# Studying speed-accuracy trade-offs in best-of-n collective decision-making
through heterogeneous mean-field modelling

Andreagiovanni Reina∗ Institute for Interdisciplinary Studies on Artificial
Intelligence (IRIDIA), Université Libre de Bruxelles, Brussels, Belgium
Thierry Njougouo  Elio Tuci  Timoteo Carletti∗ Department of mathematics and
Namur Institute for Complex Systems, naXys
University of Namur, Rue Grafé 2, B5000 Namur, Belgium
∗<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

To succeed in their objectives, groups of individuals must be able to make
quick and accurate collective decisions on the best among alternatives with
different qualities. Group-living animals aim to do that all the time. Plants
and fungi are thought to do so too. Swarms of autonomous robots can also be
programmed to make best-of-n decisions for solving tasks collaboratively.
Ultimately, humans critically need it and so many times they should be better
at it! Despite their simplicity, mathematical tractability made models like
the voter model (VM) and the local majority rule model (MR) useful to describe
in simple terms such collective decision-making processes. To reach a
consensus, individuals change their opinion by interacting with neighbours in
their social network. At least among animals and robots, options with a better
quality are exchanged more often and therefore spread faster than lower-
quality options, leading to the collective selection of the best option. With
our work, we study the impact of individuals making errors in pooling others’
opinions caused, for example, to reduce the cognitive load. Our analysis in
grounded on the introduction of a model that generalises the two existing VM
and MR models, showing a speed-accuracy trade-off regulated by the cognitive
effort of individuals. We also investigate the impact of the interaction
network topology on the collective dynamics. To do so, we extend our model
and, by using the heterogeneous mean-field approach, we show that another
speed-accuracy trade-off is regulated by network connectivity. An interesting
result is that reduced network connectivity corresponds to an increase in
collective decision accuracy.

##  I Introduction

Reaching a consensus in a group of individuals without any central authority
or coordinator requires individuals to exchange opinions and combine
conflicting information received from peers. Studying the situation in which
the group must agree on the best among a set of options, the so-called best-
of-n problem, is interesting because it helps us to both understand biological
processes and design the robotics systems of our future [1, 2]. Social insects
are an example of collectives which need to solve the best-of-n problem when
selecting the site where to nidificate [3, 4, 5]. While each insect makes an
inaccurate estimate of the quality of each site, the colony is able to filter
noise and reach a consensus on the best alternative [6]. Similarly, other more
complex animals make collective decisions on the moment and direction to flee
danger or the location where to forage [7, 8, 9]. Collective agreement is
achieved by individuals sharing their opinion with others (voting) and, in
turn, adopting the opinion expressed by others’ votes. These simple voting
rules employed by animals are an useful source of inspiration to design
algorithms of robot swarms, which make best-of-n decisions, for example, on
the shortest path to navigate [10, 11] or the most important location for
their operations [12, 13].

The group is able to select the best alternative because each individual
shares her opinion as frequently as the estimated quality, that is, better
alternatives are shared (voted) more often [14, 15]. Despite the individual
estimates being incorrect, the collective dynamics led to a consensus for the
option that, on average, is estimated of higher quality. Depending on the
effort individuals make in acquiring, processing, and sharing information in
their social network, the collective dynamics change, e.g., in the group
accuracy or the decision speed. While there are several studies analysing
voting models in decentralised networks [16, 17, 18, 19], there is no explicit
connection between the individual cognitive requirements, the social network,
and the collective decision-making performance.

Here, we build a model that explicitly considers the cognitive effort that
individuals make in acquiring and processing their neighbour’s vote. The more
cognitive effort individuals put in, the better they pool social information
(making smaller pooling errors). We investigate how speed and accuracy change
in collective decisions both as a function of the individual cognitive load
and the interaction network. Our analysis reveals that an increased cognitive
effort leads to quicker and more democratic collective decisions which however
are not necessarily accurate. Counter-intuitively, the highest levels of
collective decision accuracy can be achieved with moderate levels of cognitive
efforts, however, at the expense of longer deliberation time. Thus, our model
enriches our understanding of the classical speed-accuracy trade-off in
decision-making [20, 21, 13, 22, 23] by describing it through the lens of
individual cognitive load.

Additionally, our network analysis reveals that groups that are sparsely
connected can obtain higher collective accuracy than when they are highly
connected. Recent previous research has shown that in a number of conditions,
having reduced connectivity between the group members can improve collective
performance in terms of coordination, accuracy, or response speed [24, 25, 26,
27, 28, 29]. More precisely, fish adaptively change their interaction network
when exposed to a threat in order to maximise information transfer in the fish
school [25]. Robots that can only run simplistic algorithms can also exploit
the advantages of sparse connections to improve swarm accuracy [28, 30]. While
it is commonplace to assume that higher connectivity can improve opinion
sharing and thus lead to better coordination, these recent results show in
which conditions limited connectivity can lead to improved collective
dynamics. Our analysis uses the Heterogeneous Mean-Field theory (HMF) [31, 32,
33, 34] to show that both network connectivity and individual cognitive load
can be control parameters to regulate the speed-accuracy trade-off of group
decision-making.

##  II The model

Let us consider a population composed of $N$ agents making a binary collective
decision between two alternative options, say $A$ and $B$. Each option is
characterised by a quality, $Q_{A}$ and $Q_{B}$ (for option $A$ and $B$,
respectively); without lack of generality we hereby assume $Q_{A}>Q_{B}>0$ and
we will define the quality ratio $Q=Q_{B}/Q_{A}\in(0,1)$. Each agent, at a
given time $t$, has an opinion in favour of either option, $A$ or $B$.
Throughout the collective decision-making process, agents interact with each
other and change their opinions depending on the votes expressed by their
neighbours. Each agent votes with a frequency linearly proportional to the
estimated quality of each option, thus $Q_{A}$ and $Q_{B}$ for options $A$ and
$B$, respectively. Therefore, through mean-field approximation, we model the
change of agents’ opinions as a function of the number of agents with opinion
$A$ and $B$, denoted by $n_{A}(t)$ and $n_{B}(t)$, respectively, weighted by
the respective option’s quality and normalised by the group size $N$. Such
weighted proportions, $n_{A}^{\\#}$ and $n_{B}^{\\#}$, represent the mean-
field approximation of the votes expressed by the agents in favour of option
$A$ and $B$, respectively, and correspond to

$\begin{split}n_{A}^{\\#}=\frac{Q_{A}n_{A}/N}{Q_{A}n_{A}/N+Q_{B}n_{B}/N}&\qquad\text{and}\\\
n_{B}^{\\#}=\frac{Q_{B}n_{B}/N}{Q_{A}n_{A}/N+Q_{B}n_{B}/N}\,.\end{split}$ (1)

Agents can put different levels of effort into acquiring others’ votes and
processing them; we name such levels of effort as the agents’ cognitive load
in making their collective decisions. Different levels of cognitive load lead
to different levels of error in pooling social information. Numerous previous
studies have considered agents with a low cognitive load, in the form of the
voter model (VM) [35, 36, 37, 38, 39, 17, 40], and with high cognitive load,
in the form of the majority rule model (MR) [41, 42, 43, 44]. In the VM, an
agent samples a single vote, taken from a random agent, and adopts her
opinion; therefore the probability that an agent committed to $A$ changes her
opinion to $B$ is equal to the weighted proportion $n_{B}^{\\#}$. Instead, in
the MR, an agent has a higher cognitive load and she samples all votes
received from the other agents and selects the most-voted option. In the
majority rule model, agents do not commit pooling errors, whereas in the voter
model, agents introduce a higher pooling error by sampling a single random
vote. We build a model that generalises the two existing models and can also
interpolate, in a continuous way, the cognitive load level in the form of
pooling error among the two models and beyond. In our model, agents change
their opinion with probability

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

where $x$ is the weighted proportion of agents with a different opinion.
Therefore, an agent with an opinion in favour of option $A$ (resp. $B$) will
change her opinion to $B$ (resp. $A$) with probability
$P_{\alpha}(n_{B}^{\\#})$ (resp. $P_{\alpha}(n_{A}^{\\#})$). Note that the
assumption of fixed population, $n_{A}(t)+n_{B}(t)=N$, implies
$n^{\\#}_{A}(t)+n^{\\#}_{B}(t)=1$ and therefore, as a consequence of the
functional form of Eq. (2), we have

$P(n_{A}^{\\#})+P(n_{B}^{\\#})=1\,.$ (3)

The parameter $\alpha\geq 0$ of Eq. (2) is the pooling error which allows
controlling agents’ cognitive load. For $\alpha=0$, the model corresponds to
the MR; for $\alpha=1$ to the VM.

Figure 1: The probability $P_{\alpha}(x)$ given by Eq. (2) for representative
values of the pooling error $\alpha$, which is a parameter inversely
proportional to agents’ cognitive load. For $\alpha=0$, our model corresponds
to the majority rule model [42], for $\alpha=1$ our model corresponds to the
voter model [35, 36], for $0<\alpha<1$ our model interpolates between the two,
and for $\alpha>1$ the agents change their opinion with little attention to
others’ opinion.

Fig. 1 shows a graphical representation of $P_{\alpha}(x)$ of Eq. (2) and let
us appreciate that intermediate values of the pooling error $\alpha$ allow
interpolating between the two models. Indeed, values of $0<\alpha<1$ represent
cases in which an agent makes a higher effort than sampling a random
individual (as she does in the voter model), still the probability of changing
opinion in favour of the most-voted option is lower than the ‘perfect’ case
(zero error) of the majority rule model. These intermediate values represent
conditions in which the agent samples only a subset of the population or
approximately and imprecisely integrates others’ votes. Values of $\alpha>1$
further reduce the cognitive effort that the agent puts into taking into
consideration others’ opinions. As the value of the pooling error $\alpha$
increases, the probability $P_{\alpha}(x)$ gradually becomes more and more
independent of the actual votes. For $\alpha\gg 1$, the probability of
changing opinion approximates the flat line $P_{\alpha}(x)=0.5$, that is, the
agents make maximum levels of pooling error by randomly changing their opinion
regardless of the opinions expressed by the others.

We first consider a well-mixed population, that is, agents are the nodes of a
complete network and therefore every agent can directly exchange votes with
all the other agents. Despite being an idealised case, it allows us to build a
deep analytical understanding of our model in Section III, preliminary to the
study of a population interacting on a heterogeneous network in Section IV.

##  III Mean-field analysis

Let us introduce the proportion of agents with opinion $A$ (resp. $B$),
$a(t)=n_{A}(t)/N$ (resp. $b(t)=n_{B}(t)/N$), hence $a(t)+b(t)=1$. The
proportion of agents with opinion $A$ increases when agents with opinion $B$
change their minds and adopt opinion $A$, or decreases when agents with
opinion $A$ adopt opinion $B$. As illustrated in Appendix A, by exploiting the
well-mixed hypothesis, we obtain the time evolution of $a(t)$ in the form of
the following ordinary differential equation

$\frac{da}{dt}=-a+P_{\alpha}\left(\frac{a}{a(1-Q)+Q}\right)=:f_{\alpha}(a)\,.$
(4)

Because the population size is finite and fixed ($a+b=1$), Eq. (4) is
sufficient to fully determine the temporal dynamics of the system, without the
need to explicitly define another equation ruling the evolution of the
proportion of agents with opinion $B$.

