Cognitive contagion: How to model (and potentially counter) the spread of fake
news

Nicholas Rabb1,*, Lenore Cowen1, Jan P. de Ruiter1,2, Matthias Scheutz1,

1 Dept. of Computer Science, Tufts University, Medford, MA, United States of
America

2 Dept. of Psychology, Tufts University, Medford, MA, United States of America

* Corresponding author<EMAIL_ADDRESS>
## Abstract

Understanding the spread of false or dangerous beliefs – so-called
mis/disinformation – through a population has never seemed so urgent to many.
Network science researchers have often taken a page from epidemiologists, and
modeled the spread of false beliefs as similar to how a disease spreads
through a social network. However, absent from those disease-inspired models
is an internal model of an individual’s set of current beliefs, where
cognitive science has increasingly documented how the interaction between
mental models and incoming messages seems to be crucially important for their
adoption or rejection. We introduce a _cognitive contagion_ model that
combines a network science approach with an internal cognitive model of the
individual agents, affecting what they believe, and what they pass on. We show
that the model, even with a very discrete and simplistic belief function to
capture cognitive dissonance, both adds expressive power over existing
disease-based contagion models, and qualitatively demonstrates the appropriate
belief update phenomena at the individual level. Moreover, we situate our
cognitive contagion model in a larger _public opinion diffusion_ (POD) model,
which attempts to capture the role of institutions or media sources in belief
diffusion – something that is often left out. We conduct an analysis of the
POD model with our simple cognitive dissonance-sensitive update function
across various graph topologies and institutional messaging patterns. We
demonstrate that population-level aggregate outcomes of the model
qualitatively match what has been reported in COVID misinformation public
opinion polls. The overall model sets up a preliminary framework with which
social science misinformation researchers and computational opinion diffusion
modelers can join forces to understand, and hopefully learn how to best
counter, the spread of misinformation and “alternative facts.”

## Introduction

Understanding the spread of false or dangerous beliefs through a population
has never seemed so urgent to many. In our modern, highly networked world,
societies have been grappling with the widespread belief in conspiracies [1,
2, 3, 4, 5], increased political polarization [6, 7, 8, 9], and distrust in
scientific findings [10, 11]. Most prominent in our times are conspiracies
surrounding COVID-19 and starkly partisan distributions of beliefs regarding
scientifically-motivated safety measures.

Throughout the course of the pandemic, much effort has been spent trying to
understand why, in the face of a global pandemic, so many believed it was a
hoax, targeted political attack, result of 5G cell towers, or just that it was
not dangerous and did not justify wearing a protective mask [3, 1, 10, 12, 13,
4, 5, 14]. Understanding the spread of misinformation requires a way of
modeling and understanding both how this misinformation spreads in a
population, and also why some individuals are more or less vulnerable.

This paper introduces a new class of simple models for the spread of
misinformation that includes components from both social network theory and
cognitive theory. While social network theory and cognitive theory have both
sought to contribute to the understanding of the mechanisms that govern
individuals’ acquisition and updates of beliefs, each has traditionally
focused on different pieces of this puzzle. Social network theory has provided
interesting insights by applying techniques originally developed to model the
spread of disease to modeling the spread of misinformation [15, 16, 17, 2,
18]. Modern psychological and cognitive theory has focused on how the
relationship of suggested beliefs relates to an individual’s current set of
beliefs, and how this influences an individual’s likelihood of updating their
beliefs in the face of confirmatory or contradictory information [19, 3, 13,
20, 21, 14, 22]. We introduce a new class of models, cognitive contagion
models, that adopt a network-based contagion model from social network theory
[23, 24], but include a more individually differentiated model of belief
update and adoption that is informed by cognitive theory. We show that even
very simple versions of cognitive contagion result in interesting network
dynamics that seem to well-represent some of the real-world phenomena that
were seen in pandemic misinformation.

An additional contribution of this work is the inclusion of news or media
sources explicitly as a different type of network node from the individuals in
the network. It is suspected that information coming from media sources plays
an out-sized role in the spread of misinformation [25, 11, 1] – including how
difficult false beliefs, once adopted, are to extricate – and their inclusion
suggests some simple strategies by which misinformation might be countered.

