hubs nodes a third type of dynamics can manifest (see Fig.
5(c)): the population can converge toward a consensus for the opinion with the
lower quality. As shown in Fig. 5(c)), this happens for $\alpha\sim 1$,
$Q=0.9$, and for $\gamma\lesssim\gamma_{*}=3.31$ (the value of $\gamma^{*}$
has been computed numerically). To better emphasise this behaviour, we report
in Fig. 5(b) and (d) the position and the stability of the equilibrium
$\langle a^{*}\rangle$ as a function of the pooling error for the value
$\gamma=2.2<\gamma_{*}\sim 3.31$. Fig. 5(b) corresponds to the case where
zealots are set into the leaves nodes and the population converges to a
(almost) consensus to $A$ for $\alpha\lesssim 1.0$, while for larger values of
$\alpha$ the population faces a decision deadlock. On the other hand, once
zealots are set into hubs (Fig. 5(d)), there exists an interval of values of
$\alpha$ for which the group chooses the opinion with the lower quality. Those
results support the claim that a population of agents adopting a voter model
strategy for social exchange can be driven to adopt the opinion with the lower
quality, by zealots placed into hubs of a sufficiently heterogeneous scale-
free network, i.e., $\gamma<\gamma_{*}$.

Figure 5: Bifurcation diagrams of the HMF. We report the equilibrium $\langle
a^{*}\rangle$ given by Eq. (6) as a function of $(\alpha,\gamma)$ for a fixed
value of $Q=0.9$. Top panels (a) and (b) correspond to the strategy of placing
zealots into leaves nodes, i.e., $k_{i}=8$, while in the bottom panels (c) and
(d), zealots are assigned to large degree nodes, more precisely to nodes such
that $k_{i}\geq 60$. In panel (c) we can observe the existence of a critical
value $\gamma_{*}$ of the power-low exponent $\gamma$, below which the
equilibrium $\langle a^{*}\rangle=0$ is the unique stable one, for a given
range of $\alpha$. Panels (b) and (d) show the equilibria for a fixed value of
$\gamma=2.2<\gamma_{*}\sim 3.31$.

Fig. 6 shows that similar dynamics can be observed by considering equilibrium
$\langle a^{*}\rangle$ as a function of $(\alpha,Q)$ for a fixed value of
$\gamma$. The results presented in Fig. 6(b) show that in the case of zealots
located into hubs, for $Q$ close enough to $1$ (i.e., opinions with very
similar qualities) and $\alpha\sim 1$, the population converges to a majority
for the option with the lower quality; by decreasing $\alpha$, the system
undergoes an abrupt bifurcation passing to a population with a large majority
of agents holding an opinion in favour of the best quality. This behaviour
cannot be observed if zealots are placed into the leaves nodes (see Fig. 6
(a)). Results presented in panel 6(b) show the existence of a critical value
of $Q_{*}$ (obtained numerically) above which the system can converge toward
the lowest quality option for some values of $\alpha$.

Figure 6: Bifurcation diagrams of the HMF. We report the equilibrium $\langle
a^{*}\rangle$ given by Eq. (6) as a function of $(\alpha,Q)$ for a fixed value
of $\gamma=3.0$. Left panel (a) corresponds to zealots set into leaves nodes,
i.e., $k_{i}=8$, while the panel (b) to the strategy of placing the zealots
into the hubs, i.e., nodes with $k_{i}\geq 60$. Let us observe the presence in
panel (b) of a critical value of $Q=Q_{*}\sim 0.83$, below which the
equilibrium $\langle a^{*}\rangle=0$ is the unique stable one.

Our analytical framework confirmed the numerical results shown in Sec. III and
generalises the analysis to any scale-free network, under the assumption of
heterogeneous mean field.

##  VI How much quality is worth a zealot?

In the previous sections, we have considered the quality of the options as a
priori measure of the value of one choice over the other, or the strength with
which an option is transmitted. In the case of humans the latter role can be
played by, e.g., mass media penetration. In the same framework, zealots, can
thus be considered as individuals (or bots) repeating constantly one option
with the goal of influencing the population.