We analyse the long-term dynamics of the system by finding the equilibria of
Eq. (4) and computing their stability as a function of the model parameters.
The equilibria are found at values of $a$ that satisfy $f_{\alpha}(a)=0$. We
find that $\hat{a}^{*}=1$ and $\check{a}^{*}=0$ are always two zeros of
$f_{\alpha}(a)$. When these equilibria are stable, they correspond to a
consensus decision for either alternative: for $\hat{a}^{*}=1$, all agents
eventually have opinion $A$, that is, the population has selected the best
option (because $Q_{A}>Q_{B}$), or, when $\check{a}^{*}=0$, all agents
eventually have opinion $B$ and therefore the population has made a collective
mistake by selecting the option with the inferior quality. For a range of
values of $\alpha$ and $Q$, a third equilibrium $\tilde{a}^{*}\in(0,1)$ may
exist and it corresponds to a polarised population, in which agents with
opinion $A$ and $B$ coexist. In this case, there is not a consensus decision
but the population is in a decision deadlock. Recall that $Q<1$; therefore, we
can prove (as detailed in Appendix A) that

  1. i)

if $Q>\alpha$, both $\check{a}^{*}=0$ and $\hat{a}^{*}=1$ are stable
equilibria, and a third equilibrium $0<\tilde{a}^{*}<1$ exists and is
unstable;

  2. ii)

if $Q<\alpha<1/Q$, then $\check{a}^{*}=0$ is unstable while $\hat{a}^{*}=1$ is
stable, the third equilibrium $0<\tilde{a}^{*}<1$ does not exist;

  3. iii)

if $Q<\alpha$, both $\check{a}^{*}=0$ and $\hat{a}^{*}=1$ are unstable
equilibria, and a third equilibrium $0<\tilde{a}^{*}<1$ exists and is stable.

Note that when the third equilibrium $\tilde{a}^{*}$ exists and it is unstable
– i.e., the case (i) above – the fate of the system depends on the initial
conditions. Stated differently, the position of the third equilibrium $a^{*}$
splits the interval $[0,1]$ into two parts $[0,\tilde{a}^{*})$ and
$(\tilde{a}^{*},1]$, and if the initial conditions are such that
$a(0)\in[0,\tilde{a}^{*})$, then $a(t)\rightarrow 0$ (and thus
$b(t)\rightarrow 1$), while if $a(0)\in(\tilde{a}^{*},1]$, then
$a(t)\rightarrow 1$ (and thus $b(t)\rightarrow 0$). If there are only two
equilibria – i.e., the case (ii) above – the system converges to
$\hat{a}^{*}=1$ for any initial conditions (which correspond to the accurate
collective decision, being $Q_{A}>Q_{B}$).

In Fig. 2, we show the bifurcation diagram of the mean-field model of Eq. (4)
for $Q_{A}=1$ and $Q_{B}=0.9$, as a function of $\alpha$. We report the
equilibria and their stability (in green when stable and in red when unstable)
for values of $\alpha$ in the range $[0,2]$ (x-axis). Note that
$\check{a}^{*}=0$ (corresponding to a population fully committed to the
inferior option $B$) is a stable equilibrium for $\alpha<Q$ and unstable
otherwise. Similarly, $\hat{a}^{*}=1$ (corresponding to a population fully
committed to the best option $A$) is a stable equilibrium for $\alpha<1/Q$
(that is $1/0.9\approx 1.11$ for the used values of $Q_{A}$ and $Q_{B}$) and
unstable in the remaining range of $\alpha$. The third equilibrium
$0<\tilde{a}^{*}<1$, associated with a population where opinions $A$ and $B$
coexist, is unstable for $\alpha<Q$ (red branch), stable for $\alpha>1/Q$
(green branch), and does not exist for $Q<\alpha<1/Q$.

In Fig. 2, we also overlay to the analytical bifurcation diagram the results
obtained from the numerical stochastic simulation of a system defined on a
complete graph made of $N=500$ agents, using the same values for the opinion
quality, $Q_{A}=1$ and $Q_{B}=0.9$. For each value of $\alpha\in[0,2]$, we
numerically integrate the system for $50\,000$ time-steps and report the
average of the final proportion of agents with opinion $A$ (i.e., $\langle
n_{A}\rangle/N$) over 21 runs. When $\alpha<Q_{B}$, the mean-field theory
predicts that the system converges to $\check{a}^{*}=0$ if the initial
condition is below the value of the third unstable equilibrium
$\tilde{a}^{*}$. In agreement with the theoretical predictions, our
simulations, which have been initialised with $n_{A}(0)=100$ (i.e., $a=0.2$),
reached the final system state $n_{A}=0$ for $\alpha<Q$ (blue dots in Fig. 2).
When only two equilibria exist, the mean-field analysis predicts convergence
to $\hat{a}^{*}=1$ for any initial conditions, and indeed the blue points (up
to finite size effects) converge to $n_{A}=N$. Finally for large $\alpha$,
when the third equilibrium is stable, according to the mean-field analysis,
the system should oscillate about this third equilibrium; the numerical
results agree well with the analytical predictions, with the blue dots aligned
with the green curve.

Figure 2: Bifurcation diagrams showing the equilibria of the mean-field model
of Eq. (4) as a function of $\alpha\in[0,2]$ for $Q_{A}=1$ and $Q_{B}=0.9$.
Green dots show stable equilibria, red dots show unstable equilibria, and blue
dots show the average asymptotic values of $\langle n_{A}\rangle/N$ obtained
with stochastic numerical simulations of a population of $N=500$ agents
interacting on a complete graph and initialised at $n_{A}(0)=100$. The match
between mean-field model predictions and simulations is good.

Fig. 3a shows the stability diagram of the mean-field system of Eq. (4) as a
function of the parameters $\alpha$ and $Q$. The parameter space is divided
into three regions, determined by the stability of the system equilibria. The
three regions are delimited by the curves $\alpha=Q$ (white curve) and
$\alpha=1/Q$ (black curve), and correspond to the three stability cases
described above in this section (see Appendix C for the analytical derivation
of such curves). The large red region corresponds to model parameters by which
the population correctly chooses the opinion with the highest quality for any
initial conditions (case (ii) above, equilibrium $\check{a}^{*}=0$ is unstable
and $\hat{a}^{*}=1$ is stable). In the blue region, the population remains
undecided, composed of two sub-populations, each with a different opinion
(case (iii) above, equilibria $\check{a}^{*}=0$ and $\hat{a}^{*}=1$ are
unstable and $0<\tilde{a}^{*}<1$ is stable). In the green region, the
population can converge to the best or the worst option depending on the
initial condition, therefore there is the possibility the population may make
a collective mistake (case (i) above, equilibria $\check{a}^{*}=0$ and
$\hat{a}^{*}=1$ are stable and $0<\tilde{a}^{*}<1$ is unstable). These mean-
field predictions match well with the results shown in Fig. 3b, obtained from
the numerical simulation of a population of $N=500$ agents interacting on a
complete graph. The outcome of the decision is colour-coded in the RGB space,
colouring each pixel with an RGB colour where the red value $\text{R}\in[0,1]$
corresponds to the proportion of simulations terminating with accurate
decisions ($n_{A}=N$), the green value $\text{G}\in[0,1]$ to the proportion of
simulations terminating with incorrect decisions ($n_{A}=0$), and the blue
value $\text{B}\in[0,1]$ to the proportion of simulations terminating without
a consensus decision ($0<n_{A}<N$) after $50\,000$ time-steps.

Figs. 3a-b show that only when the agents have a high pooling error
($\alpha>1$), corresponding to a low cognitive load, the population can remain
deadlocked at indecision. This happens in the blue region which increases in
size as the cognitive load decreases (i.e., increasing pooling error $\alpha$)
or the decision problem difficulty increases (i.e., increasing quality ratio
$Q$). This is caused by agents that are rarely capable of changing their
opinion based on the real distribution of opinions in the population.
Interestingly, despite the poor skills of the agents, the population can
either select the best option and make an accurate decision or remain
undecided but it never selects the inferior option with the lower quality.
Differently, decision deadlocks do not occur for any cognitive level higher or
equal than the voter model, i.e., $\alpha\leq 1$, however, decision mistakes
can be made. For any given pooling error $\alpha<1$ (i.e., relatively high
cognitive level), as the decision problem becomes harder (i.e., the quality
ratio $Q$ increases), the possibility of making a mistake increases. The
collective mistake occurs when the system moves to a state below the unstable
equilibrium, $a<\tilde{a}^{*}$. As shown in Figs. 4a-b, the equilibrium
$\tilde{a}^{*}$ decreases to lower values with decreasing quality ratio $Q$,
leading to a lower rate of mistakes for easier (low $Q$) problems, as visible
in the green shading of Fig. 3b.

An additional result that may not be obvious is that the red region in Figs.
3a-b – which corresponds to reliably accurate decisions – is maximised for
intermediate cognitive load levels (i.e., $\alpha=1$ shown as a dashed
horizontal grey line) and it reduces as the agent cognitive load increases.
However, we can appreciate a more complete picture of the system dynamics by
also analysing the decision time. Fig. 3c shows the time-steps needed to reach
a consensus for a population of $N=500$ simulated agents interacting on a
complete graph. The comparison of Figs. 3b and 3c shows the existence of a
speed-accuracy trade-off, in which the speed to make a decision is traded with
the collective accuracy, as it has been already documented in previous work
[21, 45, 13, 23]. Therefore, increasing agents’ cognitive load allows the
population to quickly reach a consensus at the cost of possible decision
mistakes (here shown as a wider green region in Figs. 3a-b).

Figure 3: Stability diagrams and convergence time. In panel (a) we report the stability diagram of the mean-field model (4) as a function of the quality ratio $Q$ and the pooling error $\alpha$ (inversely proportional to the cognitive cost). The parameter space is divided into three regions: in the red region the population makes accurate collective decisions for any initial condition; in the blue region the population remains locked at indecision with agents’ opinions fluctuating between the two options; in the green region, a consensus for either alternative is possible depending on the initial conditions, therefore mistakes are possible. The three regions are separated by the curves $\alpha=Q$ (white curve) and $\alpha=1/Q$ (black curve). Panels (b-c) show the results from simulations of $N=500$ agents interacting on a fully connected (all-to-all) network. For each couple $(Q,\alpha)$, we perform $100$ independent simulations with random initial configurations (i.e. the initial number of agents with opinion $A$ is uniformly drawn in $[0,N]$ at each run). In (b), the RGB colour of each pixel is computed by assigning to the three values R, G, and B a value equal to the proportion of simulations that terminated at $n_{A}=500$, $n_{A}=0$, and $0<n_{A}<500$, respectively. In (c), the colormap shows the average number of time-steps needed to reach a consensus, i.e., $n_{A}=500$ or $n_{B}=500$, as a function of $Q$ and $\alpha$. |
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Figure 4: Bifurcation diagrams of the mean-field model (4) as a function of
the quality ratio $Q=Q_{B}/Q_{A}$ (i.e., the problem difficulty) for fixed
cognitive load, $\alpha=0.1$ in panel (a) and $\alpha=0.9$ in panel (b).