These beliefs, once adopted, are so difficult to extricate that over the
course of 2020, the proportion of those who did not believe in the virus
hardly changed [26]. Anecdotally, reports circulated of nurses in states with
few regulations like South Dakota, describing patients who would be dying of
COVID and refusing to believe they had it. One nurse was quoted saying, “They
tell you there must be another reason they are sick. They call you names and
ask why you have to wear all that ‘stuff’ because they don’t have COVID and
it’s not real” [27].

Modern psychological and cognitive theory have made a substantive contribution
by making a clear distinction between beliefs that update in accordance to the
evidence one receives, and others which persist despite clear, logical
contrary evidence [28, 29, 21]. This is the distinction that appears key to
understanding mechanisms governing belief in misinformation. In fact, there is
evidence that those who engage in manufacturing misinformation exploit this
research to make their messages more potent. Wiley [30] has recently revealed
that some polarized, partisan beliefs and conspiracies have been _designed_ to
persist despite contrary evidence.

While the study of individual beliefs has recently been advancing, attempting
to determine how beliefs, true or otherwise, spread through an entire
population adds yet another layer of complexity. Sociologists and political
theorists have long studied public opinion: the theoretical mechanisms by
which populations come to certain beliefs, which are notoriously difficult to
verify empirically [31, 32]. However, with the advent of social network
research, scholars are moving toward that goal, studying how information
cascades through groups [33, 34, 35, 2, 23], and the wide-scale adoption of
certain beliefs [36]. Our work seeks to take one further step: understanding
how an integrated model combining individual cognitive belief models with
social network dynamics might be employed to study how public opinion shifts.

In the social network theory realm, our work generalizes a class of _Agent-
Based Models_ (ABMs), called _Agent Based Social System_ (ABSS) [37] – more
specifically, _social contagion_ models [24]. ABM is a powerful modeling
paradigm that has both successes and future potential in a variety of areas,
including animal behavior [38, 39, 40, 41, 42], social sciences [43], and,
notably, opinion dynamics [44, 45, 24, 46].

The applications of ABMs, even within the field of opinion dynamics, are very
diverse [47], but all ABMs must balance complexity of the agent and
environment models, and strength of results [48]: A model with simple rules,
but intriguing emergent results often raises more interesting implications
than a model that strives to match the complexity of reality [49] (for a
strong example of this principle, see Schelling’s model of segregation [43]).
Amidst a vast literature of what Zhang & Vorobeychik [47] – in a review of
innovation diffusion ABMs – call “cognitive agent models,” social contagion
models appear to be the simplest models of opinion dynamics, and have
subsequently sparked robust theoretical discussion in the social sciences [50,
23, 24].

By combining the simplicity of the social contagion paradigm with some of the
cognitive literature regarding misinformation, we propose our first major
contribution: a _cognitive contagion_ model that captures the spread of
identity-related beliefs. This model is then tuned to capture, on the
individual agent-level, two major effects cited in misinformation literature:
dissonance [19], and exposure [51, 10], capturing what we call _defensive
cognitive contagion_ (DCC).

We conduct an analysis of the cognitive contagion model within our second
contribution, what we call a _public opinion diffusion_ (POD) model. We
situate agents following cognitive contagion rules in a model that includes
_institutional_ agents, addressing the crucial ontological fallacy that only
individuals play a part in these systems [52]. Media companies have played
crucial roles in COVID misinformation [1, 11], and must be captured in our
model.

Through a simple cognitive function defined at the level of individual agents,
our cognitive contagion model is more powerful than simple or complex
contagion. Moreover, the motivations behind the DCC function demonstrate that
simple and complex contagion cannot capture identity-related belief spread
between individuals. Given findings from COVID misinformation studies, results
from our POD model with DCC appear to align with population-level results
reported by opinion polls – namely that beliefs about the virus remained
starkly partisan [6, 7, 8, 9], and hardly changed throughout 2020 [26]. These
preliminary results hint at possible interventions and offer plenty of
opportunities for future studies to be conducted using these methods.

## Background

### Simple Contagion

ABMs have been widely used to model _social contagion_ effects – those which
describe the process of ideas or beliefs spreading through a population [23].
Such models attempt to explain possible processes underlying the spread of
innovations [47, 53, 54], or unpopular norms [45]. These models are extensions
of earlier work in sociology that theorized how social network structure and
simple decisions, such as the threshold effect [50], may affect group-level
behavior. Through more abundantly available computational power, these ideas
can now be simulated and their implications can be analyzed.