It can thus be interesting to consider the following problem: can a change in
option quality “compensate” the impact of the presence of zealots in the
decision outcome? Assume thus option $A$ to be better than option $B$, i.e.,
$Q_{A}>Q_{B}$, then we can find values of $\alpha\geq 1$ such that in absence
of zealots the system converges to an equilibrium
$a_{\mathrm{ini}}^{*}\lesssim 1$, namely the group achieve almost a total
consensus for the option with the highest quality [38]. Then a fraction of
zealots with opinion $B$, $0<\zeta<1$, is added to the population, and as
shown above, this induces a shift in the system equilibrium: the asymptotic
fraction of agents with opinion $A$ is smaller than without zealots,
$a_{\mathrm{tmp}}^{*}<a_{\mathrm{ini}}^{*}\lesssim 1$. The question we are
interested in is thus: would it be possible to increase the quality of opinion
$A$, $Q_{A}^{\prime}>Q_{A}$, to compensate for the presence of zealots and
restore a collective agreement to $a^{*}_{ini}$?

To answer this question we decided to simplify the model by replacing the
social network with an all-to-all coupling. This is not restrictive and allows
us to focus on the main point; a similar conclusion can be drawn also for a
generic social network. In this setting we have thus only three variables,
$a(t)$ the fraction of agents with opinion $A$, $b(t)$ the fraction of
susceptible agents with opinion $B$ and $\zeta$, the fraction of zealots
committed to opinion $B$.

In absence of zealots, we have previously shown [38] that the equilibrium
$a_{\mathrm{ini}}^{*}$ satisfies

$P_{\alpha}\left(\frac{a_{\mathrm{ini}}^{*}}{a_{\mathrm{ini}}^{*}(1-Q)+Q}\right)=a_{\mathrm{ini}}^{*}\,.$

The inclusion of zealots determines a new equilibrium, $a_{\mathrm{tmp}}^{*}$
solution of

$(1-\zeta)P_{\alpha}\left(\frac{a_{\mathrm{tmp}}^{*}}{a_{\mathrm{tmp}}^{*}(1-Q)+Q}\right)=a_{\mathrm{tmp}}^{*}\,.$

In order to have $a_{\mathrm{ini}}^{*}$ as a solution of an interacting group
that includes the same fraction $\zeta$ of zealots, we need to modify the
quality of option $A$ and thus define a new quality ratio
$Q^{\prime}=Q_{B}/Q_{A}^{\prime}$, to be able to solve

$(1-\zeta)P_{\alpha}\left(\frac{a_{\mathrm{ini}}^{*}}{a_{\mathrm{ini}}^{*}(1-Q^{\prime})+Q^{\prime}}\right)=a_{\mathrm{ini}}^{*}\,.$
(9)

Eq. (9) can be solved with respect to $Q^{\prime}$ to obtain the following
explicit expression

$\begin{cases}Q^{\prime}=\left[\frac{2}{\left(\frac{2a_{\mathrm{ini}}^{*}}{1-\zeta_{B}}-1\right)^{\frac{1}{\alpha}}+1}-1\right]\dfrac{a_{\mathrm{ini}}^{*}}{1-a_{\mathrm{ini}}^{*}}\quad\text{if
}\dfrac{a_{\mathrm{ini}}^{*}}{a_{\mathrm{ini}}^{*}(1-Q)+Q}>\frac{1}{2}\\\
Q^{\prime}=\left[\frac{2}{1-\left(1-\frac{2a_{\mathrm{ini}}^{*}}{1-\zeta_{B}}\right)^{\frac{1}{\alpha}}}-1\right]\dfrac{a_{\mathrm{ini}}^{*}}{1-a_{\mathrm{ini}}^{*}}\quad\text{if
}\dfrac{a_{\mathrm{ini}}^{*}}{a_{\mathrm{ini}}^{*}(1-Q)+Q}<\frac{1}{2}\,.\end{cases}$
(10)

In Fig. 7 we report the numerical results corresponding to the case
$\alpha=1.16$ for a generic set of parameters. The initial configuration is
associated to $Q_{A}=1$, $Q_{B}=0.9$ and no zealots, $\zeta=0$, resulting into
the equilibrium $a_{\mathrm{ini}}^{*}\sim 0.602$ and one can observe that the
numerical simulation (blue line) oscillates about such equilibrium (black
horizontal line) because of the finite size effect induced by the discrete
population. We then add a fraction $\zeta=0.2$ of zealots committed to $B$ and
the system now converges (red line) to the new equilibrium
$a_{\mathrm{tmp}}^{*}\sim 0.102$ (black dashed horizontal line). Finally, by
using Eq. (10), we compute the value of $Q_{A}^{\prime}\sim 2.076$ which
represent the new quality of option $A$, or simply the frequency with which
opinions for $A$ are spread. By increasing option $A$’s quality to
$Q_{A}^{\prime}$, the system returns to the original equilibrium
$a_{\mathrm{ini}}^{*}$ (green line).