##  IV Heterogeneous mean-field analysis

In this section, we relax the assumption of full connectivity and instead
hypothesise that each agent can only exchange her opinion with a limited
number of peers, i.e., with her neighbours. Thus, we can represent each agent
as a node of a network connected through links to a subset of other agents
from which she can receive and send information. The network of connections is
described by the adjacency matrix $\mathbf{A}$, such that $A_{ij}=1$ if and
only if agents $i$ and $j$ are connected, and $A_{ij}=0$ otherwise. We also
assume the network to be undirected, that is $A_{ij}=A_{ji}$, simple, that is
at most one link can connect two nodes, and connected, that is starting from
any node, there is a sequence of links that allows reaching any other node.

When agent $i$ is selected, she pools information from her neighbours to
change her opinion. Agent $i$ has $k_{i}=\sum_{j}A_{ij}$ neighbours, of which
$n_{i,A}$ have opinion $A$ and $n_{i,B}$ have opinion $B$, thus,
$k_{i}=n_{i,A}+n_{i,B}$. The number of $i$’s neighbours $k_{i}$ corresponds to
the degree of the node $i$. Analogous to Eq. (1), but assuming a sparse
network, the votes expressed by the neighbours of $i$ in favour of options $A$
and $B$ are, respectively,

$\begin{split}n_{i,A}^{\\#}=\frac{Q_{A}n_{i,A}/k_{i}}{Q_{A}n_{i,A}/k_{i}+Q_{B}n_{i,B}/k_{i}}&\qquad\text{and}\\\
n_{i,B}^{\\#}=\frac{Q_{B}n_{i,B}/k_{i}}{Q_{A}n_{i,A}/k_{i}+Q_{B}n_{i,B}/k_{i}}\,.\end{split}$
(5)

As further described in Appendix B, once agent $i$ interacts with her
neighbours, she adopts opinion $A$ or $B$ with probability
$P_{\alpha}(n_{i,A}^{\\#})$ or $P_{\alpha}(n_{i,B}^{\\#})$, respectively, as
we also recall that $P_{\alpha}(n_{i,A}^{\\#})=1-P_{\alpha}(n_{i,B}^{\\#})$.

By assuming the Heterogeneous Mean Field hypothesis [31, 32] to be valid, we
hypothesise that all nodes with the same degree exhibit the same behaviour.
Therefore, we can define $A_{k}$ (resp. $B_{k}$) as the number of agents,
i.e., nodes, with degree $k$ and opinion $A$ (resp. opinion $B$), and $N_{k}$
as the total number of agents with degree $k$. Hence, $A_{k}+B_{k}=N_{k}$ for
all $k$. In a similar way to the mean-field analysis of Sec. III, we define
$a_{k}=A_{k}/N_{k}$ (resp. $b_{k}=B_{k}/N_{k}$) as the proportion of agents
having opinion $A$ (resp. $B$) among all agents with degree $k$; hence, for
all $k$ we have $a_{k}+b_{k}=1$.

Let us now describe the time evolution of $a_{k}$ for a generic $k$. The
proportion $a_{k}$ increases when an agent with degree $k$ and opinion $B$
changes her opinion to $A$, or it decreases when an agent with degree $k$ and
opinion $A$ changes her opinion to $B$. To compute the frequency of these
events, we compute $n_{i,A}^{\\#}$ and $n_{i,B}^{\\#}$ given in Eq. (5) under
the HMF assumption. The HMF theory originated in epidemiology, which uses the
concept of excess degree [46] to compute the infection probability of a focal
agent. The excess degree is the number of neighbours that a neighbour of the
focal agent has, without considering the focal agent. In other words, the
excess degree of an agent is its number of neighbours minus one (see Fig. 5).
We define $p_{k}$ as the probability that a uniformly random chosen node has
degree $k$, and $q_{k}$ as the probability for a node to have excess degree
equal to $k$, which we can compute as

$q_{k}=\frac{(k+1)p_{k+1}}{\langle k\rangle}\quad\forall k\geq 0\,,$ (6)

where $\langle k\rangle=\sum_{k}kp_{k}$ is the average node degree, and
trivially $\sum_{k}q_{k}=1$. Let us consider a generic focal agent $i$ (or,
equivalently, focal node $i$) with degree $k$ and opinion $B$ (see Fig. 5);
let $q_{j_{1}}$ be the probability that an agent $i_{1}$, connected to the
focal agent $i$, has excess degree $j_{1}\geq 0$. Let $a_{j_{1}+1}$ be the
probability that agent $i_{1}$ has opinion $A$, and
$b_{j_{1}+1}=1-a_{j_{1}+1}$ the probability she has opinion $B$. By
considering all the $k$ agents connected with the focal agent $i$ we can
conclude that $q_{j_{1}}\dots q_{j_{k}}$ determines the joint probability that
each agent reachable from any of the $k$ links emerging from the former agent,
has excess degree $j_{1},\dots,j_{k}$. We can then define $\pi_{k,\ell}$ as
the probability that $\ell$ agents among the $k$ ones have opinion $B$ and
thus $k-\ell$ agents have opinion $A$. Therefore, the term $\ell$ is a linear
combination of the products of $a_{j_{m}+1}$ and $(1-a_{j_{m}+1})$, with
$m=1,\dots,k$. Hence, we can compute the weighted proportion of agents with
opinion $A$ as $n_{i,A}^{\\#}=(k-\ell)/[k-\ell+Q\ell]$, and this event happens
with probability $q_{j_{1}}\dots q_{j_{k}}\pi_{k,\ell}$. In conclusion, agent
$i$ with opinion $B$ can change opinion with probability $q_{j_{1}}\dots
q_{j_{k}}\pi_{k,\ell}P_{\alpha}\left((k-\ell)/[k-\ell+Q\ell]\right)$. See an
explanatory example for $k=2$ in Appendix B.

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Figure 5: Schematic representation of the probabilities involved in the
heterogeneous mean-field computations. The focal node $i$ has $k$ neighbours
(degree $k$), denoted by $i_{1},\dots,i_{r},\dots,i_{k}$. Each neighbour,
e.g., node $i_{r}$, has an excess degree $j_{r}$ with probability $q_{j_{r}}$
and therefore degree $j_{r}+1$. With probability $a_{j_{r}+1}$, she has
opinion $A$, and therefore with probability $1-a_{j_{r}+1}$, she has opinion
$B$.

In a similar way, we can compute the decrease rate of agents with degree $k$
and opinion $A$. In this case, the argument of the function $P_{\alpha}$ is
$\frac{Q_{B}\ell}{Q_{A}(k-\ell)+Q_{B}\ell}=\frac{Q\ell}{k-\ell+Q\ell}$, that
is, the weighted proportion of agents with opinion $B$ assuming that $\ell$
agents among them have opinion $B$. Note that, because
$P_{\alpha}(n_{A}^{\\#})+P_{\alpha}(n_{B}^{\\#})=1$, we have that

$\begin{split}P_{\alpha}\left(\frac{Q\ell}{k-\ell+Q\ell}\right)&=1-P_{\alpha}\left(1-\frac{Q\ell}{k-\ell+Q\ell}\right)\\\
&=1-P_{\alpha}\left(\frac{k-\ell}{k-\ell+Q\ell}\right)\,.\end{split}$

By combining these equations and following a series of simplifications
described in Appendix B, we describe the change in time of the proportion
$a_{k}$ as

$\frac{da_{k}}{dt}=-a_{k}+\sum_{j_{1},\dots,j_{k}}q_{j_{1}}\dots
q_{j_{k}}\sum_{\ell=0}^{k}\pi_{k,\ell}P_{\alpha}\left(\frac{k-\ell}{k-\ell+\ell
Q}\right)\,.$ (7)

To analyse the dynamics and equilibria of the system, we define

$\langle a\rangle:=\sum_{j\geq 0}q_{j}a_{j+1}\,,$ (8)

and by combining it with Eq. (7), we obtain

$\begin{split}\frac{d\langle a\rangle}{dt}=-\langle
a\rangle+\sum_{k}q_{k}\sum_{\ell=0}^{k}\binom{k+1}{\ell}\langle
a\rangle^{k+1-\ell}\left(1-\langle a\rangle\right)^{\ell}\times\\\
P_{\alpha}\left(\frac{k+1-\ell}{k+1-\ell+\ell
Q}\right):=f_{\alpha}^{(hmf)}(\langle a\rangle)\,.\end{split}$ (9)

By imposing $f_{\alpha}^{(hmf)}(\langle a^{*}\rangle)=0$, we can find (up to)
three equilibria $\langle a^{*}\rangle$. There is the equilibrium
$\langle\check{a}^{*}\rangle=0$, where $a_{k}=0$ for all $k$, which
corresponds to the system with no agent having opinion $A$ (thus, all agents
have opinion $B$). There is the equilibrium $\langle\hat{a}^{*}\rangle=1$,
where $a_{k}=1$ for all $k$, which corresponds to the system with all agents
having opinion $A$. There is also a third equilibrium
$0<\langle\tilde{a}^{*}\rangle<1$, which has a non-trivial mathematical
expression and exists only for a certain range of parameter values. This
equilibrium corresponds to the coexistence of agents with opinions $A$ and
$B$.

The stability of the three equilibria can be studied by analysing the sign of
the derivative of $f_{\alpha}^{(hmf)}$. By following the steps described in
Appendix B, the derivatives evaluated at the equilibria
$\langle\check{a}^{*}\rangle=0$ and $\langle\hat{a}^{*}\rangle=1$ are,
respectively,

$\begin{array}[]{ll}(f_{\alpha}^{(hmf)})^{\prime}(0)=&-1+\sum_{k}q_{k}(k+1)P_{\alpha}\left(\frac{1}{1+kQ}\right)\quad\text{and}\\\
(f_{\alpha}^{(hmf)})^{\prime}(1)=&-1+\sum_{k}q_{k}(k+1)P_{\alpha}\left(\frac{Q}{k+Q}\right)\,.\end{array}$
(10)

Therefore, we can appreciate that the stability of both equilibria is not only
determined by the parameters $\alpha$ and $Q$, as in the mean-field case of
Sec. III, but also by the network structure, via the probability of excess
degree $q_{k}$.

###  IV.1 Scale-free networks

The heterogeneous mean-field analysis allows us to study the influence of the
network topology on group dynamics. Here, we consider scale-free networks with
a degree distribution that follows a power law with exponent $\gamma>2$.
Therefore, $p_{k}=c_{\gamma}/k^{\gamma}$ (where $c_{\gamma}:=\left(\sum_{k\geq
1}1/k^{\gamma}\right)^{-1}>0$ is a normalising constant), the excess degree
probability is $q_{k}=\frac{1}{\langle
k\rangle}\frac{c_{\gamma}}{(k+1)^{\gamma-1}}$, and the average degree is
$\langle k\rangle=\sum_{k\geq 1}kc_{\gamma}/k^{\gamma}$. Using the same colour
code of Fig. 3a, Fig. 6 shows the stability diagrams for
$\gamma\in\\{2.2,2.6,3.1\\}$ as a function of the parameters $\alpha$ and $Q$,
defining the individual cognitive load and the decision-problem difficulty,
respectively. We can appreciate that, as the exponent $\gamma$ increases, the
size of the red region increases and the size of the green region decreases
(on the other hand, the change in the blue region is unnoticeable). While in
the green region the collective decision depends on the initial system
condition (thus opening the group to possible errors), in the red region the
population cannot make mistakes but only makes accurate collective decisions
for any initial condition. Therefore, the results of Fig. 6 suggest that when
opinions are exchanged in networks with high $\gamma$, the population is more
accurate in collective decision-making. The same conclusion can be reached by
observing the stability diagrams of Fig. 7, in which the problem difficulty is
fixed (quality ratio $Q\in\\{0.5,0.8,0.9\\}$) and the stability regions are
computed as a function of $\alpha$ and $\gamma$.

