11institutetext:  Department of Computer Science, UFMG, Brazil
22institutetext: Department of Computer Science, University of Minnesota
Duluth, USA 33institutetext: CNRS-LIX, École Polytechnique de Paris, France
44institutetext: Pontificia Universidad Javeriana Cali, Colombia

# A Multi-Agent Model for Opinion Evolution under Cognitive Biases

Mário S. Alvim 11 Artur Gaspar da Silva 11 Sophia Knight 22 Frank Valencia
Mário S. Alvim and Artur Gaspar da Silva were partially supported by CNPq,
CAPES, and FAPEMIG. Frank Valencia’s contribution to this work is partially
supported by the SGR project PROMUEVA (BPIN 2021000100160) supervised by
Minciencias.4455

###### Abstract

We generalize the DeGroot model for opinion dynamics to better capture
realistic social scenarios. We introduce a model where each agent has their
own individual _cognitive biases_. Society is represented as a directed graph
whose edges indicate how much agents influence one another. Biases are
represented as the functions in the square region $[-1,1]^{2}$ and categorized
into four sub-regions based on the potential reactions they may elicit in an
agent during instances of _opinion disagreement_. Under the assumption that
each bias of every agent is a _continuous_ function within the region of
receptive but resistant reactions ($\mathbf{R}$), we show that the society
converges to a consensus if the graph is strongly connected. Under the same
assumption, we also establish that the entire society converges to a unanimous
opinion if and only if the _source components_ of the graph—namely, strongly
connected components with no external influence—converge to that opinion. We
illustrate that convergence is not guaranteed for strongly connected graphs
when biases are either discontinuous functions in $\mathbf{R}$ or not included
in $\mathbf{R}$. We showcase our model through a series of examples and
simulations, offering insights into how opinions form in social networks under
cognitive biases.

###### Keywords:

Cognitive bias, Multi-Agent Systems, Social Networks

##  1 Introduction

In recent years, the significance and influence of social networks have
experienced a remarkable surge, capturing widespread attention and shaping
users’ opinions in substantial ways. The _dynamics of opinion/belief
formation_ in social networks involves individuals expressing their opinions,
being exposed to the opinions of others, and adapting or reinforcing their own
views based on these interactions. Modeling these dynamics allows us to gain
insights into how opinions form, spread, and evolve within social networks.

The DeGroot multi-agent model [5] is one of most prominent formalisms for
opinion formation dynamics in social networks. Society is represented as a
directed graph whose edges indicate how much individuals (called _agents_)
influence one another. Each agent has an opinion represented as a value in
$[0,1]$ indicating the strength of their agreement with an underlying
proposition (e.g., “ _vaccines are safe_ ”). They repeatedly update their
opinions with the weighted average of their opinion differences (level of
_disagreement_) with those who influence them. The DeGroot model is valued for
its tractability, derived from its connection with matrix powers and Markov
chains, and it remains a significant focus of study providing a comprehensive
understanding of opinion evolution[6].

Nevertheless, the DeGroot model has an important caveat: It assumes
_homogeneity_ and _linearity_ of opinion update. In social scenarios, however,
two agents may update their opinions differently depending on their individual
_cognitive biases_ on disagreement—i.e., how they interpret and react towards
the level of disagreements with others. This results in more complex updates
that may involve non-linear even non-monotonic functions. For example, an
individual under _confirmation (cognitive) bias_ [2] may ignore the opinion of
those whose level of disagreement with them is over a certain threshold. In
fact, much of the unpredictability in opinion formation is due to users’
_biases_ in their belief updates, where users sometimes tend to reinforce
their original beliefs, instead of questioning and updating their opinions
upon encountering new information. Indeed, rather than perfect rational
agents, users are often subject to cognitive biases.

In an earlier FORTE paper [1], we introduced a DeGroot-like model with a _non-
linear_ update mechanism tailored for a specific type of confirmation bias.
The model was shown to be tractable and it provides insights into the effect
of this cognitive bias in opinion dynamics. Nevertheless, it also assumes
homogeneity of opinion update, and choosing a particular function to represent
the bias, although natural, may seem somewhat ad-hoc.

To address the above-mentioned caveat, in this paper we introduce a
generalization of the DeGroot model that allows for _heterogeneous_ and _non-
linear_ opinion updates. Each agent has their own individual cognitive biases
on levels of disagreement. These biases are represented as arbitrary functions
in the square region $[-1,1]^{2}$. The model then unifies disparate belief
update styles with bias into a single framework which takes _disagreement_
between agents as the central parameter. Indeed, standard cognitive biases of
great importance in social networks such as backfire effect [8], authority
bias [9], and confirmation bias [2], among others, can be represented in the
framework.

We classify the biases in $[-1,1]^{2}$ into four sub-regions
($\mathbf{M},\mathbf{R},\mathbf{B},\mathbf{I}$) based on the cognitive
reactions they may cause in an agent during instances of _opinion
disagreement_. For example, agents that are malleable, easily swayed, exhibit
fanaticism or prompt to follow authoritative figures can be modelled with
biases in $\mathbf{M}$. Agents that are receptive to other opinions, but
unlike malleable ones, can exhibit some skepticism to fully accepting them can
be modelled with biases in the region $\mathbf{R}$. Individuals that become
more extreme when confronted with opposing opinions can be modelled by biases
in $\mathbf{B}$. Finally insular agents can be modelled with the bias in
$\mathbf{I}$.

_Consensus_ is a central property for opinion formation dynamics. Indeed the
inability to converge to consensus is often a sign of a polarized society. In
this paper we use the above-mentioned region classification to provide the
following insightful theoretical results for consensus.

  * •

Assuming that each bias of every agent is a _continuous_ function in
$\mathbf{R}$, the society converges to a consensus if that society is strongly
connected. This implies that a strongly connected society can converge to a
consensus if its members are receptive but resistant to the opinions of
others.

  * •

Under the same assumption, we also establish that the entire society converges
to a unanimous opinion if and only if the _source components_ of the graph,
i.e., strongly connected components with no external influence, converge to
that opinion. This implies that upon agreeing on an opinion, closed and
potentially influential groups, can make all individuals converge to that
opinion in a society whose members are receptive but resistant.

  * •

We show that convergence is not guaranteed for strongly connected graphs when
biases are either discontinuous functions in $\mathbf{R}$ or not included in
$\mathbf{R}$.

We also demonstrate our model with examples and computer simulations that
provide insights into opinion formation under cognitive biases. The open code
for these simulations can be found at
https://github.com/bolaabcd/polarization2.

##  2 An Opinion Model with Cognitive Biases

The DeGroot model [5] is a well-known model for social learning. In this
formalism each individual (_agent_) repeatedly updates their current opinion
by averaging the opinion values of those who influence them. But one of its
limitation is that the model does not provide a mechanism for capturing the
_cognitive biases_ under which each individual may interpret and react to the
opinion of others.

In this section we introduce a generalization of the DeGroot model with a
mechanism to express arbitrary cognitive bias based on opinion disagreement.

###  2.1 Influence Graph.

In social learning models, a _community/society_ is typically represented as a
directed weighted graph with edges between individuals (agents) representing
the direction and strength of the influence that one carries over the other.
This graph is referred to as the _Influence Graph_.

######  Definition 1 (Influence Graph).

An ($n$-agent) _influence graph_ is a directed weighted graph $G=(A,E,I)$ with
$A=\\{1,\ldots,n\\}$ the vertices, $E\subseteq A\times A$ the edges, and
$I:A\times A\to[0,1]$ a weight function s.t. $I(i,j)=0$ iff $(i,j)\notin E$.

The vertices in $A$ represent $n$ agents of a given community or network. The
set of edges $E\subseteq A\times A$ represents the (direct) influence relation
between these agents; i.e., $(i,j)\in E$ means that agent $i$ influences agent
$j$. The value $I(i,j)$, for simplicity written $I_{i,j}$, denotes the
strength of the influence: $0$ means no influence and a higher value means
stronger influence. We use $A_{i}$ to denote the set $\\{i\ |\ (j,i)\in E\\}$
of agents that have a direct influence over agent $i$.

######  Remark 2.1.

In contrast to [1], we do not require agents to have nonzero self-influence.
Furthermore, since we do not require the sum of influences over a given agent
to be 1 (unlike [5]), we will use the following notation for _proportional
influence_ of $j$ over $i$: $\overline{I_{j,i}}=\frac{I_{j,i}}{\sum_{k\in
A_{i}}I_{k,i}}$ if $(j,i)\in E$, else $\overline{I_{j,i}}=0$.