There are two popular types of social contagion models used in ABMs: _simple_
and _complex contagion_. Both model the spread of behaviors, norms, or ideas
through a population. For simplicity, we will refer to behaviors or norms as
“beliefs” going forward, as it is plausible to argue that both are generated
by beliefs that an individual holds, explicit or otherwise. The simple
contagion model assumes that behaviors or norms can spread in a manner akin to
a disease [44, 33, 34, 23]. Simply being connected to an individual who holds
a belief engenders a probability, $p$, that the belief may spread to you. This
can even be true given different belief strengths or polarities for the same
proposition. More formally, given two nodes in the model $u$ and $v$, with
each having respective beliefs $b_{u}$ and $b_{v}$, when node $u$ is playing
out its decision process, the probability of adopting belief $b_{v}$ can be
modeled as:

$P(b_{u}=b_{v}\ |\ b_{u})=p.$ (1)

It is important to note that in belief contagion models, the probabilities
assigned to adopting a new belief given a prior one are over an event set of
only two outcomes: adopt the new belief, or keep the prior one. Thus, if there
are several possible beliefs to adopt from a set $B$, the sums of
probabilities of adopting some $b_{v}$ given a prior of $b_{u}$, for all
values in $B$, do not necessarily add to 1. Rather, because each agent
interaction is only between two belief values, even if they both come from a
larger set, the interaction is represented as a Bernoulli process. Each
instance below in which we motivate probabilities of adopting a belief given a
prior is adherent to the same logic. This is graphically depicted in Fig. 1.

Fig 1: An illustration of simple contagion. Given an agents $u$ and $v$,
contagion of $b_{v}=0$ to $u$ happens with probability $p=0.15$. The
probability of not changing beliefs becomes 0.85.

This simple equation can also be used to show how many infected neighbors are
necessary to nearly guarantee any agent being infected. Suppose an agent $u$
with neighbors $N(u)$ and a simple contagion probability of $p$. With each
infected neighbor $N_{i}(u)$, $u$’s chance of infection increases to
$1-p^{|N_{i}(u)|}$. If the model assumes some “high-likelihood” threshold for
infection, $\delta=0.95$, as a near guarantee of infection, some $|N_{i}(u)|$
will raise the infection chance such that $(1-p)^{|N_{i}(u)|}\geq\delta$. The
requisite number of neighbors becomes:

$|N_{i}(u)|\geq\lceil\frac{\log(1-\delta)}{\log(1-p)}\rceil$ (2)

For example, if $p=0.5$ and $\delta=0.95$, then $|N_{i}(u)|\geq\lceil
4.32\rceil\geq 5$ – $u$ would need at least five infected neighbors to almost
guarantee infection.

Of course, since beliefs are not actually transmitted through airborne
pathogens that incite infected individuals to believe something, there are
abundant sociological hypotheses as to why this phenomenon may _appear_
infectious [23]. There are other contagion models that have put forward
alternative explanations.

### Complex Contagion

Complex contagion rather imagines that the spread of beliefs is predominantly
governed by a ratio of consensus of those whom any agent is connected to [24].
In perhaps its simplest variant, given an agent to focus on (ego) they may
believe something if a certain proportion of their neighbors in the network
(alters) believe it. One of the most famous examples of this type of model is
in Schelling’s segregation model [43]. Formally, given an ego $u$ and set of
neighbors $N(u)$, the probability of adopting belief $b$ can be represented
as:

$P(b_{u}=b\ |\ b_{u})=\begin{cases}1,&\frac{1}{|N(u)|}\sum\limits_{v\in
N(u)}{d(v,b)}\geq\alpha,\\\ 0,&otherwise\end{cases},$ (3)

where $d(v,b)$ is a simple indicator function which returns 1 if $b_{v}=b$ –
i.e. if neighbor $v$ believes $b$ – and where $\alpha\in[0,1]$ is a threshold
indicating the ratio of believing neighbors necessary for $u$ to adopt the
belief. In this model, the agent $u$ is guaranteed to adopt $b$ if a
sufficient ratio of its neighbors believe $b$. We depict this graphically in
Fig. 2.

Fig 2: An illustration of complex contagion. Given an agent $u$, contagion
would require $\alpha=0.4$ of $u$’s neighbors to have some belief $b_{v}$.
Since $2/5$ of $u$’s neighbors have belief $b_{v}=0$, $u$ adopts $b_{u}=0$
with probability 1. The chance of adopting any other neighbor beliefs is 0.