As $\gamma$ increases, the probability of having nodes with a large degree
decreases as $1/k^{\gamma}$ and most of the nodes have a very low degree (one
or two neighbours). Therefore, these results suggest that collective accuracy
can improve as connectivity decreases, i.e., in networks that are sparse with
nodes with mostly low degrees. The counter-intuitive nature of this result –
groups perform better communicating on sparse networks – can be explained by
complementing the analysis with the decision time. In fact, panels (c), (f),
and (i) of Fig. 7 show that as the probability of having nodes with a large
degree decreases (i.e., increasing $\gamma$), the average decision time
slightly increases. This result supports and extends beyond the case
$\alpha=1$ the results presented in [24]. Therefore, once again, improved
accuracy is coupled with slower decisions, and in this case the speed-accuracy
trade-off can be regulated by both the individual cognitive load (pooling
error $\alpha$) and the network connectivity (exponent $\gamma$).

---
Figure 6: Stability diagrams and convergence time for collective decision-
making on scale-free networks as a function of the pooling error $\alpha$ and
quality ratio $Q$. We present the results for three values of the exponent
$\gamma$ regulating network connectivity: top row $\gamma=2.2$, central row
$\gamma=2.6$, bottom row $\gamma=3.1$. Left column panels – i.e., (a), (d),
and (g) – show the convergence diagram of the mean-field model (9). The
parameter space is divided into the same three regions of Fig. 3a using the
same colour code. Central and right column panels show the results of
simulations (100 independent for each $(Q,\alpha)$ configuration) of $N=500$
agents interacting on a scale-free network with random initial configurations
(i.e., $n_{A}(t=0)\sim\mathcal{U}(0,N)$) for $50\,000$ time steps. Central
column panels – i.e., (b), (e), and (h) – show the outcome of the collective
decision-making process using the same RGB colour code as Fig. 3b. Right
column panels – i.e., (c), (f), and (i) – show the average number of time-
steps needed to reach a consensus, i.e., $n_{A}=500$ or $n_{B}=500$. The white
area indicates the absence of data, as the system never reaches a consensus.
Figure 7: Stability diagrams and convergence time for collective decision-
making on scale-free networks as a function of the pooling error $\alpha$ and
network’s power-law exponent $\gamma$. We report results for three values of
the quality ratio $Q=Q_{B}/Q_{A}$ which encodes the decision problem
difficulty: top row $Q=0.5$ (easy problem), central row $Q=0.8$ (medium
problem), bottom row $Q=0.9$ (difficult problem), with $Q_{A}=1$. Colour code
and experimental design are the same as the one described in the caption of
Fig. 6.

As a cautionary note, we also want to highlight the limitations of the HMF
approach. While the model predictions are confirmed by agent-based simulations
on the tested scale-free networks (Figs. 6 and 7), the model is not able to
predict accurately the dynamics of a population interacting on $2m$-regular
graphs. In particular, we investigate the dynamics of nodes interacting on
ring networks where all nodes are connected to their first $m$ neighbour nodes
on the “left” and $m$ on the “right”, in such a way all nodes have the same
degree $k=2m$. In Appendix C, we study the heterogeneous mean-field model for
the case of $2m$-regular ring networks and derive the equations describing the
bifurcation points that determine the stability changes of the system. Fig. 8
shows stability diagrams (left column panels) for $m\in\\{2,3,10\\}$ that are
qualitatively similar to the ones computed for the other types of networks
(complete graph and scale-free networks). However, the central column panels
of Fig. 8 show that the results of the numerical simulations only partially
agree with the theory predictions. The agreement improves as the connectivity
increases ($m=10$), however, the dynamics for low values of $m$ are different.
In particular, we can appreciate that the agent-based simulations are able to
make accurate decisions (red area) for a much larger range of parameters $Q$
and $\alpha$ than what the theory predicted. It is unclear whether the
discrepancy is due to the regularity of the graph which the HMF model does not
capture or to other factors (e.g., low connectivity), and it can be the focus
of future research. Despite the current HFM model cannot grasp the dynamics on
$2m$-regular ring networks, the simulation results on this type of networks
(Fig. 8b, e, h) confirm our intuition that collective decision-making accuracy
increases as the network connectivity reduces. Population operating on
networks with low $m$ have consistently a higher accuracy. Interestingly, for
$2m$-regular ring networks, it seems that there is no trade-off between
decision speed and accuracy but, instead, rings with low $m$ enable both quick
and accurate decisions.

Figure 8: Stability diagrams and convergence time for collective decision-
making on $2m$-regular ring networks as a function of the pooling error
$\alpha$ and the quality ratio $Q$. We present the results for three values of
the parameter $m$ regulating network connectivity: top row $m=2$, central row
$m=3$, bottom row $m=10$. Left column panels – i.e., (a), (d), and (g) – show
the convergence diagram of the HMF model (9). The parameter space is divided
into the same three regions of Fig. 3a using the same colour code. Central and
right column panels show the results of simulations (100 independent for each
$(Q,\alpha)$ configuration) of $N=500$ agents interacting on a ring network
with random initial configurations (i.e., $n_{A}(t=0)\sim\mathcal{U}(0,N)$)
for $50\,000$ time steps. Central column panels – i.e., (b), (e), and (h) –
show the outcome of the collective decision-making process using the same RGB
colour code as Fig. 3b. Right column panels – i.e., (c), (f), and (i) – show
the average number of time-steps needed to reach a consensus, i.e.,
$n_{A}=500$ or $n_{B}=500$. The white area indicates the absence of data, as
the system never reaches a consensus.

##  V Conclusions

Modelling collective decision-making processes can provide useful insights
into the understanding of the living world and design new decision protocols
for more efficient group decisions. The voter model and variations of them,
thanks to their mathematical simplicity, have been effectively used to model
decision-making at every level of biological complexity, from ecological
dynamics of plant communities [39] to coordinated motion in fish [37] to
house-hunting in honeybees [5] to human group dynamics [47, 48]. In the best-
of-$n$ problem, the group has to select the best option among a discrete set
of $n$ alternatives and these models describe how opinions spread from one
individual to another through voting interactions on a social network. This
study presents a model capable of generalising a set of popular existing
voter-like models for collective decision-making in the best-of-2 scenario.
Our analysis focuses on understanding the impact of two control parameters –
the pooling error and the network connectivity – on the collective performance
in terms of decision accuracy and time.

Both pooling error and the network connectivity regulate a speed-accuracy
trade-off. By reducing the pooling error, and therefore demanding a higher
cognitive effort from the individuals to correctly process the social
information, the population can make quicker and more democratic decisions,
however with reduced accuracy due to the more frequent selection of the option
with the lowest quality, compared with models with higher pooling error and
thus lower individual cognitive effort. Instead, by reducing network
connectivity, and therefore reducing the average number of neighbours of each
individual, collective accuracy is improved at the cost of higher decision
time. These results improve our understanding of the role of individual costs
and network connectivity in collective decision-making.

By measuring the losses, and even the benefits, of reducing computation and
communication costs, our analysis can be useful to support the design of
autonomous robot systems capable of operating without human supervision.
Reduced costs can save energy, money, and, in general, resources both at
design and run time, i.e., designing robots with simpler circuitry can be
cheaper and consuming less energy in social interactions can improve
efficiency. Erroneous computations and limited connectivity can not only be
cheaper but also increase performance in terms of decision accuracy (due to a
longer deliberation time during which the decision, rather than being rushed,
is more accurately made). Recent previous results also observed that there are
certain conditions where reduced connectivity between the agents of the group
– robots, animals, or humans – can give important group-level advantages,
e.g., better responsiveness to environmental changes [28, 30, 26], evading a
predator or avoiding dangers [25, 27], or generate higher cultural diversity
and innovations [49, 50]. We believe that the results of our analysis can also
have important implications in the study and design of group decision-making
in human societies, which can be biased and manipulated through targeted
interventions on how information is exchanged and aggregated in the social
networks [51, 52].

Despite abstracting components of the process, the type of analysis proposed
in our study can give useful predictions. Future work can look into extending
the analysis to measure the impact of different types of network topologies,
e.g., Erdös-Rényi’s random graphs, Barabási-Albert’s scale-free networks,
Watts-Strogats’s small-word network, and random geometric graphs. We also
believe that an interesting extension of the work could apply the present
analysis to study the dynamics of heterogeneous populations, comprising
individuals that follow different voting rules or that have different levels
of conformism with others (e.g., populations comprising stubborn agents [53,
54]).

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Supplementary Material.
Studying speed-accuracy trade-offs in best-of-n collective decision-making
through heterogeneous mean-field modelling

##  Appendix A The mean-field model

In this appendix, we first indicate the steps needed to deduce the mean-field
system of Eq. (4) and then we find the equilibria of the system and study
their stability.

Let us introduce the proportion of agents with opinion $A$ (resp. $B$),
$a(t)=n_{A}(t)/N$ (resp. $b(t)=n_{B}(t)/N$), hence $a(t)+b(t)=1$. The
proportion of agents with opinion $A$ increases because agents with opinion
$B$ change their minds and adopt opinion $A$, or decreases if agents with
opinion $A$ adopt opinion $B$; therefore, we can write the change of $a$ in a
small time interval $dt\rightarrow 0$ as

$\frac{da}{dt}=bP_{\alpha}\left(a^{\\#}\right)-aP_{\alpha}\left(b^{\\#}\right)\,.$
(11)

We recall the quantities $n_{A}^{\\#}$ and $n_{B}^{\\#}$ defined in Eq. (1),
representing the votes expressed for option $A$ and $B$, respectively, which
are weighted by the quality. Therefore, we can define the weighted proportions

$a^{\\#}=\frac{a}{a+Qb}\quad\text{ and }\quad b^{\\#}=\frac{Qb}{a+Qb}\,,$ (12)

where $Q$ is the ratio $Q_{B}/Q_{A}$. Given the functional form of
$P_{\alpha}$ given in Eq. (2), we can conclude that

$P_{\alpha}\left(a^{\\#}\right)+P_{\alpha}\left(b^{\\#}\right)=1\,.$

Hence, we can derive the mean-field model of Eq. (4) as

$\frac{da}{dt}=(1-a)P_{\alpha}\left(a^{\\#}\right)-a\left(1-P_{\alpha}\left(a^{\\#}\right)\right)=P_{\alpha}\left(\frac{a}{a(1-Q)+Q}\right)-a=:f_{\alpha}(a)\,.$

The equilibria of this system are determined by the zeros of $f_{\alpha}(a)$,
namely we are looking for values $a^{*}\in[0,1]$ such that

$f_{\alpha}(a^{*})=0\,,\quad\text{i.e.,}\quad
P_{\alpha}\left(\frac{a^{*}}{a^{*}(1-Q)+Q}\right)-a^{*}=0\,.$ (13)

The equilibrium stability is determined by the sign of the derivative computed
on the equilibrium, $f^{\prime}(a^{*})$. To simplify the analysis, let us
introduce a new variable

$x=\frac{a}{a(1-Q)+Q}\,.$ (14)

Observe that $x$ is well defined, indeed $a(1-Q)+Q\neq 0$ for $a\in[0,1]$ and
moreover $x=0$ if $a=0$ and $x=1$ if $a=1$. In conclusion, Eq. (14) defines a
bijective map from $[0,1]$ into $[0,1]$. By inverting the relation (14) we can
write

$a=\frac{xQ}{1-x(1-Q)}\,,$ (15)

hence solving Eq. (13) is equivalent to solve

$P_{\alpha}\left(x^{*}\right)=\frac{x^{*}Q}{1-x^{*}(1-Q)}\,,$ (16)

that is to determine the intersections between the function
$P_{\alpha}\left(x\right)$ and the hyperbola $g(x)=\frac{xQ}{1-x(1-Q)}$. Two
solutions are trivially found: $\check{x}^{*}=0$ and $\hat{x}^{*}=1$. However,
for some choice of the parameters $Q$ and $\alpha$, a third solution
$\tilde{x}^{*}$ can also exist.