###  2.2 General Opinion Update.

Similar to the DeGroot-like models in [6], we model the evolution of agents’
opinions about some underlying _statement_ or _proposition_. For example, such
a proposition could be “ _vaccines are unsafe_ ,” “ _human activity has little
impact on climate change_ ,”, “ _AI poses a threat to humanity_ ”, or “
_Reviewer 2 is wonderful_ ”.

The _state of opinion_ (or _belief state_) of all the agents is represented as
a vector in $[0,1]^{|A|}$. If $B$ is a state of opinion, $B_{i}\in[0,1]$
denotes the _opinion_ (_belief_ , or _agreement_) value of agent $i\in A$
regarding the underlying proposition. If $B_{i}=0$, agent $i$ completely
disagrees with the underlying proposition; if $B_{i}=1$, agent $i$ completely
agrees with the underlying proposition. Furthermore, the higher the value of
$B_{i}$, the stronger the agreement with such a proposition.

At each time unit $t\in\mathbb{N}$, every agent $i\in A$ updates their
opinion. We shall use $B^{t}$ to denote the state of opinion at time
$t\in\mathbb{N}.$ We can now define a general DeGroot-like opinion model as
follows.

######  Definition 2 (Opinion Model).

An _Opinion Model_ is a tuple $(G,B^{0},\mu_{G})$ where $G$ is an $n$-agent
influence graph, $B^{0}$ is the initial state of opinion, and
$\mu_{G}:[0,1]^{n}\rightarrow[0,1]^{n}$ is a state-transition function, called
_update function_. For every $t\in\mathbb{N}$, the state of opinion at time
$t+1$ is given by $B^{t+1}=\mu_{G}(B^{t})$.

The update functions can be used to express any deterministic and discrete
transition from one opinion state to the next, possibly taking into account
the influence graph. This work singles out and characterizes a meaningful
family of update functions extending the basic DeGroot model with cognitive
biases that are based on opinion _disagreement_. Intuitively, these update
functions specify the reaction of an agent to the opinion disagreements with
each of their influencers. To build some intuition, we first recall the update
function of the DeGroot model.

Below we omit the index from the update function $\mu_{G}$ if no confusion
arises.

###  2.3 DeGroot Update

The standard DeGroot model [5] is obtained by the following update function:

$\mu(B)_{i}=\sum_{j\in A_{i}}\overline{I_{j,i}}B_{j}$ (1)

for every $i\in A$. Thus, in the DeGroot model each agent updates their
opinion by taking the weighted average of the opinions of those who influence
them. We can rewrite Eq.1 as follows:

$\mu(B)_{i}=B_{i}+\sum_{j\in A_{i}}\overline{I_{j,i}}(B_{j}-B_{i}).$ (2)

Notice that DeGroot update is _linear_ in the agents’ opinions and can be
expressed in terms of _disagreement_ : The opinion of every agent $i$ is
updated taking into account the weighted average of their _opinion
disagreement_ or _opinion difference_ with those who influence them.

Intuitively, if $j$ influences $i$, then $i$’s opinion would tend to move
closer to $j$’s. The _disagreement term_ $(B_{j}-B_{i})\in[-1,1]$ in Eq.2
realizes this intuition. If $(B_{j}-B_{i})$ is a negative term in the sum, the
disagreement can be thought of as contributing with a magnitude of
$|B_{j}-B_{i}|$ (multiplied by $\overline{I_{j,i}}$) to _decreasing_ $i$’s
belief in the underlying proposition. Similarly, if $(B_{j}-B_{i})$ is
positive, the disagreement contributes with the same magnitude but to
_increasing $i$’s_ belief.

###  2.4 Disagreement-Bias Update

Now we generalize DeGroot updates by defining a class of update functions that
also allows for _non-linear_ updates, and for each agent to react differently
to opinion disagreement with distinct agents. We capture this reaction by
means of bias functions $\beta_{i,j}:[-1,1]\to[-1,1]$, where $(j,i)\in E$, on
opinion disagreement stating how the bias of $i$ towards the opinion of $j$,
$\beta_{i,j}$, affects $i$’s new opinion.

In the following definition we use the clamp function for the interval $[0,1]$
which is defined as $[r]_{0}^{1}=\min(\max(r,0),1)$ for any $r\in\mathbb{R}$.

######  Definition 3 (Bias Update).

Let $(G,B^{0},\mu_{G})$ be an opinion model with $G=(A,E,I)$. The function
$\mu_{G}$ is a _(disagreement) bias update_ if for every $i\in A$,

$\mu_{G}(B)_{i}=\left[B_{i}+\sum_{j\in
A_{i}}\overline{I_{j,i}}\beta_{i,j}(B_{j}-B_{i})\right]^{1}_{0}$ (3)

where each $\beta_{i,j}$ with $(j,i)\in E$, called _the (disagreement) bias_
from $i$ towards $j$, is an endo-function111The biases we wish to capture can
be seen as distortions of disagreements, themselves disagreements. It seems
then natural to choose $[-1,1]$ as the domain and co-domain of the bias
function. on $[-1,1]$. The model $(G,B^{0},\mu_{G})$ is a _(disagreement)
bias_ opinion model if $\mu_{G}$ is a disagreement bias update function.

The clamp function $[\cdot]_{0}^{1}$ guarantees that the right-hand side of
Eq.3 yields a valid belief value (a value in $[0,1]$). Intuitively, the
function $\beta_{i,j}$ represents the direction and magnitude of how agent $i$
reacts to their disagreement $B_{j}-B_{i}$ with agent $j$. If
$\beta_{i,j}(B_{j}-B_{i})$ is a negative term in the sum of Eq.3, then the
bias of agent $i$ towards $j$ contributes with a magnitude of
$|\beta_{i,j}(B_{j}-B_{i})|$ (multiplied by $\overline{I_{j,i}}$) to
_decreasing_ $i^{\prime}$s belief in the underlying proposition. Conversely,
if $\beta_{i,j}(B_{j}-B_{i})$ is positive, it contributes to _increasing_
$i^{\prime}$s belief with the same magnitude.

Below we identify some particular examples of the cognitive biases that can be
captured with disagreement-bias opinion models.

######  Example 1 (Some Cognitive Biases).

Clearly, the classical DeGroot update function Eq.2 can be recovered from Eq.3
by letting every bias $\beta_{i,j}$ be the identity on disagreement: i.e.,
$\beta_{i,j}=\mathtt{degroot}$ where $\mathtt{degroot}(x)=x$.

_Confirmation Bias._ We now illustrate some form of _confirmation bias_[2]
where agents are more _receptive_ to opinions that are closer to theirs. An
example of confirmation bias can be obtained by letting
$\beta_{i,j}=\mathtt{conf}(x)=x(1+\delta-|x|)/(1+\delta)$ for a very small
non-negative constant $\delta$.222The confirmation bias function from [1] uses
$\delta=0$ In the following plots and simulations we fix $\delta=1\times
10^{-4}$. This bias causes $i$ to pay less attention to the opinion of $j$ as
their opinion distance $|x|=|B_{j}-B_{i}|$ tends to 1.

_Backfire Effect._ Let us now consider another important cognitive bias called
backfire effect[8]. Under this effect an agent strengthens their position of
disagreement with another agent if their opinions are significantly distant. A
form of backfire effect can be obtained by letting
$\beta_{i,j}=\mathtt{backf}$ where $\mathtt{backf}(x)=-x^{3}$. Notice that
unlike the DeGroot update, this bias contributes to changing $i$’s opinion
with a _magnitude_ of $|\mathtt{backf}(B_{j}-B_{i})|$ (multiplied by
$\overline{I_{j,i}}$) _but in the opposite direction_ of the opinion of $j.$
This potentially makes the new opinion of agent $i$ _more_ distant from that
of $j$.