It may seem tempting to imagine, given the simple contagion variation in Eq.
2, that the behavior of complex contagion can also be modeled with the
“sufficient number of neighbors” idea. However, that notion does not capture
the ratio effect that complex contagion models. Complex contagion says nothing
about the _innate infectiousness_ of any belief, but rather the infectiousness
of _the connections surrounding_ any agent.

The focus on a “portion” of believing neighbors being required to propagate a
belief spawned new questions and investigations. This type of belief contagion
has been argued to explain why some norms may spread despite them being
disagreed with on an individual level, such as collegiate drinking behavior
[45]. It elegantly captures phenomena associated with group dynamics such as
peer pressure. It also can be used to model diffusion of health information or
technological innovations [55].

Both these contagion models can be generalized to allow heterogeneous sets of
agents whose update rules are different for agents of different types (for
example, more or less susceptible to infection). However, while these two
popular types of contagion can effectively model some classes of belief
contagion, others cannot be captured by their mechanisms – even with
heterogeneous agents. Notably, models where there is an internal model of what
a given individual agent _already believes_ that dynamically affects what
beliefs they spread and adopt cannot be described by either simple or complex
contagion. We now introduce a more powerful model, cognitive contagion, that
allows the modeling and updating of individualized internal belief states.

## Cognitive Contagion Model

The goal of this work is to show that even the simplest cognitive contagion
model adds expressive power and leads to interesting dynamics of belief
propagation that cannot arise in the simple or complex contagion models. In
our simple example, we consider belief in a single proposition $B$ (for
example, $B$ could be, “COVID is a hoax,” or, “mask-wearing does not help
protect against spreading or contracting COVID”). In cognitive contagion
models, as distinct from simple or complex contagion models, $u$’s probability
of believing a message from $v$ is influenced by an internal model of $u$’s
beliefs. For our simple cognitive contagion model, we model this initially as
a single internal variable $b_{u}$, with $-1\leq b_{u}\leq 1$ – representing
$u$’s prior inclination based on its internal state to believe a message from
$v$. Without loss of generality, we use -1 to indicate strong disbelief and 1
to indicate strong belief.

$b_{u}$ can be a continuous variable with the interval from strong disbelief
to strong belief, or it can take on discrete values. For simplicity, and also
in harmony with frequently used 7-point Likert scales in public opinion
surveys (e.g. [6, 25, 11], and justified by [56]) for the remainder of this
paper we choose 7 discrete equally spaced values for belief in $B$ as follows:
we represent the strength of the belief in proposition $B$ with elements from
the set $B=\\{b\ |\ 0\leq b\leq 6\\},b\in\mathbb{Z}$; 0 represents strong
disbelief, 1 disbelief, 2 slight disbelief, 3 uncertainty, 4 slight belief, 5
belief, and 6 strong belief. Importantly, this representation captures the
polarity of the proposition as well: belief strength of the affirmative of $B$
(if $b\geq 4$), and the negation of $B$ (if $b\leq 2)$. From here on, we will
capture the idea of belief polarity – belief in or against a proposition $B$ –
by simply saying “belief strength.”

We include this cognitive model for an individual agent within a message-
passing ABM: At each time step $t$, agents have the chance to receive
messages, and to believe and share them with neighbors. We will further
clarify the role of messages in spreading beliefs below, when we describe our
diffusion model. But it should be noted upfront that regardless of being
passed by a message, or by simple network connection exposure, we can compare
beliefs from two agents, $u$ and $v$ the same way. We further note that
cognitive theory shows evidence that, for beliefs that are core to an
individual’s identity (such as political or ideological beliefs), exposure to
evidence that is too incongruous with an individual’s existing belief causes
individuals to disregard evidence in an attempt to reduce cognitive dissonance
[19]. Therefore, we later choose an update rule where an agent $u$ is only
likely to believe messages when encoded belief values for proposition $B$ are
not too far from $u$’s prior beliefs.