To find the intersections between the two functions, let us define
$\Delta(x)=P(x)-g(x)$, and search for which values $\Delta(x)=0$. As indicated
above, for $x=0$ and $x=1$, we have $\Delta(0)=\Delta(1)=0$. To determine the
existence of (at least) a third root let us consider the derivative of
$\Delta(x)$ at $x=0$ and $x=1$. A straightforward computation returns
$\Delta^{\prime}(0)=\alpha-Q$ and $\Delta^{\prime}(1)=\alpha-1/Q$. A
sufficient condition to have a third solution is thus $\Delta^{\prime}(0)>0$
and $\Delta^{\prime}(1)>0$ or $\Delta^{\prime}(0)<0$ and
$\Delta^{\prime}(1)<0$. The reason is that the function $\Delta(x)$ is
continuous and if it approaches both $x=0$ and $x=1$ with increasing (or both
decreasing) derivatives, it should cross (at least once) the $0$ line at some
point.

Let us now consider the stability of the three equilibria. The equilibrium
$\check{x}^{*}=0$ is stable if and only if $\Delta^{\prime}(0)<0$, namely if
$\alpha<Q$. On the other hand, the equilibrium $\hat{x}^{*}=1$ is stable if
and only if $\Delta^{\prime}(1)<0$, namely $\alpha<1/Q$. From Eq. (15) one can
obtain the value of the variable $a$ given $x$, we can thus draw the following
conclusions:

  * •

Let $Q>1$:

    * –

if $\alpha<1/Q$, then we also have $\alpha<Q$, thus $\check{a}^{*}=0$ and
$\hat{a}^{*}=0$ are stable equilibria, and the third equilibrium
$0<\tilde{a}^{*}<1$ exists but it is unstable;

    * –

if $1/Q<\alpha<Q$, then $\check{a}^{*}=0$ is stable, $\hat{a}^{*}=0$ is
unstable, and the third equilibrium $0<\tilde{a}^{*}<1$ does not exist;

    * –

if $Q<\alpha$, then $\check{a}^{*}=0$ and $\hat{a}^{*}=0$ are unstable
equilibria, and the third equilibrium $0<\tilde{a}^{*}<1$ exists and is
stable.

  * •

Let $Q<1$:

    * –

if $\alpha<Q$, then we also have $\alpha<1/Q$, thus $\check{a}^{*}=0$ and
$\hat{a}^{*}=0$ are stable equilibria, and the third equilibrium
$0<\tilde{a}^{*}<1$ exists but is unstable;

    * –

if $Q<\alpha<1/Q$, then $\check{a}^{*}=0$ is unstable, $\hat{a}^{*}=0$ is
stable, and the third equilibrium $0<\tilde{a}^{*}<1$ does not exist;

    * –

if $Q<\alpha$, then $\check{a}^{*}=0$ and $\hat{a}^{*}=0$ are unstable
equilibria, and the third equilibrium $0<\tilde{a}^{*}<1$ exists and is
stable.

##  Appendix B The heterogeneous mean-field model

In this section, we present the detailed computation needed to derive through
heterogeneous mean-field theory Eqs. (7) and (9), presented in the main text.
Let us assume that agents are connected via a network and they can exchange
opinions only with neighbours to which they are directly corrected. Given an
agent $i$, her neighbours are defined as the nodes $j$ for which $A_{ij}=1$,
where $\mathbf{A}$ is the $N\times N$ adjacency matrix. Observe that
$A_{ij}=0$ if agents $i$ and $j$ are not connected and therefore cannot
directly exchange opinions.

Let us introduce the quantities $n_{i,A}$ and $n_{i,B}$ that indicate the
number of neighbours of agent $i$ with opinion $A$ and $B$, respectively.
Formally, we can define them as

$n_{i,A}=\sum_{j}A_{ij}\hat{A}_{j}\quad\text{ and }\quad
n_{i,B}=\sum_{j}A_{ij}\hat{B}_{j}\,,$ (17)

where $\hat{A}_{j}=1$ (resp. $\hat{B}_{j}=1$) if agent $j$ has opinion $A$
(resp. $B$), and zero otherwise.

An agent $i$ with opinion $A$ (resp. $B$) changes her opinion to $B$ (resp.
$A$) with probability defined by the nonlinear function
$P_{\alpha}(n_{i,B}^{\\#})$ (resp. $P_{\alpha}(n_{i,A}^{\\#})$) presented in
Eq. (2), with argument the weighted proportion $n_{i,B}^{\\#}$ (resp.
$n_{i,A}^{\\#}$) of $i$’s neighbours with opinion $B$ (resp. $A$). The weights
of $n_{i,A}^{\\#}$ and $n_{i,B}^{\\#}$ are proportional to the quality of
opinions $A$ and $B$, respectively, and can be mathematically defined as

$n_{i,A}^{\\#}=\frac{Q_{A}n_{i,A}}{Q_{A}n_{i,A}+Q_{B}n_{i,B}}\quad\text{and}\quad
n_{i,B}^{\\#}=\frac{Q_{B}n_{i,B}}{Q_{A}n_{i,A}+Q_{B}n_{i,B}}\,.$ (18)

We recall that $k_{i}=\sum_{j}A_{ij}$ is the degree of the node $i$ and
trivially $k_{i}=n_{i,A}+n_{i,B}$. Let us observe that by defining
$Q=Q_{B}/Q_{A}$ we can rewrite the previous relations as:

$n_{i,A}^{\\#}=\frac{n_{i,A}}{n_{i,A}+Qn_{i,B}}=\frac{n_{i,A}}{(1-Q)n_{i,A}+Qk_{i}}=\frac{n_{i,A}/k_{i}}{(1-Q)n_{i,A}/k_{i}+Q}\quad\text{and}\quad
n_{i,B}^{\\#}=1-n_{i,A}^{\\#}\,,$ (19)

namely Eq. (5) in the main text.

Let us now assume the validity of the Heterogeneous Mean Field hypothesis
(HMF) [31, 32] and let thus aggregate agents according to their opinion and
degree, namely we define $A_{k}$ and $B_{k}$ to be the number of agents with
degree $k$ and opinion $A$ or $B$, respectively. Then, setting $N_{k}$ to be
the number of agents with degree $k$, we have $A_{k}+B_{k}=N_{k}$ for all $k$.
Let us introduce $a_{k}=A_{k}/N_{k}$ and $b_{k}=B_{k}/N_{k}$ as the proportion
of agents with degree $k$ and opinion $A$ or $B$, respectively. The goal is to
express the probability of changing opinion by using the HMF.

Let us consider an agent $i$ with opinion $B$ and assume she has $k$
neighbours; and we want to compute the probability that $\ell$ neighbours have
opinion $B$ and $k-\ell$ opinion $A$ (with $\ell\in\\{0,\dots,k\\}$), so that
we can compute the weighted proportions of Eq. (18) as
$n_{i,A}^{\\#}=(k-\ell)/[k-\ell+Q\ell]$ and $n_{i,B}^{\\#}=\ell
Q/[k-\ell+Q\ell]$. In the spirit of the HMF hypothesis, we determine the
probability that a node has degree $k^{\prime}$ by only knowing that it is
connected to a node with degree $k$; the latter is given by the excess degree
$q_{k^{\prime}}$, namely
$q_{k^{\prime}}={({k^{\prime}}+1)p_{{k^{\prime}}+1}}/{\langle k\rangle}$,
where $p_{k}$ is the proportion of nodes with degree $k$ and $\langle
k\rangle$ the average network degree.

#### Example for $k=2$

Before computing the formula for a general degree $k$, let us present an
example for $k=2$ which helps us to explain our reasoning. Assume the focal
agent $i$ has degree $k_{i}=2$ and the two neighbours have excess degree
$j_{1}\geq 0$ and $j_{2}\geq 0$, then there are three possible cases:

  * •

Both neighbours have opinion $A$. This happens with probability

$q_{j_{1}}a_{j_{1}+1}q_{j_{2}}a_{j_{2}+1}\,,$

  * •

Both neighbours have opinion $B$. This happens with probability

$q_{j_{1}}(1-a_{j_{1}+1})q_{j_{2}}(1-a_{j_{2}+1})\,.$

  * •

One neighbour has opinion $A$ and one neighbour has opinion $B$. This happens
with probability

$q_{j_{1}}a_{j_{1}+1}q_{j_{2}}(1-a_{j_{2}+1})+q_{j_{1}}(1-a_{j_{1}+1})q_{j_{2}}a_{j_{2}}\,.$

Let us define $\pi_{2,\ell}$ be the probability that $\ell\in\\{0,1,2\\}$
agents have opinion $B$ and thus $2-\ell$ opinion $A$. Then the previous three
cases can be summarised into a single formula

$q_{j_{1}}q_{j_{2}}\pi_{2,\ell}\,,\quad\forall\ell\in\\{0,1,2\\}\,.$

We can now compute the probability that the focal agent $i$ changes her
opinion to $A$:

  * •

In the case both neighbours have opinion $A$,

$q_{j_{1}}q_{j_{2}}\pi_{2,0}P_{\alpha}\left(\frac{2-0}{2-0+0Q}\right)=q_{j_{1}}q_{j_{2}}\pi_{2,0}P_{\alpha}\left(1\right)\,.$

  * •

In the case both neighbours have opinion $B$,

$q_{j_{1}}q_{j_{2}}\pi_{2,2}P_{\alpha}\left(\frac{2-2}{2-2+2Q}\right)=q_{j_{1}}q_{j_{2}}\pi_{2,2}P_{\alpha}\left(0\right)\,.$

  * •

In the case one of the two neighbours has opinion $A$ the other has opinion
$B$,

$q_{j_{1}}q_{j_{2}}\pi_{2,1}P_{\alpha}\left(\frac{2-1}{2-1+1Q}\right)=q_{j_{1}}q_{j_{2}}\pi_{2,1}P_{\alpha}\left(\frac{1}{1+Q}\right)\,.$

#### The general case

The reasoning presented in the example for $k=2$ can be repeated for a general
$k$. Then $q_{j_{1}}\dots q_{j_{k}}$ evaluates the joint probability that each
node reachable from any of the $k$ links emerging from the focal node, has
excess degree $j_{1},\dots,j_{k}$. We can define $\pi_{k,\ell}$, to be the
probability that $\ell$ nodes among the $k$ neighbours have opinion $B$ and
thus $k-\ell$ opinion $A$. Therefore, the term $\pi_{k,\ell}$ is a linear
combination of products of $a_{j_{m}+1}$ and $(1-a_{j_{m}+1})$, with
$m=1,\dots,k$. Finally, the probability $q_{j_{1}}\dots q_{j_{k}}\pi_{k,\ell}$
is multiplied by the function $P_{\alpha}$ with argument the weighted
proportion of agents with opinion $A$ or $B$, that is
$\frac{k-\ell}{k-\ell+Q\ell}$ or $\frac{\ell Q}{k-\ell+Q\ell}$.