_Authority Bias._ Another common cognitive bias in social networks is the
authority bias [9] under which individuals tend to blindly follow
authoritative or influential figures often to the extreme. Let
$\beta_{i,j}=\mathtt{fan}$ be the _sign_ function, i.e.,
$\mathtt{fan}(x)=x/|x|\mbox{ if }x\neq 0,\mbox{ otherwise }\mathtt{fan}(x)=0$.
This bias illustrates a case of die-hard _fanaticism_ of $i$ towards $j$.
Intuitively, when confronted with any disagreement $x=B_{j}-B_{i}\neq 0$, this
bias contributes to changing $i$’s opinion with the _highest magnitude_ ,
i.e., $|\beta_{i,j}(x)|=1$, in the _direction_ of the opinion of $j$.

Finally we illustrate a bias that, unlike the previous, causes agents to
ignore opinions of others. We call it the _insular_ bias
$\beta_{i,j}=\mathtt{ins}$ and it is defined as the zero function
$\mathtt{ins}:x\mapsto 0.$ ∎

Figure 1: Bias functions from Ex.1 in the region $[-1,1]^{2}$:
$\mathtt{degroot}$ (in green), $\mathtt{conf}$ (in blue), $\mathtt{backf}$ (in
yellow), $\mathtt{fan}$ (in red), $\mathtt{ins}$ (in orange).

The particular bias function examples of Ex.1 are depicted in the _square
region_ $[-1,1]^{2}$ in Fig.1. The functions may seem somewhat ad hoc but in
Section 3 we identify a broad _family_ of _bias functions_ in the region
$[-1,1]^{2}$ that guarantees a property of central interest in multi-agent
opinion evolution; namely, whether all the agents will converge to the same
opinion, i.e. _convergence to consensus_.

######  Remark 2.2.

We conclude this section by noting that unlike the DeGroot model, in Eq.3 we
allow agents to react with a distinct bias function to each of their
influencers. This broadens the range of captured opinion dynamics and we
illustrate this in the next section with an example exhibiting agents with
different bias functions including those in Ex.1. This, however, comes at a
price; the update function can be non-linear in the agents’ opinions (see
e.g., functions $\mathtt{backf}$ and $\mathtt{conf}$). Thus, the analysis of
opinion convergence using Markov chain theory for linear-system evolution as
done for the DeGroot model is no longer applicable. In Section 3 we study
opinion convergence using methods from real analysis.

###  2.5 Vaccine Example

Let us suppose that the proposition of interest is _“vaccines are unsafe”_ and
$G=(A,E,I)$ is as in Fig.2. Suppose that initally the agents $1,2,3$ are
_anti-vaxers_ with opinion values $1.0,0.9,0.8$ about the proposition. In
contrast, agents $4,5,6$ are initially _pro-vaxers_ , with opinion values
$0.2,0.1,0.0$ about the proposition. Thus, the initial state of opinion is
$B^{0}=(1.0,0.9,0.8,0.2,0.1,0.0)$.

Notice that although agent $1$ is the most extreme anti-vaxer, agent $6$, the
most extreme vaxer, has the highest possible influence over them. As we shall
illustrate below, depending on the bias of $1$ towards $6$, this may have a
strong impact on the evolution of the opinion of agent $1$.

1234560.60.40.40.40.60.60.21.01.01.01.01.01.01.0

Figure 2: Influence graph for vaccine example.

We now consider the evolution of their opinion under different update
functions obtained by combining biases from Ex.1. In Fig.3 we show the
evolution of opinions of vaxxers and anti-vaxxers using combinations of the
bias functions from Fig.1. Consider Fig.3(a). Agent $2$ reaches the extreme
opinion value $1.0$ rather quickly because of their die-hard fanaticism
towards the opinion of $1$ (i.e., $\beta_{2,1}=\mathtt{fan}$). As the
influence of $6$ on agent $1$ backfires ($\beta_{1,6}=\mathtt{back}$), agent
$1$ stays with belief value $1.0$. Eventually, all the other biases contribute
to changing the belief value of the influenced agents towards $1.0$. Indeed,
the agents converge to a consensus that vaccines are unsafe.

In Fig.3(b), the influence of $3$ on agent $5$ backfires, since
$\beta_{5,3}=\mathtt{backf}$. This makes their disagreement increase, moving
agent $5$’s opinion closer to $0$. On the other hand, the opinion of agent $6$
is influenced at the same time by the belief values of $5$ and $4$ as in the
DeGroot model $(\beta_{6,5}=\beta_{6,4}=\mathtt{degroot})$ so her opinion
stays between theirs.

Notice that in Fig.3(c) agent $5$ reacts to $3$ with die-hard fanaticism
($\beta_{5,3}=\mathtt{fan}$) while $3$’s belief value does not converge to
$0.0$ or $1.0$. Thus we obtain the looping behaviour of agent $5$. The
fanaticism of agent $5$ propagates also to agent $6$ since he is influenced by
agent $5$ by $\mathtt{degroot}$ bias.

Finally, notice the behaviours in Fig.3 when all the agents have the same
bias. In particular, Fig.3(g) suggests convergence to consensus when all the
agents are under confirmation bias. In fact convergence to consensus is indeed
guaranteed for this example as we shall later see in this paper.

The above illustrates that different types of biases can have strong impact on
opinion evolution for a given influence graph. In the next section, we will
identify meaningful families of bias as functions in the region $[-1,1]^{2}$.

(a) $\beta_{2,1}=\mathtt{fan}$, $\beta_{1,6}=\mathtt{backf}$,
$\beta_{4,2}=\beta_{1,2}=\mathtt{degroot}$, otherwise
$\beta_{i,j}=\mathtt{conf}$.

(b) $\beta_{2,1}=\mathtt{fan}$, $\beta_{1,6}=\beta_{5,3}=\mathtt{backf}$,
$\beta_{6,5}=\beta_{6,4}=\mathtt{degroot}$, otherwise
$\beta_{i,j}=\mathtt{conf}$.

(c) $\beta_{5,3}=\mathtt{fan}$, $\beta_{4,3}=\mathtt{backf}$,
$\beta_{6,5}=\beta_{1,6}=\mathtt{degroot}$, otherwise
$\beta_{i,j}=\mathtt{conf}$.

(d) Each $\beta_{i,j}=\mathtt{degroot}$.

(e) Each $\beta_{i,j}=\mathtt{fan}$.

(f) Each $\beta_{i,j}=\mathtt{backf}$.

(g) Each $\beta_{i,j}=\mathtt{conf}$.

Figure 3: Simulations for $G$ in Fig.2 with $B^{0}=(1.0,0.9,0.8,0.2,0.1,0.0)$
using different biases. Each plot represents the evolution in time of the
opinion of the agent in Fig.2 with the same color.

##  3 Bias Region and Consensus

Consensus is a property of central interest in social learning models. Indeed,
failure to converge to a consensus is often an indicator of polarization in a
society.

######  Definition 4 (Consensus).

Let $(G,B^{0},\mu_{G})$ be an opinion model with $G=(A,E,I)$. We say that the
subset of agents $A^{\prime}\subseteq A$ _converges to an opinion value
$v\in[0,1]$_ iff for every $i\in A^{\prime}$, $\lim_{t\to\infty}B^{t}_{i}=v.$
We say $A^{\prime}\subseteq A$ _converges to consensus_ iff $A^{\prime}$
converges to an opinion value $v$ for some $v$.

In this section we identify a broad and meaningful region of $[-1,1]^{2}$
where all the _continuous_ disagreement bias functions guarantee that agents
converge to consensus under certain topological conditions on the influence
graph.

###  3.1 Bias Regions

In what follows we say that a bias $\beta_{i,j}$ _is in a region_
$R\subseteq[-1,1]^{2}$ if its function graph is included in $R$, i.e., if
$\\{(x,\beta_{i,j}(x))\ |\ x\in[-1,1]\\}\subseteq R$. We now identify regions
of $[-1,1]^{2}$ that capture several notions of cognitive bias.