As a simple example, this could be represented by a binary threshold function.
Given an agent $u$ with belief strength in $B$, $b_{u}$, and an incoming
belief from $v$ with strength $b_{v}$, the following equation could govern
whether agent $u$ updates its belief:

$P(b_{u}=b_{v}\ |\ b_{u})=\begin{cases}1,&|b_{u}-b_{v}|\leq\gamma,\\\
0,&otherwise\end{cases},$ (4)

where $\gamma$ is a distance threshold. Each agent has some existing belief
strength in the proposition $B$, but will be unwilling to change their belief
strength if a neighbor’s belief strength is too far from theirs. Perhaps an
agent $u$ who disbelieves the proposition ($b_{u}=1$) will not switch
immediately to strongly believing it without passing through an intermediary
step of uncertainty. Given a neighbor $v$ sharing belief $b_{v}=6$, agent $u$
should not adopt this belief strength, because the difference in belief
strengths is clearly greater than $\gamma$. Simple contagion would fall short
because agent $u$ may simply randomly become “infected” with belief strength 6
by $v$ with some probability $p$. A complex contagion would similarly falter
if agent $u$ were entirely surrounded by alters with belief strength 6. It
would inevitably switch belief strengths regardless of some threshold $\alpha$
as in Eq. (3).

Fig 3: An illustration of cognitive contagion with the DSS contagion function
described in Eq. 7. (a) (Top) Given an agent $u$ with $b_{u}=6$ and $v$ with
$b_{v}=0$, the chance of contagion is $<$ 0.001. (b) (Bottom) Given an agent
$u$ with $b_{u}=6$ and $v$ with $b_{v}=5$, the chance of contagion is 0.982.

As mentioned above, this manner of belief update could model the update of
beliefs that are core to an individual’s identity, such as political or
ideological beliefs [57, 58, 28]. Rather than updating based on evidence
presented, exposure to evidence that is too incongruous with an individual’s
existing belief may have no effect due to rationalization processes activated
by cognitive dissonance [19].

In addition to the effects related to cognitive dissonance, there are other
effects that have been reported to be involved in belief update. Pertinent to
the misinformation literature, two that center the incoming belief itself are
the illusory truth effect [51, 22] and the mere-exposure effect [59]. These
effects emphasize that the number of exposures to a piece of information can
motivate belief in it.

Regardless of effect, it is clear that some social contagion processes cannot
be captured without modeling some sort of representation of an agent’s
cognition. It is for this reason that we have developed the _cognitive
contagion_ model. We will motivate an alternative model of contagion that
focuses on giving each agent in the network a very simple cognitive model. In
general, given an agent $u$ with belief $b_{u}$, during its update step, the
likelihood of updating its belief strength from $b_{u}$ to $b$, given their
prior belief, can be formalized as:

$P(b_{u}=b\ |\ b_{u})=\beta(b_{u},b).$ (5)

Because this equation is so general, there is a need to motivate a meaningful
choice of $\beta$ function, and analyze how its effects differ from simple and
complex contagion. There are obviously many choices for such a function, but
the key lies in the fact that it compares beliefs between two agents, rather
than being driven by network structure or mere chance. Below, we will describe
our process of choosing a $\beta$ function in order to adequately model the
misinformation effects that motivated our study.

## Public Opinion Diffusion Model

Of course, a cognitive contagion model could be implemented on top of many
different ABMs, so we will describe one that captures the misinformation
problem, and that we can use over multiple experiments to arrive at a
cognitive contagion model suited to the problem. Our _public opinion
diffusion_ (POD) ABM will be designed to capture the effects of misinformation
spread by media companies, since in the aforementioned COVID misinformation
studies, media played a pivotal role in people’s belief or disbelief in safety
protocols [25, 60, 61]. Often, ABMs do not model so-called “levels” of social
systems: ontological distinctions in abstraction (e.g. those between
individuals and institutions) [52]. The POD model addresses this by including
the _institutional_ level in the form of media agents. A visual description of
the model is included in Fig. 4.

Fig 4: A visual description of one time step of the POD model. In the left
panel, (A) depicts the initial setup of a small network with institutional
agent $i_{1}$ with subscribers $s_{1},s_{2},s_{3}$. All agents in the network
are labeled with their belief strength. The right panel, (B) depicts one time
step $t=0$ of agent $i_{1}$ sending messages $M_{1}(t=0)=(m_{0},m_{1})$. (_i_)
shows the initial sending of $m_{0}=4$ to subscribers, and (_ii_) shows
$s_{1}$ and $s_{3}$ believing the message and propagating it to their
neighbors. (_iii_) and (_iv_) show the same for $m_{1}=3$, but only $s_{3}$
believes $m_{1}$.