Let us observe that because of property (3) (i.e.,
$P(n_{A}^{\\#})+P(n_{B}^{\\#})=1$), we have that

$P_{\alpha}\left(\frac{Q\ell}{k-\ell+Q\ell}\right)=1-P_{\alpha}\left(1-\frac{Q\ell}{k-\ell+Q\ell}\right)=1-P_{\alpha}\left(\frac{k-\ell}{k-\ell+Q\ell}\right)\,.$

Thus, we can wrap together the above expressions and obtain the time evolution
of $a_{k}$ for a generic degree $k$:

$\frac{da_{k}}{dt}=(1-a_{k})\sum_{j_{1},\dots,j_{k}}q_{j_{1}}\dots
q_{j_{k}}\sum_{\ell=0}^{k}\pi_{k,\ell}P_{\alpha}\left(\frac{k-\ell}{k-\ell+\ell
Q}\right)-a_{k}\sum_{j_{1},\dots,j_{k}}q_{j_{1}}\dots
q_{j_{k}}\sum_{\ell=0}^{k}\pi_{k,\ell}\left[1-P_{\alpha}\left(\frac{k-\ell}{k-\ell+\ell
Q}\right)\right]\,.$ (20)

Let us explain each term on the right-hand side. The leftmost term,
$(1-a_{k})$, is the probability that the focal agent has degree $k$ and does
not have opinion $A$, she hence has opinion $B$. The term $q_{j_{1}}\dots
q_{j_{k}}$ evaluates the joint probability that each node reachable from any
of the $k$ links emerging from the focal node, has excess degree
$j_{1},\dots,j_{k}$; the sum $\sum_{j_{1},\dots,j_{k}}$ allows to consider all
the possibilities. For a given choice of $j_{1},\dots,j_{k}$, the next term,
$\pi_{k,\ell}$, determines the probability that $\ell$ nodes among the $k$
ones have opinion $B$ and thus $k-\ell$ have opinion $A$. The sum
$\sum_{\ell=0}^{k}$ allows to consider all the possibilities from $\ell=0$,
all agents have opinion $A$, to $\ell=k$, all agents have opinion $B$.
Finally, the term $P_{\alpha}\left(\frac{k-\ell}{k-\ell+\ell Q}\right)$ is the
probability the focal agent with opinion $B$ changes her mind because there
are $k-\ell$ agents with opinion $A$ and $\ell$ agents with opinion $B$. The
remaining terms denote the opposite process where the selected agent has
opinion $A$, with probability $a_{k}$, and she changes opinion after an
interaction with her neighbours with opinion $B$. As already observed, we used
the property (3) for the function $P_{\alpha}$ to rewrite the rightmost term.

Eq. (20) can be split into four parts

$\displaystyle\frac{da_{k}}{dt}$ $\displaystyle=$
$\displaystyle\sum_{j_{1},\dots,j_{k}}q_{j_{1}}\dots
q_{j_{k}}\sum_{\ell=0}^{k}\pi_{k,\ell}P_{\alpha}\left(\frac{k-\ell}{k-\ell+\ell
Q}\right)-a_{k}\sum_{j_{1},\dots,j_{k}}q_{j_{1}}\dots
q_{j_{k}}\sum_{\ell=0}^{k}\pi_{k,\ell}P_{\alpha}\left(\frac{k-\ell}{k-\ell+\ell
Q}\right)+$ $\displaystyle-$ $\displaystyle
a_{k}\sum_{j_{1},\dots,j_{k}}q_{j_{1}}\dots
q_{j_{k}}\sum_{\ell=0}^{k}\pi_{k,\ell}+a_{k}\sum_{j_{1},\dots,j_{k}}q_{j_{1}}\dots
q_{j_{k}}\sum_{\ell=0}^{k}\pi_{k,\ell}P_{\alpha}\left(\frac{k-\ell}{k-\ell+\ell
Q}\right)\,,$

and we can observe that the rightmost terms on the first and second line do
simplify each other by returning

$\frac{da_{k}}{dt}=\sum_{j_{1},\dots,j_{k}}q_{j_{1}}\dots
q_{j_{k}}\sum_{\ell=0}^{k}\pi_{k,\ell}P_{\alpha}\left(\frac{k-\ell}{k-\ell+\ell
Q}\right)-a_{k}\sum_{j_{1},\dots,j_{k}}q_{j_{1}}\dots
q_{j_{k}}\sum_{\ell=0}^{k}\pi_{k,\ell}\,.$ (21)

We trivially have $\sum_{\ell=0}^{k}\pi_{k,\ell}=1$ and, by assuming absence
of correlations among nodes degrees, we also have
$\sum_{j_{1},\dots,j_{k}}q_{j_{1}}\dots
q_{j_{k}}=\sum_{j_{1}}q_{j_{1}}\dots\sum_{j_{k}}q_{j_{k}}=1$, hence we can
simplify Eq. (21) into:

$\frac{da_{k}}{dt}=\sum_{j_{1},\dots,j_{k}}q_{j_{1}}\dots
q_{j_{k}}\sum_{\ell=0}^{k}\pi_{k,\ell}P_{\alpha}\left(\frac{k-\ell}{k-\ell+\ell
Q}\right)-a_{k}\,.$ (22)

As already indicated in the main text, we define $\langle
a\rangle:=\sum_{j}q_{j}a_{j+1}$. By using combinatorial and assuming
probability independence, we can show that

$\sum_{j_{1},\dots,j_{k}}q_{j_{1}}\dots
q_{j_{k}}\pi_{k,\ell}=\binom{k}{\ell}\langle a\rangle^{k-\ell}\left(1-\langle
a\rangle\right)^{\ell}\,;$

the rough idea is that in $\pi_{k,\ell}$ there are $k-\ell$ events with
probability $a_{j_{m}+1}$, thus $\ell$ with $(1-a_{j_{m}+1})$, and the
binomial coefficient computes all possible permutations. We can thus rewrite
Eq. (22) as

$\frac{da_{k}}{dt}=-a_{k}+\sum_{\ell=0}^{k-1}\binom{k}{\ell}\langle
a\rangle^{k-\ell}\left(1-\langle
a\rangle\right)^{\ell}P_{\alpha}\left(\frac{k-\ell}{k-\ell+\ell Q}\right)\,,$
(23)

where we removed from the sum the term $\ell=k$ because it contains
$P_{\alpha}\left(0\right)=0$. By rewriting the previous equation with
$k\rightarrow k+1$, and by multiplying both sides by $q_{k}$ and summing over
$k$ to bring out $\langle a\rangle$ we get Eq. (9) in the main text, namely:

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

where the right hand side defines the function $f_{\alpha}^{(hmf)}(\langle
a\rangle)$.

#### Stability analysis

Let us now consider the zeros of $f_{\alpha}^{(hmf)}(\langle a\rangle)$, hence
the equilibria of the system. Because the sum over $\ell$ ranges from $\ell=0$
and $\ell=k$, and because the involved terms are of the form $\langle
a\rangle^{k+1-\ell}$, they all vanish once $\langle a\rangle=0$, hence
$f_{\alpha}^{(hmf)}(\langle a\rangle)=0$. The same holds true for $\langle
a\rangle=1$, indeed

$f_{\alpha}^{(hmf)}(1)=-1+\sum_{k}q_{k}\sum_{\ell=0}^{k}\binom{k+1}{\ell}\left(1-\langle
a^{*}\rangle\right)\rvert_{\langle
a^{*}\rangle=1}^{\ell}P_{\alpha}\left(\frac{k+1-\ell}{k+1-\ell+\ell
Q}\right)=-1+\sum_{k}q_{k}\,,$

where we used the fact that all the terms $\left(1-\langle
a^{*}\rangle\right)\rvert_{\langle a^{*}\rangle=1}^{\ell}$ vanish except the
one with $\ell=0$, for which we also have $\binom{k+1}{0}=1$ and
$P_{\alpha}\left(\frac{k+1}{k+1}\right)=1$. The conclusion follows by
recalling that $\sum_{k}q_{k}=1$.

The stability of the above equilibria can be determined by considering the
derivative of $f_{\alpha}^{(hmf)}$ at $0$ and $1$ that is given by

$\begin{split}\left(f_{\alpha}^{(hmf)}\right)^{\prime}(\langle
a^{*}\rangle)=-1+\sum_{k}q_{k}\sum_{\ell=0}^{k}\binom{k+1}{\ell}P_{\alpha}\left(\frac{k+1-\ell}{k+1-\ell+\ell
Q}\right)\times\\\ \left[(k+1-\ell)\langle
a^{*}\rangle^{k-\ell}\left(1-\langle a^{*}\rangle\right)^{\ell}-\ell\langle
a^{*}\rangle^{k+1-\ell}\left(1-\langle
a^{*}\rangle\right)^{\ell-1}\right]\,,\end{split}$ (24)

hence

$\left(f_{\alpha}^{(hmf)}\right)^{\prime}(0)=-1+\sum_{k}q_{k}\binom{k+1}{k}P_{\alpha}\left(\frac{1}{1+kQ}\right)=-1+\sum_{k}q_{k}(k+1)P_{\alpha}\left(\frac{1}{1+kQ}\right)\,,$
(25)

and

$\displaystyle\left(f_{\alpha}^{(hmf)}\right)^{\prime}(1)$ $\displaystyle=$
$\displaystyle-1+\sum_{k}q_{k}\left[\binom{k+1}{0}P_{\alpha}\left(\frac{k+1}{k+1}\right)(k+1)-\binom{k+1}{1}P_{\alpha}\left(\frac{k}{k+Q}\right)\right]$
(26) $\displaystyle=$
$\displaystyle-1+\sum_{k}q_{k}(k+1)\left[1-P_{\alpha}\left(\frac{k}{k+Q}\right)\right]=-1+\sum_{k}q_{k}(k+1)P_{\alpha}\left(\frac{Q}{k+Q}\right)\,.$

In Fig. 9 we report four examples of the function
$f_{\alpha}(x)=x+f_{\alpha}^{(hmf)}(x)$ for four values of $\alpha$ for a
scale-free network with exponent $\gamma=2.2$. Observe that, differently from
Fig. 1, the function is smooth even for $\alpha=0$ (red curve). Additionally,
the presence of three intersections of $f_{\alpha}(x)$ with the line $y=x$,
hence three zeros for $f_{\alpha}^{(hmf)}(x)$, indicates the presence of three
system equilibria. For $\alpha=1$ (yellow line), there are only two line
intersections, at $x=0$ and $x=1$, indicating the existence of only two
equilibria.