######  Definition 5 (Bias Regions).

Let $\mathbf{S}$ be the square region $[-1,1]^{2}.$ Let the (sub)regions
$\mathbf{M},\mathbf{R},\mathbf{B},\mathbf{I}\subseteq\mathbf{S}$, named
_Malleability_ , _Receptive-Resistant_ , _Backfire_ and _Insular_ , be defined
as follows:

$\displaystyle\mathbf{M}=$ $\displaystyle\\{(x,y)\in\mathbf{S}\ |(x<0\mbox{
and }y\leq x)\mbox{ or }(x>0\mbox{ and }y\geq x)\mbox{ or }x=0\\}$
$\displaystyle\mathbf{R}=$ $\displaystyle\\{(x,y)\in\mathbf{S}\ |\ (x<0\mbox{
and }x<y<0)\mbox{ or }(x>0\mbox{ and }0<y<x)\mbox{ or }x=y=0\\}$
$\displaystyle\mathbf{B}=$ $\displaystyle\\{(x,y)\in\mathbf{S}\ |\ (x<0\mbox{
and }0<y)\mbox{ or }(x>0\mbox{ and }y<0)\mbox{ or }x=y=0\\}$
$\displaystyle\mathbf{I}=$ $\displaystyle\\{(x,y)\in\mathbf{S}\ |\ y=0\\}.$

Figure 4: Bias Regions: Malleability ($\mathbf{M}$, in blue), Receptive-
Resistant ($\mathbf{R}$, in yellow), Backfire ($\mathbf{B}$, in red), Insular
($\mathbf{I}$, the dotted line $y=0$).

The regions are depicted in Fig.4. Notice that if a point $(x,y)$ of a bias
$\beta_{i,j}$ is in the _Malleability_ region $\mathbf{M}$ (i.e.,
$y=\beta_{i,j}(x)$ and $(x,y)\in\mathbf{M}$) it means that for a disagreement
$x=B_{j}-B_{i}$ between $j$ and $i$, the bias will contribute with a magnitude
$|y|\geq|x|$ (multiplied by $\overline{I_{j,i}}$) to changing the opinion of
$i$ in the direction of $j$’s opinion. Since $|y|\geq|x|$, depending on the
value of $\overline{I_{j,i}}$, the opinion of $i$ may move to match $j$’s
opinion or even further (which can make $i$’s new opinion even more extreme
than that of $j$). Individuals that blindly follow authoritative or
influential figures, easily swayed agents, fanaticism, among others, can be
modelled by bias functions in this region. Indeed the function $\mathtt{fan}$
from Ex.1 is in $\mathbf{M}$ (see Fig.1). The identity bias function
${\mathtt{degroot}}$ is also in $\mathbf{M}.$

Like in the case above, if a point $(x,y)$ of a bias $\beta_{i,j}$ is in the
_Receptive-Resistant_ region $\mathbf{R}$, it also means that for a
disagreement $x=B_{j}-B_{i}\neq 0$ between $j$ and $i$, the bias contributes
to changing the opinion of $i$ in the direction of $j$’s opinion.
Nevertheless, the magnitude of contribution is not as high as the previous
case, namely it is $|y|$ with $|x|>|y|>0$. Individuals that are receptive to
other opinions but, unlike malleable ones, may demonstrate some resistance,
reluctance, or skepticism to fully accept them, can be modelled in this
region. The confirmation bias function
$\mathtt{conf}(x)=x(1+\delta-|x|)/(1+\delta)$ from Ex.1, where $\delta>0$ is a
very small constant, is in $\mathbf{R}$ (see Fig.1).

In fact, it is worth noticing that for any constant $\delta>0$, the resulting
bias function $\beta_{i,j}(x)=x(1+\delta-|x|)/(1+\delta)$ is in $\mathbf{R}$.
In the limit, however, we have
$\lim_{\delta\to\infty}x(1+\delta-|x|)/(1+\delta)=x={\mathtt{degroot}(x)}$
which is not in $\mathbf{R}$ but in $\mathbf{M}$. Therefore, $\delta$ could be
viewed as a _parameter of receptiveness_ ; the higher the value of $\delta$,
the more receptive and less resistant agent $i$ is toward $j$’s opinion. In
the limit, agent $i$ is not resistant and behaves as a malleable agent towards
$j$.

Contrary to the previous two cases, if a point $(x,y)$ of a bias $\beta_{i,j}$
is in the _Back-Fire_ region $\mathbf{B}$, it means that for a disagreement
$x=B_{j}-B_{i}\neq 0$ between $j$ and $i$, the bias contributes to changing
the opinion of $i$ _but in the opposite direction_ of $j$’s opinion. This bias
can then cause the disagreement between $i$ and $j$ to grow. Individuals that
become more extreme when confronted with a different opinion can be modelled
by bias functions in this region. Indeed, the function $\mathtt{backf}$ from
Ex.1 is in $\mathbf{B}$ (see Fig.1).

Finally, if a point $(x,y)$ of a bias $\beta_{i,j}$ is in the Insular region
$\mathbf{I}$, it means that $y=0$, thus for a disagreement $x=B_{j}-B_{i}\neq
0$ between $j$ and $i$, the bias causes $i$ to completely ignore the opinion
of $j$. Individuals that are stubborn or closed-minded can be modelled with
the function in this region. In fact, the function $\mathtt{ins}$ from Ex.1 is
the only function in $\mathbf{I}$ (see Fig.1).

We conclude this section with a proposition stating that we can dispense with
the clamp function whenever all the bias functions are in the $\mathbf{R}$
region.

######  Proposition 3.1 (Update with Bias in $\mathbf{R}$).

Given a Bias Opinion Model $(G,B^{0},\mu_{G})$ with $G=(A,E,I)$, if for all
$(a,b)\in E$ we have $\beta_{b,a}\in\mathbf{R}$, then for all
$B\in[0,1]^{|A|}$ and $i\in A$: $\mu_{G}(B)_{i}=B_{i}+\sum_{j\in
A_{i}}\overline{I_{j,i}}\beta_{i,j}(B_{j}-B_{i}).$

The proof of this proposition can be found in the Appendix.

###  3.2 Consensus under Receptiveness in Strongly Connected Graphs

Our first main result states the convergence to consensus for strongly
connected societies when all bias functions are continuous and in the
Receptive-Resistant Region defined in 3.1. We need some standard notions from
graph theory.

Recall that a _path_ from $i$ to $j$ in $G=(A,E,I)$ is a sequence
$i_{0}i_{1}\ldots i_{m}$ such that $i=i_{0}$, $j=i_{m}$ and
$(i_{0},i_{1}),(i_{1},i_{2}),\ldots(i_{m-1},i_{m})$ are edges in $E$. The
graph $G$ is _strongly connected_ iff there is path from any agent to any
other. We can now state our first consensus result.

######  Theorem 3.1 (Consensus I).

Let $(G,B^{0},\mu)$ be a bias opinion model with a strongly connected graph
$G=(A,E,I)$. Suppose that for every $(j,i)\in E$, $\beta_{i,j}$ is a
continuous function in $\mathbf{R}$. Then the set of agents $A$ converges to
consensus.

Hence, the continuous bias functions in $\mathbf{R}$ guarantee consensus in
strongly connected graphs, regardless of initial beliefs. Intuitively, the
theorem says that a strongly connected community/society will converge towards
consensus if its members are receptive but resistant to the opinions of
others.

Notice that the Vaccine Example in Sec.2.5 with all agents under confirmation
bias satisfy the conditions of Th.3.1, so their convergence to consensus is
guaranteed. In fact, the opinion difference between any two agents grows
smaller rather rapidly (Fig.3(g) illustrates this). In contrast, Fig.5
illustrates an example, with different a different bias also in $\mathbf{R}$,
where the opinion difference grows smaller much slowly. But since such an
example also satisfies the conditions of Th.3.1, convergence to consensus is
guaranteed.

(a) Bias function $\beta_{1,2}(x)=\beta_{2,1}(x)=\begin{cases}0&\text{if
}x=0\\\ \frac{x}{|x|}\cdot e^{-\frac{1}{|x|}}&\text{if }x\neq 0\end{cases}$

(b) Each plot represents the evolution in time of the opinion of the agent in
Fig.6(b) with the same color.

Figure 5: Simulations for $G$ in Fig.6(b) with $B^{0}=(0.0,1.0)$ using a bias
function in region $\mathbf{R}$, with very slow convergence. Each plot
represents the evolution in time of the opinion of the agent in Fig.6(b) with
the same color.

Before outlining the proof of this theorem, we elaborate on its conditions.

#### Discontinuous Bias.

Requiring continuity for the bias functions in Th.3.1 seems reasonable; small
changes in an opinion disagreement value $x=B_{j}-B_{i}$ should result in
small changes in $i$’s biased reaction to $x$. Nevertheless, if we relaxed the
continuity requirement, we would have the following counter-example.