Figure 7: Numerical study of the relevance of quality versus zealots. We fixed
$\alpha=1.16$ and $Q_{B}=0.9$, the blue line correspond to $Q_{A}=1$ and
$\zeta=0$, the red line to $Q_{A}=1$ and $\zeta=0.2$ and the green line to
$Q_{A}^{\prime}\sim 2.0756$ and $\zeta=0.2$.

One can prove [38] that if $Q<1$ and $\alpha<1$, in absence of zealots the
system, intialized with half of agents committed to $A$ and half to $B$, will
converge to a full consensus to $A$, namely $a_{\mathrm{ini}}^{*}=1$. In the
previous section we have shown that in presence of zealots, the system cannot
converge to a full $A$ consensus, hence we can never find a new value for the
quality $Q_{A}$ able to return the equilibrium $a_{\mathrm{ini}}^{*}=1$ once
zealots are present. To avoid this issue, we presented the above example by
assuming $\alpha\geq 1$, values for which an equilibrium
$a_{\mathrm{ini}}^{*}<1$ can be achieved.

##  VII Conclusions

In this paper, we presented the results of a study focused on a best-of-$n$
collective decision-making problem, with $n=2$ options of different quality.
We analysed this problem through agent-based simulations and a mathematical
model based on the heterogeneous mean-field approach. In this study, the
interactions among the individuals are defined using a social network whose
nodes are agents, or decision-markers, and edges the possible interactions
among them. The original contribution of this paper is in highlighting the
mutual effects existing between the following parameters of the model: i) the
lower or higher heterogeneity of the scale-free network modelling the
interactions between the agents; ii) the number of agents within the
population that never change opinion (i.e., the zealots); iii) the pooling
error which enables the control of agents’ cognitive load; iv) the ratio in
quality between the two options corresponding to the combination of the cost
and benefit to each option, and v) the degree centrality (or the number of
social connections) of the zealots. We have studied populations in which
individuals are connected according to a scale-free network, and they select
their opinion using decision mechanisms that differ in terms of their
cognitive load, allowing to interpolate among existing models such as the
voter model and the majority model. For each case, we have varied the number
(or, in the mean-field model, the fraction) of zealots, all committed to the
lowest quality option. The mathematical analysis of the combined effect of the
considered parameters has been performed by determining the system equilibria
and their stability.

The results have shown that the combined effect of these parameters has
generated an articulated landscape characterised by different outcomes of the
collective decision-making process. We have shown that, when susceptible
agents employ opinion selection mechanisms characterised by pooling error
$\alpha\sim 1$ (representing the voter model), both the number and the degree
centrality of the zealots are elements that can induce the population to
converge to the lowest quality option. In particular, the higher the number of
zealots or the larger their degree centrality, the stronger their influence on
the opinion dynamics. We have also shown that these effects are influenced by
the nature of the opinion selection mechanisms employed by the susceptible
agents. For example, the effect of the number of zealots on the opinion
dynamics is not observed when susceptible agents make no pooling error, i.e.,
$\alpha=0$ (representing the majority model). We have also shown that the
connectivity structure modulates the influence of the zealots, indeed when the
network becomes increasingly sparse and less heterogeneous, the effect of
zealots is mitigated or largely reduced. In the future, we aim to extend to
the proposed analytical model collective to best-of-$n$ decision-making by
considering the concept of group interactions (also known as higher-order
interactions[2]) among the decision-makers and study how such higher-order can
change the option dynamics.

#### Acknowledgements.

T.N. thanks the University of Namur for the financial support. A.R.
acknowledges support from the Belgian F.R.S.-FNRS, of which he was a Chargé de
Recherches, and support from the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy - EXC2117 -
422037984.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

### Data availability

No data was used for the research described in the article.

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