---
Figure 9: The function $f_{\alpha}(x)=x+f_{\alpha}^{(hmf)}(x)$ for some values
of $\alpha$ for a scale free network made by $N=400$ nodes, $\gamma=2.2$,
$k_{min}=3$, $k_{max}=189$, the quality ratio is $Q=0.9$. The dashed line
represents the identity curve and thus its intersections with the function
$f_{\alpha}(x)$ determine the equilibria of the system.

#### Limitations of the heterogeneous mean-field model

We conclude this appendix by studying the limitation of the heterogeneous
mean-field model, more precisely we look for (family of) networks for which
there is a disagreement between the dynamics predicted by the HMF model and
the numerical agent-based simulations. More precisely, we look at the case in
which the HMF model’s equilibrium $\langle\check{a}^{*}\rangle=0$ is unstable,
i.e., Eq. (25) is positive, and $\langle\hat{a}^{*}\rangle=1$ is stable, i.e.,
Eq. (26) is negative. Therefore, the prediction is consistent convergence for
any initial state to the latter stable equilibrium
$\langle\hat{a}^{*}\rangle=1$; however, numerical simulations do not always
terminate with all agents with opinion $A$.

For any given $Q<1$ there exists $\bar{k}$ such that $kQ>1$ for all
$k\geq\bar{k}$ and thus $kQ<1$ for all $k<\bar{k}$. Hence

$kQ>1\Rightarrow\frac{1}{1+kQ}<\frac{1}{2}\Rightarrow
P_{\alpha}\left(\frac{1}{1+kQ}\right)=\frac{1}{2}-\frac{1}{2}\left(1-\frac{2}{1+kQ}\right)^{\alpha}\,,$

and

$kQ<1\Rightarrow\frac{1}{1+kQ}>\frac{1}{2}\Rightarrow
P_{\alpha}\left(\frac{1}{1+kQ}\right)=\frac{1}{2}+\frac{1}{2}\left(\frac{2}{1+kQ}-1\right)^{\alpha}\,.$

Eq. (25) rewrites thus

$\displaystyle\left(f_{\alpha}^{(hmf)}\right)^{\prime}(0)$ $\displaystyle=$
$\displaystyle-1+\frac{1}{2}\sum_{k\geq\bar{k}}q_{k}(k+1)\left[1-\left(1-\frac{2}{1+kQ}\right)^{\alpha}\right]+\frac{1}{2}\sum_{k<\bar{k}}q_{k}(k+1)\left[1+\left(\frac{2}{1+kQ}-1\right)^{\alpha}\right]=$
$\displaystyle=$
$\displaystyle-1+\frac{1}{2}\sum_{k}q_{k}(k+1)+\frac{1}{2}\sum_{k<\bar{k}}q_{k}(k+1)\left(\frac{2}{1+kQ}-1\right)^{\alpha}-\frac{1}{2}\sum_{k\geq\bar{k}}q_{k}(k+1)\left(1-\frac{2}{1+kQ}\right)^{\alpha}\,.$

By using the definition of $q_{k}$, we can compute

$\sum_{k\geq 0}q_{k}(k+1)=\sum_{k\geq 0}\frac{(k+1)p_{k+1}}{\langle
k\rangle}(k+1)=\frac{1}{\langle k\rangle}\sum_{k\geq
1}k^{2}p_{k}=\frac{1}{\langle k\rangle}\sum_{k\geq 0}k^{2}p_{k}=\frac{\langle
k^{2}\rangle}{\langle k\rangle}\,,$

hence

$\displaystyle\left(f_{\alpha}^{(hmf)}\right)^{\prime}(0)=-1+\frac{1}{2}\frac{\langle
k^{2}\rangle}{\langle
k\rangle}+\frac{1}{2}\sum_{k<\bar{k}}q_{k}(k+1)\left(\frac{2}{1+kQ}-1\right)^{\alpha}-\frac{1}{2}\sum_{k\geq\bar{k}}q_{k}(k+1)\left(1-\frac{2}{1+kQ}\right)^{\alpha}\,.$

To compute $\left(f_{\alpha}^{(hmf)}\right)^{\prime}(0)$ we observe that if
$k\geq 1>Q$ then $Q/(k+Q)<1/2$ and thus we can conclude

$\displaystyle\left(f_{\alpha}^{(hmf)}\right)^{\prime}(1)$ $\displaystyle=$
$\displaystyle-1+q_{0}+\frac{1}{2}\sum_{k\geq
1}q_{k}(k+1)\left[1-\left(1-\frac{2Q}{k+Q}\right)^{\alpha}\right]=$
$\displaystyle=$ $\displaystyle-1+\frac{q_{0}}{2}+\frac{1}{2}\frac{\langle
k^{2}\rangle}{\langle k\rangle}-\frac{1}{2}\sum_{k\geq
1}q_{k}(k+1)\left(1-\frac{2Q}{k+Q}\right)^{\alpha}\,.$

For sake of definitiveness let us assume $1/2<Q<1$ and thus $\bar{k}=2$. Hence
$\left(f_{\alpha}^{(hmf)}\right)^{\prime}(0)$ simplifies into

$\displaystyle\left(f_{\alpha}^{(hmf)}\right)^{\prime}(0)=-1+\frac{1}{2}\frac{\langle
k^{2}\rangle}{\langle
k\rangle}+q_{1}\left(\frac{2}{1+Q}-1\right)^{\alpha}-\frac{1}{2}\sum_{k\geq
2}q_{k}(k+1)\left(1-\frac{2}{1+kQ}\right)^{\alpha}\,.$

Finally let us consider a $2$-ring network where, i.e., each node is connected
with its two neighbours, hence $\langle k\rangle=2$, $\langle k^{2}\rangle=4$,
$p_{2}=1$ and $p_{k}=0$ for all $k\neq 2$ and thus $q_{1}=1$ and $q_{k}=0$ if
$k\neq 1$. The previous equation simplifies to give

$\displaystyle\left(f_{\alpha}^{(hmf)}\right)^{\prime}(0)=-1+\frac{1}{2}\frac{4}{2}+\left(\frac{2}{1+Q}-1\right)^{\alpha}=\left(\frac{2}{1+Q}-1\right)^{\alpha}>0\,,$

namely $\langle\check{a}^{*}\rangle=0$ is unstable under the assumption of
HMF. Similarly the equation for $\left(f_{\alpha}^{(hmf)}\right)^{\prime}(1)$
rewrites

$\left(f_{\alpha}^{(hmf)}\right)^{\prime}(1)=-1+\frac{1}{2}\frac{4}{2}-\frac{1}{2}q_{1}2\left(1-\frac{2Q}{1+Q}\right)^{\alpha}=-\left(\frac{1-Q}{1+Q}\right)^{\alpha}<0\,,$

namely $\langle\hat{a}^{*}\rangle=1$ is stable according to the HMF theory.

In Fig. 10 we show the results of numerical simulation of $500$ agents
exchanging opinions on a $2$-ring, i.e., each agent has two neighbours. Each
point is asymptotic value after $500\,000$ time steps, of $\langle
n_{A}^{*}\rangle/N$ averaged over $401$ independent simulations as a function
of $\alpha$ for $Q=0.9$. Simulations have different initial conditions
(initial opinions distributed differently on the network) but always with
$250$ agents with opinion $A$ and $250$ agents with opinion $B$, i.e., half of
the population committed to each option. One can observe that for $\alpha<1$
the simulations converge to $\langle n_{A}^{*}\rangle/N=1$ and thus the claim
of the predictions of the HMF model are confirmed. Whereas the good match
between theory and simulations is no longer valid for $\alpha>1$. Theory
predicts a single stable equilibria for full consensus for $A$, while the
simulated system remains locked at indecision at $0<\tilde{a}*<1$ with only a
part of the agents with opinion $A$ and the rest with opinion $B$.

---
Figure 10: Asymptotic state for the agent based model on a $2$-ring. In the
main panel we report the distribution of the asymptotic state $\langle
n_{A}^{*}\rangle/N$ for for $N=500$ agents interacting in a $2$-ring as a
function of the parameter $\alpha$ for a fixed $Q=0.9$. For each given
$\alpha$ we repeat the simulation $n_{iter}=400$ times and we then we display
the probability distribution of the obtained values (the darker the higher the
probability). On the left panel we report the case small $\alpha$, here
$\alpha=0.59$, and we can appreciate the fact the all the simulations returned
$\langle n_{A}^{*}\rangle/N=1$; on the right panel we show the case of large
$\alpha$, here $\alpha=1.5$, and we can observe a large spreading of values
about the mean (blue dots in the main panel) but also the presence of a small
peak corresponding to $\langle n_{A}^{*}\rangle/N=1$ and an even smaller for
$\langle n_{A}^{*}\rangle/N=0$.

The different behaviour for $\alpha>1$ is due to finite-size effects and can
be explained as follows. Let us assume to have a ring with $N$ agents and
assume that all agents but one have opinion $A$, we are interested in the
possibility that also this last $B$-agent changes her mind and becomes
$A$-agent, in this way the system reaches the equilibrium $\langle
n_{A}^{*}\rangle/N=1$, i.e., all $A$. This process should be compared with the
one where one agent $A$ becomes a $B$-agent, the ratio of the probabilities of
those two events determines the stability (or not) of the state all $A$.

The probability that the event $B\rightarrow A$ happens is the combination of
the probability of selecting the $B$-agent, hence probability $1/N$, and the
probability that she will receive a message from a neighbour committed to $A$.
Because there is only one $B$-agent, her two neighbours have opinion $A$ and
thus $P_{\alpha}(1)=1$. In summary, $P(B\rightarrow A)=1/N$.

The probability that the event $A\rightarrow B$ happens is the combination of
the probability of selecting one of the two agents $A$ sitting in the ring
next to the unique agent $B$, hence probability $2/N$, and the probability
that the selected agent $A$ will select the message from the $B$-agent. The
weighted proportion of agents $B$ that are neighbours of the selected agent
$A$ is $P_{\alpha}\left(\frac{Q}{1+Q}\right)$. Being $Q\in[0,1]$, the quantity
$\frac{Q}{1+Q}$ is smaller than $1/2$ and thus
$P_{\alpha}\left(\frac{Q}{1+Q}\right)=\frac{1}{2}-\frac{1}{2}\left(\frac{1-Q}{1+Q}\right)^{\alpha}$.
In summary, $P(A\rightarrow
B)=\frac{2}{N}\left[\frac{1}{2}-\frac{1}{2}\left(\frac{1-Q}{1+Q}\right)^{\alpha}\right]$.