Consider a strongly connected graph with two agents with $I_{1,2}=I_{2,1}=1$,
agent $1$ influences agent $2$ with the bias functions
$\beta_{1,2}=\beta_{2,1}=f$, satisfying $f(x)=\frac{x}{2}$ if
$x\in[-\frac{1}{2},\frac{1}{2}]$, $f(x)=\frac{x-0.5}{8}$ if
$x\in(\frac{1}{2},1]$ and $f(x)=\frac{x+0.5}{8}$ if $x\in[-1,-\frac{1}{2})$.
If one agent starts with belief value $1.0$ and the other $0.0$, then they
will not converge to consensus (their belief values will approach
$\frac{3}{4}$ and $\frac{1}{4}$, but will never reach those values). Figure 6
illustrates this example.

(a) Bias function
$\beta_{1,2}(x)=\beta_{2,1}(x)=\begin{cases}\frac{x+0.5}{8}&\text{if}-1\leq
x<-\frac{1}{2}\\\ \frac{x}{8}&\frac{1}{2}\leq x\leq\frac{1}{2}\\\
\frac{x-0.5}{8}&\text{if}\frac{1}{2}<x\leq 1\end{cases}$

121.0

(b) Influence Graph ($I_{1,2}=I_{2,1}=1.0$).

(c) Each plot represents the evolution in time of the opinion of the agent in
Fig.6(b) with the same color.

Figure 6: Counter-example to consensus for two agents with non-continuous bias
functions in $\mathbf{R}$ and with $B^{0}=(1.0,0.0)$.

(a) Bias function $\beta_{1,2}(x)=\beta_{2,1}(x)=\frac{\arctan x}{\arctan 1}$

(b) Each plot represents the evolution in time of the opinion of the agent in
Fig.6(b) with the same color.

Figure 7: Counter-example for consensus when all bias function are continuous
but allowed to have points in $\mathbf{M}$, with initial belief vector
$B^{0}=(0.0,0.001)$ and influence graph 6(b).

#### Bias Outside $\mathbf{R}$.

Notice that Th.3.1 requires bias functions to be in the responsive-resistant
region $\mathbf{R}$. We consider counter-examples where we allow bias
functions outside this region in Th.3.1. If we allowed continuous bias
functions outside $\mathbf{R}$ with points in the backfire region
$\mathbf{B}$, then the scenario in Fig.3(f) provides a counter-example to
consensus. If we allow continuous bias functions outside $\mathbf{R}$ with
points in region $\mathbf{M}$, then the scenario in Fig.7 is a counter-example
to consensus: notice how the absolute value of their disagreement begins at
$0.001$ and increases until it reaches $1$. Finally, it is clear that if we
allowed the only function in $\mathbf{I}$, the insular bias, with the graph in
Fig.6(b) and initial beliefs $B^{0}=(0,1)$, consensus will never be reached
since the agents will ignore each other.

###  3.3 Proof Outline of Th.3.1.

In this Section we outline the proof of Th.3.1. In the process we single out
the central properties of the behaviour of agents that are receptive and yet
resistant to disagreement. The complete proof can can be found in the
appendix.

Let $(G,B^{0},\mu)$ be as in the statement of Th.3.1. Suppose
$B=\mu^{t}(B^{0})$ is the state of opinion at some time $t\geq 0$ where
consensus has not yet been reached: i.e., assume $\min(B)\neq\max(B)$ where
$\min(B)$ and $\max(B)$ are the minimum and maximum opinion values in $B$. By
assumption, all the biases $\beta_{i,j}$ are in $\mathbf{R}$. Thus
$\beta_{i,j}(x)=y$, where $x=B_{j}-B_{i}$, contributes to update the opinion
of $i$ in the direction of the opinion of $j$ but with a magnitude $|y|>0$
strictly smaller than $|x|$ if $|x|>0$ (or equal to $0$ if $|x|=0$). Using
this and Prop.3.1333This follows from the known property that weighted
averages of any set of values are always between the minimum and the maximum
of those values., we show the new (updated) opinion of each $i$, $\mu(B)_{i}$,
is bounded as follows:

######  Lemma 3.1 (Update Bounds).

For each $i\in A$, $\min(B)\leq\mu(B)_{i}\leq\max(B)$.

We use the above lemma to prove that the bounded sequences of minimum and
maximum opinion values at each time, $\\{\min(B^{t})\\}_{t\geq 0}$ and
$\\{\max(B^{t})\\}_{t\geq 0}$, are monotonically non-decreasing and
monotonically non-increasing. Thus by the Monotone convergence theorem [10],
they both converge. Therefore, by the Squeeze theorem [10], to prove Th.3.1,
it suffices to show that $\\{\min(B^{t})\\}_{t\geq 0}$ and
$\\{\max(B^{t})\\}_{t\geq 0}$ converge to the same value.

We first prove the following lemma which intuitively states that the number of
extreme agents decrease with time.

######  Lemma 3.2 (Extreme Agents Reduction).

Suppose that $\min(B)\neq\max(B)$ and let $M=\max(B)$. If $G$ has a path
$i_{1}\ldots i_{n}$ such that $B_{i_{n}}=M$ and $B_{i_{1}}<M$, then $|\\{j\in
A:B_{j}\geq M\\}|>|\\{j\in A:\mu(B)_{j}\geq M\\}|$. A symmetric property
applies to the minimum.

To see the lemma’s intuition, notice that since $G$ is strongly connected and
$\min(B)<\max(B)$, $G$ indeed has a path $i_{1}\ldots i_{n}$ such that
$B_{i_{n}}=M=\max(B)$ and $B_{i_{1}}<M$. In the path some agent $i_{k}$ whose
belief value is equal to $M$ will be influenced by some agent with smaller
belief value. Thus, since the bias functions are in $\mathbf{R}$, the opinion
of $i_{k}$ will change in the direction of the smaller value, and thus will
strictly decrease. Also, no agent that had a smaller belief value will reach
the current maximum, as the bias functions are in the region $\mathbf{R}$.

Thus, because of Lem.3.2 and $G$ being strongly connected, we conclude that
the maximum (minimum) belief value will eventually decrease (increase). I.e.,

######  Corollary 1.

Suppose that $\min(B)\neq\max(B).$ Then there exist $s,t>0$ such that
$\max(\mu^{s}(B))<\max(B)$ and $\min(\mu^{t}(B))>\min(B).$

We now apply Bolzano-Weierstrass theorem [10]444Every infinite bounded
sequence in $\mathbb{R}^{n}$ has a convergent sub-sequence. to find a sub-
sequence $\\{B^{t}\\}_{t\in\Delta}$ of $\\{B^{t}\\}_{t\in\mathbb{N}}$ that
converges to some $B^{\infty}$. Notice that $\\{\max(B^{t})\\}_{t\in\Delta}$
converges to $\max(B^{\infty})$ and it is a sub-sequence of the convergent
sequence $\\{\max(B^{t})\\}_{t\in\mathbb{N}}$, so
$\\{\max(B^{t})\\}_{t\in\mathbb{N}}$ should also converge to
$\max(B^{\infty})$. Since each bias function $\beta_{i,j}$ is continuous, the
update function $\mu$ is continuous. Therefore,
$\\{\mu(B^{t})\\}_{t\in\Delta}$ converges to $\mu(B^{\infty})$, and thus
$\\{\max(\mu(B^{t}))\\}_{t\in\Delta}$ converges to $\max(\mu(B^{\infty}))$.
But since the sequence
$\\{\max(\mu(B^{t}))\\}_{t\in\Delta}=\\{\max(B^{t+1})\\}_{t\in\Delta}$ is a
sub-sequence of the convergent sequence $\\{\max(B^{t})\\}_{t\in\mathbb{N}}$,
both must converge to the same value, hence
$\max(B^{\infty})=\max(\mu(B^{\infty}))$. Similarly, we can show that
$\min(B^{\infty})=\min(\mu(B^{\infty}))$. It can thus be shown that if we
repeatedly apply $\mu$ to $B^{\infty}$, the maximum should not change, and the
same applies to the minimum. More precisely, we conclude the following.

######  Corollary 2.

$\max(B^{\infty})=\max(\mu^{t}(B^{\infty}))$ and
$\min(B^{\infty})=\min(\mu^{t}(B^{\infty}))$ for each $t\geq 0$.