As previously mentioned, we will be using a message-passing ABM: at each time
step $t$, agents have the chance to receive messages $m$ from the set of all
possible messages $\mathcal{M}$ – whose spread begins with media agents, which
we call _institutional agents_ – and to believe and share them with neighbors.
We chose a message-passing model as opposed to the simple diffusion models
often found in simple and complex contagion models because it allows us to
capture the notion that beliefs are spread by explicit communication rather
than simply by being connected to an agent.

Our model consists of $N$ agents in a graph, $G=(V,E)$ where each agent’s
initial belief strength $b_{u},0\leq u\leq N$ is drawn from a uniform
distribution over the set of possible belief strengths for proposition $B$,
$B=\\{b,0\leq b\leq 6\\},b\in\mathbb{Z}$. There is a separate set of
institutional agents $I$ which have directed edges to a set of “subscribers”
$S\subseteq V$ if some parameter $\epsilon\leq|b_{u}-b_{i}|,u\in V,i\in I$ \-
i.e. an agent in the network will subscribe to an institution if its belief
strength from $B$ is sufficiently close to the belief strength of that
institution. For all of our experiments, we will fix $\epsilon$ at 0.
Institutional agents are designed to model media companies or notable public
figures which begin the mass spread of ideas through the population. The
belief strength of an institutional agent can be thought of as a perceived
ideological “leaning” that would cause people with different prior beliefs to
trust different media organizations.

At each time step, $t$, each institution $i$ will send a list of messages to
each of its subscribers, represented by the function
$M_{i}:t\to(m_{0},m_{1},...,m_{j}),m_{j}\in\mathcal{M},j\geq 0$. In this
simple example, the set of possible messages $\mathcal{M}$ will only encode
one proposition, $B$, so for simplicity, we can set $\mathcal{M}=B$.
Additionally, institutions will only send one message per time step. Whenever
a message is received, an agent will “believe” it based on the contagion
method being utilized, where $b_{mj}$ is the strength of belief in proposition
$B$ encoded by the message, and $b_{u}$ is the agent’s belief strength for
$B$. If agent $u$ believes message $m_{j}$, then its belief strength is
updated to be $b_{mj}$. When an agent believes a message, it shares the
original message, $m_{j}$, with all its neighbors. It should be noted that
because agent $u$ would change its belief strength to $b_{mj}$, agents will
always share beliefs that are congruous to prior belief strengths – cohering
to our cognitive contagion model outlined above. After a neighbor receives a
message, the cycle continues: It has a chance to believe the message, and if
believed, spread it to its neighbors. To avoid infinite sharing, each agent
will only believe and share a given message once based on a unique identifier
assigned to it when it is broadcast by the institutional agent.

Our model and experiments were implemented using NetLogo [62] and Python 3.5.
Code is made available on GitHub 111https://github.com/RickNabb/cognitive-
contagion.

## Contagion Experiments

### Experiment Design

We wish to show that a model of cognitive contagion, on top of this ABM, can
capture the observed effects of identity-based belief spread better than
existing models of simple or complex contagion. To do so, we will lay out a
series of experiments to test each contagion model given the same initial
conditions.

For each model, we test three conditions where one institutional agent,
$i_{1}$, attempts to spread different combinations of messages over time. We
will refer to the first message set as _single_ , as the institutional agent
simply broadcasts one message for the entirety of the simulation;
$M_{i}(t)=(6),1\leq t\leq 100$. The second set, we will call _split_ , as the
institution switches from messages of $M_{i}(t)=(6),1\leq t\leq 50$ to
$M_{i}(t)=(0),51\leq t\leq 100$ halfway through the simulation. We call the
final set _gradual_ because the institution starts out spreading messages of
$M_{i}(t)=(6)$, but at every interval of ten time steps, switches to
$M_{i}(t)=(5)$, $M_{i}(t)=(4)$, etc. until finishing the last 30 time steps by
broadcasting $M_{i}(t)=(0)$.

Fig 5: A visual depiction of the different message set conditions used in our
experiments: _single_ (top), _split_ (middle), and _gradual_ (bottom), set
against a 100-step simulation from $t=0$ to $t=100$.