Finally, we can compute the ratio of the probabilities for the two events

$\frac{P(A\rightarrow B)}{P(B\rightarrow
A)}=1-\left(\frac{1-Q}{1+Q}\right)^{\alpha}\,.$

Because $\frac{1-Q}{1+Q}<1$, we can conclude that if $\alpha<1$, then
$\frac{P(A\rightarrow B)}{P(B\rightarrow A)}$ is small and thus the system has
a large probability to evolve toward a consensus for $A$, the event
$B\rightarrow A$ is much more probable than $A\rightarrow B$. On the other
hand, if $\alpha>1$, then the above probability approaches $1$ and thus both
events are (almost) equally probable. Therefore, despite the equilibrium of a
consensus for $A$ being stable, it is difficult to reach it because the two
events ($A\rightarrow B$ and $B\rightarrow A$) are equally likely to happen
and the system can fluctuate indefinitely.

##  Appendix C The $2m$-regular graph

Let us consider now $2m$-regular graphs, $m\geq 1$, namely networks where all
the nodes have the same degree $2m$. Observe that $1$-dimensional rings where
each node is connected to $m$ left and $m$ right neighbours fall in this
class, that however contains more general structures. By construction we
trivially have $p_{k}=1$ if $k=2m$ and $p_{k}=0$ otherwise, then $\langle
k\rangle=2m$, which implies that $q_{k}=1$ if $k=2m-1$ and $0$ otherwise,
indeed

$q_{k}=\frac{k+1}{\langle
k\rangle}p_{k+1}=\begin{cases}\frac{(2m-1)+1}{2m}p_{2m}=1&\text{if
$k+1=2m$}\\\ 0&\text{if $k+1\neq 2m$}\,.\end{cases}$

Therefore, the weighted average proportion of agents with opinion $A$ simply
becomes $\langle a\rangle=a_{2m}$. From the definition (9), we can simplify
the function $f_{\alpha}^{(hmf)}(x)$ and obtain

$f_{\alpha}^{(hmf)}(x)=-x+\sum_{\ell=0}^{2m-1}\binom{2m}{\ell}x^{2m-\ell}\left(1-x\right)^{\ell}P_{\alpha}\left(\frac{2m-\ell}{2m-\ell+\ell
Q}\right)\,.$ (27)

The derivatives of (27) evaluated at $\check{x}^{*}=0$ and $\hat{x}^{*}=1$ are
given by (see also (25) and (26)):

$(f_{\alpha}^{(hmf)})^{\prime}(0)=-1+2mP_{\alpha}\left(\frac{1}{1+(2m-1)Q}\right)\quad\text{and}\quad(f_{\alpha}^{(hmf)})^{\prime}(1)=-1+2mP_{\alpha}\left(\frac{Q}{2m-1+Q}\right)\,.$

In conclusion, the equilibrium $\check{a}^{*}_{2m}=0$, i.e., all agents have
opinion $B$, is stable if and only if

$(f_{\alpha}^{(hmf)})^{\prime}(0)=-1+2mP_{\alpha}\left(\frac{1}{1+(2m-1)Q}\right)<0\,,$

and similarly $\hat{a}^{*}_{2m}=1$, i.e., all agents have opinion $A$, is
stable if and only if

$(f_{\alpha}^{(hmf)})^{\prime}(1)=-1+2mP_{\alpha}\left(\frac{Q}{2m-1+Q}\right)<0\,.$

When $Q<1/(2m-1)$, then $1/[(2m-1)Q+1]>1/2$, therefore, using the definition
of $P_{\alpha}$ we get:

$(f_{\alpha}^{(hmf)})^{\prime}(0)=-1+2m\left[\frac{1}{2}+\frac{1}{2}\left(\frac{2}{1+(2m-1)Q}-1\right)^{\alpha}\right]=-1+m\left[1+\left(\frac{2}{1+(2m-1)Q}-1\right)^{\alpha}\right]\geq-1+m\,,$

and because $m\geq 2$, the latter expression is positive for all $\alpha\geq
0$. In conclusion, the equilibrium $\check{a}^{*}_{2m}=0$ is unstable for all
$\alpha\geq 0$ and $Q<1/(2m-1)$.

Let us now consider the case $Q>1/(2m-1)$. By definition of $P_{\alpha}$ we
get

$(f_{\alpha}^{(hmf)})^{\prime}(0)=-1+2m\left[\frac{1}{2}-\frac{1}{2}\left(1-\frac{2}{1+(2m-1)Q}\right)^{\alpha}\right]=-1+m\left[1-\left(1-\frac{2}{1+(2m-1)Q}\right)^{\alpha}\right]\,,$

and now the quantity on the right-hand side can have both signs. Let us define
$\hat{\alpha}(Q)$ as the value of $\alpha$ for which the right-hand side
vanishes for a fixed $Q>1/(2m-1)$, then we can straightforwardly obtain

$\hat{\alpha}(Q)=\frac{\log\left(1-\frac{1}{m}\right)}{\log\left(1-\frac{2}{1+(2m-1)Q}\right)}\,.$
(28)

By looking at its definition we can conclude that $\hat{\alpha}(1)=1$ and that
$\hat{\alpha}(Q)\rightarrow 0$ for $Q\rightarrow 1/(2m-1)$ (from values larger
than $1/(2m-1)$). Given $Q>1/(2m-1)$, then
$(f_{\alpha}^{(hmf)})^{\prime}(0)>0$ for all $\alpha>\hat{\alpha}(Q)$; this
means that the equilibrium $\check{a}^{*}_{2m}=0$ is unstable. The function
$\hat{\alpha}(Q)$ is drawn in white in Fig. 8 and it delimits the red region
where $(f_{\alpha}^{(hmf)})^{\prime}(0)>0$ ($\check{a}^{*}_{2m}=0$ is
unstable) and the green region where $(f_{\alpha}^{(hmf)})^{\prime}(0)<0$
($\check{a}^{*}_{2m}=0$ is stable).

Let us now consider the stability of the equilibrium $\hat{a}^{*}_{2m}=1$.
Because $m\geq 1$ and $Q<1$, we always have $Q/(2m-1+Q)<1/2$, hence by
definition of $P_{\alpha}$ we obtain

$(f_{\alpha}^{(hmf)})^{\prime}(1)=-1+2m\left[\frac{1}{2}-\frac{1}{2}\left(1-\frac{2Q}{2m-1+Q}\right)^{\alpha}\right]=-1+m\left[1-\left(1-\frac{2Q}{2m-1+Q}\right)^{\alpha}\right]\,.$
(29)

Eq. (29) can also have either positive and negative values. Let
$\tilde{\alpha}(Q)$ the value of $\alpha$ for which the right-hand side of Eq.
(29) vanishes for a fixed $Q$, then

$\tilde{\alpha}(Q)=\frac{\log\left(1-\frac{1}{m}\right)}{\log\left(1-\frac{2Q}{2m-1+Q}\right)}\,.$
(30)

We have $\tilde{\alpha}(1)=1$ and $\tilde{\alpha}(Q)\rightarrow\infty$ if
$Q\rightarrow 0^{+}$. The function $\tilde{\alpha}(Q)$ is drawn in black in
Fig. 8 and it delimits the red region where
$(f_{\alpha}^{(hmf)})^{\prime}(1)<0$ ($\hat{a}^{*}_{2m}=1$ is stable) and the
blue region where $(f_{\alpha}^{(hmf)})^{\prime}(1)>0$ ($\hat{a}^{*}_{2m}=1$
is unstable).

#### Generalise to complete graphs

Let us conclude this part by showing the previous analysis returns the results
obtained by using the mean-field hypothesis once we assume the underlying
network to be a complete graph. To simplify the setting we will assume the
network to be composed of $N=2N^{\prime}+1$ nodes and $m=N^{\prime}$, hence
each node has $2N^{\prime}$ neighbours. Let us observe that one trivially has
$p_{k}=1$ if $k=N-1=2N^{\prime}$ and $p_{k}=0$ otherwise, and $\langle
k\rangle=N-1=2N^{\prime}$, which implies that $q_{k}=1$ if
$k=N-2=2N^{\prime}-2=2(m-1)$ and $0$ otherwise. From Eq. (8) we can obtain
$\langle a\rangle=a_{N-1}=n_{A}/N$, namely there is only one variable that is
the proportion of agents with opinion $A$. From the Eq. (27) we can get

$f_{\alpha}^{(hmf)}(a_{N-1})=-a_{N-1}+\sum_{\ell=0}^{2N^{\prime}}\binom{2N^{\prime}}{\ell}(a_{N-1})^{2N^{\prime}-\ell}\left(1-a_{N-1}\right)^{\ell}P_{\alpha}\left(\frac{n_{A}}{n_{A}+n_{B}Q}\right)\,,$

where we recall that $\ell=n_{B}$, to be the number of agents with opinion
$B$, and $2N^{\prime}-\ell=n_{A}$, the number of agents with opinion $A$. Let
us observe that we also added the term $\ell=2N^{\prime}$ in the sum, whose
contribution vanishes because $P_{\alpha}$ does. By using $n_{A}+n_{B}=N$ we
can rewrite the previous equation as

$\displaystyle f_{\alpha}^{(hmf)}(a_{N-1})$ $\displaystyle=$ $\displaystyle-
a_{N-1}+\sum_{\ell=0}^{2N^{\prime}}\binom{2N^{\prime}}{\ell}(a_{N-1})^{2N^{\prime}-\ell}\left(1-a_{N-1}\right)^{\ell}P_{\alpha}\left(\frac{n_{A}}{n_{A}(1-Q)+NQ}\right)$
$\displaystyle=$ $\displaystyle-
a_{N-1}+\sum_{\ell=0}^{2N^{\prime}}\binom{2N^{\prime}}{\ell}(a_{N-1})^{2N^{\prime}-\ell}\left(1-a_{N-1}\right)^{\ell}P_{\alpha}\left(\frac{a_{N-1}}{a_{N-1}(1-Q)+Q}\right)\,,$

where in the last step we divided by $N$ the number of agents to obtain the
proportion. Being the term involving $P_{\alpha}$ independent from $\ell$, we
eventually obtain

$\begin{split}f_{\alpha}^{(hmf)}(a_{N-1})&=-a_{N-1}+P_{\alpha}\left(\frac{a_{N-1}}{a_{N-1}(1-Q)+Q}\right)\sum_{\ell=0}^{2N^{\prime}}\binom{2N^{\prime}}{\ell}(a_{N-1})^{2N^{\prime}-\ell}\left(1-a_{N-1}\right)^{\ell}=\\\
&=-a_{N-1}+P_{\alpha}\left(\frac{a_{N-1}}{a_{N-1}(1-Q)+Q}\right)\,,\end{split}$

where the Newton property for the binomial has been used to eventually get the
same function we obtained under the mean field assumption (4).

Let us now rewrite $\hat{\alpha}(Q)$ given by Eq. (28) under the above
assumption of complete graph, namely

$\hat{\alpha}(Q)=\frac{\log\left(1-\frac{1}{N^{\prime}}\right)}{\log\left(1-\frac{2}{1+(2N^{\prime}-1)Q}\right)}\,,$

then letting the number of nodes to be very large,
$N=2N^{\prime}+1\rightarrow\infty$, then we obtain

$\hat{\alpha}(Q)\sim Q+\dots\,,$

namely the curve separating the convergence to a consensus for option $A$ (red
region in Fig. 8) to the region where mistakes are possible (green region in
Fig. 8) converges to the line $\hat{\alpha}(Q)=Q$ in the limit of infinitely
many agents, in agreement with the results reported in Fig. 3.