Consequently, if $\min(B^{\infty})\neq\max(B^{\infty})$ then Cor.1 and Cor.2
lead us to a contradiction. Therefore, $\min(B^{\infty})=\max(B^{\infty})$ and
thus, $\\{\min(B^{t})\\}_{t\geq 0}$ and $\\{\max(B^{t})\\}_{t\geq 0}$ converge
to the same value $\max(B^{\infty})$ as wanted. ∎

###  3.4 Consensus Under Receptiveness in Arbitrary Graphs

Recall that Th.3.1 applies to strongly connected influence graphs. Our second
main result applies to arbitrary influence graphs. First we need to recall the
notion of strongly-connected components of a graph.

A _strongly-connected component_ of $G$ is a _maximal subset_ $S\subseteq A$
such that for each two $i,j\in S$, there is path from $i$ to $j$. A strongly-
connected component $S$ is said to be a _source component_ iff there is no
edge $(i,j)\in E$ such that $i\in A\setminus S$ and $j\in S$. We use
$\mathcal{S}(G)$ to denote the set of source components of $G$.

Intuitively, a source component of a graph can be thought of as a closed group
that is not externally influenced but may influence individuals outside the
group. The following theorem gives a characterization of consensus with biases
in $\mathbf{R}$ for arbitrary graphs in terms of source components.

######  Theorem 3.2 (Consensus II).

Let $(G,B^{0},\mu)$ be a bias opinion model with $G=(A,E,I)$. Suppose that for
every $(j,i)\in E$, $\beta_{i,j}$ is a continuous function in $\mathbf{R}$.
Then the set of agents $A$ converges to consensus iff there exists $v\in[0,1]$
such that every source component $S\in\mathcal{S}(G)$ converges to opinion
$v.$

The above theorem, whose proof can be found in the Appendix, provides the
following intuitive yet insightful remark. Namely, upon agreeing on an
opinion, the closed and potentially influential groups, can make all
individuals converge to that opinion in a society whose members are receptive
but resistant.

##  4 Concluding Remarks and Related Work

We introduced a generalization of the DeGroot Model where agents interact
under different biases. We identified the notion of bias on disagreement and
made it the focus our model. This allowed us to identify families of biases
that capture a broader range of social dynamics. We also provided theoretical
results characterizing the notion of consensus for a broad family of cognitive
biases.

The relevance of biased reasoning in human interactions has been studied
extensively in [12], [9], [11], and others. There is a great deal of work on
formal models for belief change in social networks; we focus on the work on
_biased_ belief update, which is the focus of this paper. Some models were
previously proposed to generalize the DeGroot model and introduce bias, for
instance [4], [3] and [14] analyse the effects of incorporating a bias factor
for each agent to represent biased assimilation: how much of the external
opinions the agent will take into consideration. [13] extends the model [4] to
include the effect of backfire-effect as well. The main difference of these
models to our model is that biases are not incorporated in those models in
terms of the disagreement level between agents, but either as an exponential
factor that reduces the impact of the opinion of neighbours or by dynamically
changing the weights of the DeGroot model. Thus, our model brings a new point
of view to how distinct types of biases can be represented and identified.

In [7], it is proved that “constricting” update functions, roughly, functions
where the extreme agents move closer to each other, lead to convergence in
strongly connected social networks. This is similar to our theorem, indeed,
the functions in our $\mathbf{R}$ region are easily shown to be constricting.
However, their social network model is more abstract than ours and further
from real social networks, and they do not directly analyse biases as a
function of disagreement.

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##  Appendix 0.A Appendix

######  Remark 0.A.1 (Fixed bias opinion model throughout the appendix).

For all our formal results we assume a fixed bias opinion model
$(G,B^{0},\mu)$ with $G=(A,E,I)$. Also, we assume an arbitrary
$B\in[0,1]^{|A|}$.

######  Remark 0.A.2 (All bias functions are in $\mathbf{R}$).

For all our formal results, we assume that all bias functions are in region
$\mathbf{R}$. Formally, this means that for all $x\in[-1,1]$ and $(j,i)\in E$,
we have $0>\beta_{i,j}(x)>x$ if $x<0$ and $0<\beta_{i,j}(x)<x$ if $x>0$ and
$\beta_{i,j}(x)=0$ if $x=0$.

######  Remark 0.A.3 (All bias functions are continuous).

Also, for all our results, we assume continuity of all bias functions. This
means that for all $(j,i)\in E$ and for any sequence
$\\{x_{n}\\}_{n\in\mathbb{N}}$ satisfying $\forall
n\in\mathbb{N}:x_{n}\in[-1,1]$ and $\lim_{n\rightarrow\infty}x_{n}=L$ for some
$L\in[-1,1]$, we have
$\lim_{n\rightarrow\infty}\beta_{i,j}(x_{n})=\beta_{i,j}(L)$.

######  Lemma 0.A.1.

If $M\in[0,1]$ such that $M\geq B_{k}$ for any $k\in A$, then for any
$(j,i)\in E$, if $B_{j}\neq B_{i}$, then
$\beta_{{i},{j}}(B_{j}-B_{i})<M-B_{i}$. Otherwise,
$\beta_{{i},{j}}(B_{j}-B_{i})\leq M-B_{i}$. A symmetrical property applies for
the minimum.

###### Proof.

Let $(j,i)\in E$ be arbitrary. By Rmk.0.A.2, if $B_{j}-B_{i}<0$, we have
$\beta_{{i},{j}}(B_{j}-B_{i})<0\leq M-B_{i}$, and if $B_{j}-B_{i}>0$, we have
$\beta_{{i},{j}}(B_{j}-B_{i})<B_{j}-B_{i}\leq M-B_{i}$. This proves the first
part of the lemma. If $B_{j}-B_{i}=0$, $\beta_{{i},{j}}(B_{j}-B_{i})=0\leq
M-B_{i}$ (because $(0,0)$ is the only point in $\mathbf{R}$ with $x=0$). This
proves the second part of the lemma.

The proof for the minimum is symmetrical. ∎

######  Lemma 0.A.2.

Let $K_{i}=B_{i}+\sum_{j\in A_{i}}\overline{I_{j,i}}\beta_{i,j}(B_{j}-B_{i})$.
Then for any $i\in A$, $\min(B)\leq K_{i}\leq\max(B)$.

###### Proof.

$\displaystyle K_{i}$ $\displaystyle=B_{i}+\sum_{j\in
A_{i}}\overline{I_{j,i}}\beta_{{i},{j}}(B_{j}^{t}-B_{i}^{t})\qquad\text{(by
definition)}$ $\displaystyle\leq B_{i}+\sum_{j\in
A_{i}}\overline{I_{j,i}}(\max(B)-B_{i})\qquad\text{(Lem.\ref{aux1A} with
$M=\max(B)$)}$

If $A_{i}=\emptyset$, we have $K_{i}=B_{i}\leq\max(B)$. If not, we have
$K_{i}\leq B_{i}+\max{B}-B_{i}=\max(B)$. We can find a symmetrical proof for
the minimum ($\min(B)\leq K_{i}$). ∎

######  Proposition 0.A.1 (Update with Bias in $\mathbf{R}$, Prop.3.1 of
section 3).

For all $i\in A$:

$\mu(B)_{i}=B_{i}+\sum_{j\in A_{i}}\overline{I_{j,i}}\beta_{i,j}(B_{j}-B_{i})$

###### Proof.

Let $i\in A$ be an arbitrary agent. By using Lem.0.A.2 (notice that
$\mu(B)_{i}=\left[K_{i}\right]^{1}_{0}$) and the fact that $B\in[0,1]^{|A|}$,
we conclude that $\mu(B)_{i}=K_{i}$, which is exactly what the proposition
says. ∎

######  Lemma 0.A.3 (Update Bounds, Lem.3.1 of Sec.3).

For each $i\in A$, $\min(B)\leq\mu(B)_{i}\leq\max(B)$.

###### Proof.

This follows directly from the combination of Prop.0.A.1 and Lem.0.A.2. ∎

######  Lemma 0.A.4.

There are $U,L\in[0,1]$ such that $\lim_{t\rightarrow\infty}\max(B^{t})=U$ and
$\lim_{t\rightarrow\infty}\min(B^{t})=L$.

###### Proof.

It follows directly from Lem.0.A.3 that $\max(B^{t})$ and $\min(B^{t})$ are
respectively monotonically non-increasing and monotonically non-decreasing
with respect to $t$. Then, for any integer $t\geq 0$ we have
$\max(B^{t+1})\leq\max(B^{t})$ and $\min(B^{t+1})\geq\min(B^{t})$. Thus, the
result follows from the monotone convergence theorem[10]. ∎

######  Lemma 0.A.5.

The function $\mu$ is continuous.

###### Proof.

Note that we can compute $\lim_{Y\rightarrow B}\mu(Y)$ from it’s individual
values: let $i\in A$. Then, by Lem.0.A.1, we get $\mu(Y)_{i}=Y_{i}+\sum_{j\in
A_{i}}\overline{I_{j,i}}\beta_{i,j}(Y_{j}-Y_{i})$. Also, $Y\rightarrow B$
implies that $Y_{i}\rightarrow B_{i}$ for all $i\in A$. Then,
$\lim_{Y\rightarrow B}\mu(Y)_{i}=\lim_{Y\rightarrow B}(Y_{i}+\sum_{j\in
A_{i}}\overline{I_{j,i}}\beta_{i,j}(Y_{j}-Y_{i}))=B_{i}+\sum_{j\in
A_{i}}\overline{I_{j,i}}\lim_{Y\rightarrow B}\beta_{i,j}(Y_{j}-Y_{i})$.
Because all $\beta_{i,j}$ are continuous (Rem.0.A.3), this is equal to
$B_{i}+\sum_{j\in
A_{i}}\overline{I_{j,i}}\beta_{{i},{j}}(B_{j}-B_{i})=\mu(B)$, thus concluding
our proof, as $B$ is arbitrary. ∎

######  Lemma 0.A.6 (Extreme Agents Reduction, Lem.3.2 from Sec.3).

Suppose that $\min(B)\neq\max(B)$ and let $M=\max(B)$ If $G$ has a path
$i_{1}\ldots i_{m}$ such that $B_{i_{m}}=M$ and $B_{i_{1}}<M$, then $|\\{j\in
A:B_{j}\geq M\\}|>|\\{j\in A:\mu(B)_{j}\geq M\\}|$. A symmetrical property
applies to the minimum.

###### Proof.

First note that by Lem.0.A.3, the maximum can not increase, which means that
$\max(\mu(B))\leq\max(B)$, and also by Lem.0.A.1 the clamp function has no
impact on the value of $\mu(B)$.

Now note that if $B_{i}<\max(B)$, then if $A_{i}\neq\emptyset$, we can use
Lem.0.A.1 to conclude $(\mu(B))_{i}=B_{i}+\sum_{j\in
A_{i}}\overline{I_{j,i}}\beta_{{i},{j}}(B_{j}-B_{i})<B_{i}+\sum_{j\in
A_{i}}\overline{I_{j,i}}(\max(B)-B_{i})=B_{i}+(\sum_{j\in
A_{i}}\overline{I_{j,i}})(\max(B)-B_{i})=B_{i}+\max(B)-B_{i}=\max(B)$. Also,
if $A_{i}=\emptyset$, then $(\mu(B))_{i}=B_{i}<\max(B)$. This means that the
number of agents with belief value greater than or equal to $\max(B)$ cannot
increase. We proceed to prove that at least one agent with belief $\max(B)$
will decrease it’s belief value after the application of $\mu$.

Let $i_{1}i_{2}i_{3}...i_{m}$ be the path declared in the assumption. Let
$1\leq y\leq n$ be the smallest value such that $B_{i_{y}}=\max(B)$. Then
$B_{i_{y-1}}<\max(B)$ by construction, and $A_{y}\neq\emptyset$ (because $y$
is influenced by $y-1$). We proceed to prove that $(\mu(B))_{i_{y}}<\max(B)$,
which means $i_{y}$ is an agent with belief $\max(B)$ that will decrease it’s
belief after the application of $\mu$, which concludes our proof. Let $y-1=u$

$\displaystyle(\mu(B))_{i_{y}}$ $\displaystyle=B_{i_{y}}+\sum_{j\in
A_{i_{y}}}\overline{I_{j,i_{y}}}\beta_{{i_{y}},{j}}(B_{j}-B_{i_{y}})$
$\displaystyle=B_{i_{y}}+\overline{I_{i_{u},i_{y}}}\beta_{{i_{y}},{i_{u}}}(B_{i_{u}}-B_{i_{y}})+\sum_{j\in
A_{i_{y}}\backslash\\{i_{u}\\}}\overline{I_{j,i_{y}}}\beta_{i_{y},j}(B_{j}-B_{i_{y}})$
$\displaystyle\leq
B_{i_{y}}+\overline{I_{i_{u},i_{y}}}\beta_{{i_{y}},{i_{u}}}(B_{i_{u}}-B_{i_{y}})+(\sum_{j\in
A_{i_{y}}\backslash\\{i_{u}\\}}\overline{I_{j,i_{y}}})(\max(B)-B_{i_{y}})$
$\displaystyle=B_{i_{y}}+\overline{I_{i_{u},i_{y}}}\beta_{{i_{y}},{i_{u}}}(B_{i_{u}}-B_{i_{y}})$
$\displaystyle<B_{i_{y}}$ $\displaystyle=\max(B)$

The proof for the minimum is symmetrical. ∎

######  Remark 0.A.4 (Composition of $\mu$).

We consider that $\mu^{0}(B)=B$ and $\mu^{k}{B}=\mu^{k-1}(\mu(B))$ for any
integer $k>0$.

######  Lemma 0.A.7.

Suppose that $\min(B)\neq\max(B)$ and let $M=\max(B)$. If $G$ has a path
$i_{1}\ldots i_{m}$ such that $B_{i_{m}}=M$ and $B_{i_{1}}<M$, then
$\max(\mu^{|A|-1}(B))<\max(B)$. A symmetrical property applies to the minimum.

###### Proof.

Note that $B$ has at most $|A|-1$ agents with belief value $\max(B)$,
otherwise $\min(B)=\max(B)$. Then, by Lem.0.A.6, in $\mu(B)$ we have at most
$|A|-2$ agents with belief value $\max(B)$. Let $k>0$ be the smallest integer
such that $\mu^{k}(B)$ has no agent with belief $\max(B)$, and note that this
means that $\mu^{k-1}(B)$ has at least one agent with belief $\max(B)$. We
first prove that that $k\leq|A|-1$:

We know $\mu^{k-1}(B)$ has at least one agent with belief $\max(B)$, and by
Lem.0.A.6, for all $k\geq y>0$ we know that the number of agents in
$\mu^{y-1}(B)$ with belief $\max(B)$ is at least one plus the number of agents
with that belief in $\mu^{y}(B)$. This means that for all $k\geq z\geq 0$,
$\mu^{z}(B)$ has at least $k-z$ agents with belief value $\max(B)$. If (for
the sake of contradiction) $k\geq|A|$, $\mu^{0}(B)=B$ would have at least
$k-0=k\geq|A|$ agents with belief greater than or equal to $\max(B)$, and
because this would mean $B$ is consensus, we have a contradiction with
$\max(B)\neq\min(B)$.

Now note that, by Lem.0.A.3 if $\mu^{k}(B)$ has no agent with belief
$\max(B)$, then this means that $\max(\mu^{k}(B))<\max(B)$, and it also means
that for all $z\geq k$, $\max(\mu^{z}(B))\leq\max(\mu^{k}(B))<\max(B)$. So, as
we just proved $k\leq|A|-1$,
$\max(\mu^{|A|-1}(B))\leq\max(\mu^{k}(B))<\max(B)$, as desired.

∎

######  Remark 0.A.5 (Source Components).

A _strongly-connected component_ of $G$ is a _maximal subset_ $S\subseteq A$
such that for each two $i,j\in S$, there is path from $i$ to $j$. A strongly-
connected component $S$ is said to be a _source component_ iff no agent in $S$
is influenced by an agent outside $S$, i.e., iff there is no edge $(i,j)\in E$
such that $i\in A\setminus S$ and $j\in S$. From now on we use
$\mathcal{S}(G)$ to be the set of source components of $G$.

######  Theorem 0.A.1 (Consensus I, Thm.3.1 of Sec.3).

Let $(G,B^{0},\mu)$ be a bias opinion model with a strongly connected graph
$G=(A,E,I)$. Suppose that for every $(j,i)\in E$, $\beta_{i,j}$ is a
continuous function in $\mathbf{R}$. Then the set of agents $A$ converges to
consensus.

###### Proof.

By Bolzano–Weierstrass theorem[10], for all integers $t\geq 0$, as all values
of $B^{t}$ are bounded in $[0,1]$, there is a convergent infinite subsequence
of the sequence $\\{B^{t}\\}_{t\in\mathbb{N}}$. Let this be
$\\{B^{t}\\}_{t\in\Delta}$, and let the value to which it converges be
$B^{\infty}$. Now let $\Gamma^{k}=\\{B^{t+k}\\}_{t\in\Delta}$, and note that
this is equal to $\\{\mu^{k}(B^{t})\\}_{t\in\Delta}$. By Lem.0.A.5, this
sequence converges to $\mu^{k}(B^{\infty})$, because the composition of
continuous functions is also continuous. We will use $k=|A|-1$ in this
theorem.

Let $\lim_{t\rightarrow\infty}\max(B^{t})=U$ (this limit exists because of
Lem.0.A.4), and, for the sake of contradiction, suppose $B^{\infty}$ is not
consensus. Then, note that, as assume that the graph is strongly connected we
can guarantee that the path assumption of Lem.0.A.7 is satisfied for
$B=B^{\infty}$, with $i_{1}\in A$ such that
$B_{i_{1}}^{\infty}=\min(B^{\infty})$ and $i_{m}\in A$ such that
$B_{i_{m}}^{\infty}=\max(B^{\infty})$. Then, by Lem.0.A.7,
$\max(\mu^{|A|-1}(B^{\infty}))<\max(B^{\infty})$.

We can reach a contradiction of this using the fact that
$\\{\max(B^{t})\\}_{t\in\Delta}$ (which converges to $\max(B^{\infty})$) is a
subsequence of the convergent sequence $\\{\max(B^{t})\\}_{t\in\mathbb{N}}$.
As the latter converges to $U$, so does the former, so $U=\max(B^{\infty})$.
Also, $\\{\mu^{|A|-1}(B^{t})\\}_{t\in\Delta}$ converges to
$\mu^{|A|-1}(B^{\infty})$ as $\mu$ is continuous. We also notice that
$\\{\max(\mu^{|A|-1}(B^{t}))\\}_{t\in\Delta}=\\{\max(B^{t+|A|-1})\\}_{t\in\Delta}$
converges to the value $\max(\mu^{|A|-1}(B^{\infty}))$ and is a subsequence of
the convergent sequence $\\{\max(B^{t})\\}_{t\in\mathbb{N}}$, so both converge
to the same value $\max(\mu^{|A|-1}(B^{\infty}))=U$. So we conclude that
$\max(\mu^{|A|-1}(B^{\infty}))=U=\max(B^{\infty})$ which is a contradiction
with the inequality $\max(\mu^{|A|-1}(B^{\infty}))<\max(B^{\infty})$. Thus,
$B^{\infty}$ must be consensus.

Let $v$ be the value such that for all $i\in A$, $B_{i}^{\infty}=v$. As
$\\{\max(B^{t})\\}_{t\in\Delta}$ (which converges to $\max(B^{\infty})=v$) is
a subsequence of the convergent sequence $\\{\max(B^{t})\\}_{t\in\mathbb{N}}$,
both converge to the same value, so $\lim_{t\rightarrow\infty}\max(B^{t})=v$.
The same applies for the minimum, so
$\lim_{t\rightarrow\infty}\max(B^{t})=v=\lim_{t\rightarrow\infty}\min(B^{t})$,
and as $\max(B^{t})\leq B_{i}^{t}\leq\min(B^{t})$ for all $i\in A$
(Lem.0.A.3), by the Squeeze Theorem[10], $A$ converges to consensus.

∎

######  Theorem 0.A.2 (Consensus II, Thm. 3.2 of Sec.3).

Let $(G,B^{0},\mu)$ be a bias opinion model with $G=(A,E,I)$. Suppose that for
every $(j,i)\in E$, $\beta_{i,j}$ is a continuous function in $\mathbf{R}$.
Then the set of agents $A$ converges to consensus iff there exists $v\in[0,1]$
such that every source component $S\in\mathcal{S}(G)$ converges to opinion
$v$.

###### Proof.

If there is no $v\in[0,1]$ such that all source components converges to $v$,
then $A$ does not converge to consensus by definition.

If there is such a $v\in[0,1]$, then we only need to show that the belief
value of agents outside of source components converge to $v$.

By Bolzano–Weierstrass theorem[10], for all integers $t\geq 0$, as all values
of $B^{t}$ are bounded in $[0,1]$, there is a convergent infinite subsequence
of the sequence $\\{B^{t}\\}_{t\in\mathbb{N}}$. Let this be
$\\{B^{t}\\}_{t\in\Delta}$, and let the value to which it converges be
$B^{\infty}$. Now let $\Gamma^{k}=\\{B^{t+k}\\}_{t\in\Delta}$, and note that
this is equal to $\\{\mu^{k}(B^{t})\\}_{t\in\Delta}$. By Lem.0.A.5, this
sequence converges to $\mu^{k}(B^{\infty})$, because the composition of
continuous functions is also continuous. We will use $k=|A|-1$ in this
theorem.

Let $\max(B^{\infty})=U$, and for the sake of contradiction, suppose
$B^{\infty}$ is not consensus. Then at least one of the following is
satisfied: $\min(B^{\infty})\neq v$, $\max(B^{\infty})\neq v$. We will assume
$\max(B^{\infty})\neq v$, if that’s not the case we can get a symmetric proof
with $\min(B^{\infty})\neq v$.

Also note that, as we assume that the belief value of all agents in source
components converge to $v$, then their belief values will also converge to $v$
in the subsequence $\\{B_{i}^{t}\\}_{t\in\Delta}$. So if $i\in S$ for some
$S\in\mathcal{S}(G)$, then $B_{i}^{\infty}=v$. So can guarantee that the path
assumption of Lem.0.A.7 is satisfied, with $i_{1}\in S$ for some
$\mathcal{S}(G)$ (so $(\mu^{l}(B^{\infty}))_{i_{1}}=v$) and $i_{m}\in A$ such
that $(\mu^{l}(B^{\infty}))_{i_{m}}=\max(\mu^{l}(B^{\infty}))$. Then, by
Lem.0.A.7, $\max(\mu^{|A|-1}(B^{\infty}))<\max(B^{\infty})$.

We can reach a contradiction of this using the fact that
$\\{\max(B^{t})\\}_{t\in\Delta}$ (which converges to $\max(B^{\infty})$) is a
subsequence of the convergent sequence $\\{\max(B^{t})\\}_{t\in\mathbb{N}}$.
As the latter converges to $U$, so does the former, so $U=\max(B^{\infty})$.
Also, $\\{\mu^{|A|-1}(B^{t})\\}_{t\in\Delta}$ converges to
$\mu^{|A|-1}(B^{\infty})$ as $\mu$ is continuous. We also notice that
$\\{\max(\mu^{|A|-1}(B^{t}))\\}_{t\in\Delta}=\\{\max(B^{t+|A|-1})\\}_{t\in\Delta}$
converges to the value $\max(\mu^{|A|-1}(B^{\infty}))$ and is a subsequence of
the convergent sequence $\\{\max(B^{t})\\}_{t\in\mathbb{N}}$, so both converge
to the same value $\max(\mu^{|A|-1}(B^{\infty}))=U$. So we conclude that
$\max(\mu^{|A|-1}(B^{\infty}))=U=\max(B^{\infty})$ which is a contradiction
with the inequality $\max(\mu^{|A|-1}(B^{\infty}))<\max(B^{\infty})$. Thus,
$B^{\infty}$ must be consensus.

Let $v$ be the value such that for all $i\in A$, $B_{i}^{\infty}=v$. As
$\\{\max(B^{t})\\}_{t\in\Delta}$ (which converges to $\max(B^{\infty})=v$) is
a subsequence of the convergent sequence $\\{\max(B^{t})\\}_{t\in\mathbb{N}}$,
both converge to the same value, so $\lim_{t\rightarrow\infty}\max(B^{t})=v$.
The same applies for the minimum, so
$\lim_{t\rightarrow\infty}\max(B^{t})=v=\lim_{t\rightarrow\infty}\min(B^{t})$,
and as $\max(B^{t})\leq B_{i}^{t}\leq\min(B^{t})$ for all $i\in A$
(Lem.0.A.3), by the Squeeze Theorem[10], $A$ converges to consensus.

∎