These specific sets of messages were chosen to expose distinct effects given
each set, specifically with cognitive contagion in mind. Based on research
about how identity-related beliefs update, a proper model would not allow
agents with a belief strength significantly different from the message to
update their belief. With the _single_ message set, we wish to provide the
simplest case: only one belief strength message is being spread. We
hypothesize that simple contagion will simply spread the messages to all
agents, regardless of prior belief strength, and all agents will eventually
update their belief strength to that in the message. We also anticipate that
complex contagion, being so reliant on prior belief strengths of an agent’s
neighbors, will not be so straightforward. Assuming a nearly uniform
distribution of agent neighbor belief strengths, the threshold chosen would
likely make the difference between all agents updating their beliefs, or no
agents updating their beliefs. A proper cognitive contagion model that
captures our desired effect should see only belief updates from agents whose
belief strengths are already close to the belief strength encoded in the
message.

The _split_ message set, on the other hand, should have different effects. We
hypothesize that simple contagion will see all agents believe the first
belief, then all agents believe the second, while complex contagion may spread
the initial belief but not the second. For cognitive contagion, if the
function we choose successfully models our target phenomena, only agents
within the same threshold as in the _single_ condition should believe the
first message, then virtually no agents should believe the second – except a
few who may switch based on exposure effects.

Finally, we hypothesize that the _gradual_ message set should be the only
which allows cognitive contagion to sway the entire network. Because agents
will only believe messages that are relatively close to their prior beliefs,
it logically follows that the only way to move beliefs from one pole to
another is incrementally. We further hypothesize that simple contagion will
sway the entire agent population to adopt the belief strength of each message
in turn, and that complex contagion may sway the entire population to one
specific belief strength, but then not be able to change any agent belief
strengths after such a contagion.

We also keep certain contagion variables static between experiments and
conditions. In each case, we will fix the simple contagion probability, $p$ to
be 0.15, and fix $\alpha$, the complex contagion neighbor threshold to be
0.35. The former was chosen to allow a slower spread of belief strengths, as
higher values would make the spread too fast to properly analyze. The latter
was chosen as it, too, would ideally avoid being so low that contagion happens
immediately and in all cases, or so high that it never occurs. In preliminary
experiments, the chosen values best satisfied these goals.

As a final experimental condition to vary independently of message set, we
will run experiments on a host of different graph topologies, keeping the
number of nodes and prior distribution of belief strengths constant. We test
each contagion model on four networks topologies:

  * •

The Erdős-Rényi (ER) random graph [63]

  * •

The Watts-Strogatz (WS) small world network [64]

  * •

The Barabási-Albert (BA) preferential attachment network [65]

  * •

The Multiplicative Attribute Graph (MAG) [66]

We explicate rationale behind choosing each below. Running experiments across
different graph topology should allow us to determine how much graph structure
affects contagion. We hypothesize that simple contagion may not vary much over
graph type, that complex contagion will vary the most because it is the most
dependent on neighborhood structure, and that cognitive contagion should vary
least, as it relies more on single instances of neighbor belief strengths
rather than an aggregate. Additionally, for each graph and contagion type,
experiments are run ten times and results are shown as averages over the total
simulations, with variances displayed in supplemental materials (S1-S2) where
applicable. Ten simulations appeared adequate in preliminary experiments
because variance was fairly low, and did not seem to require hundreds or more
simulations.

But first, we need to choose a cognitive contagion function that best suits
our empirically-based goals. After choosing such a function, it will be
compared against simple and complex contagion in each condition.

### Motivating Choice of $\beta$ Functions

Unsurprisingly, depending on the choice of $\beta$, the network should display
significantly different dynamics. That choice should depend on what type of
phenomena is being modeled. We will explore a variety of choices for this
function and compare the outcome of their respective cognitive contagion
against the qualitative target phenomena.

For our specific purposes, we want to tune the choice of function to capture
the effect of updating beliefs core to an individual’s identity: in a manner
where incoming belief strengths must be close to existing belief strengths to
yield an update. If possible, it would also be useful to be able to choose a
function which also captures aspects of exposure effects: that beliefs do have
a small chance to update that grows as the agent is exposed to a belief over
time. However, agents should prioritize the dissonance effect. That is, they
should only experience an exposure effect if incoming messages are already
somewhat close to existing beliefs. The dissonance effect should take
precedence.

Keeping these two target phenomena in mind, we explore three classes of
functions: linear, threshold, and logistic.

#### Linear Functions

To begin with perhaps the simplest function, an inverse linear function would
capture the effect of making beliefs that are further apart have a smaller
probability of updating. Moreover, we can generalize this inverse function by
adding some parameters to add a bias and scalar to the denominator. The
equation comes out as follows:

$P(b_{u}=b\ |\ b_{u})=\beta(b_{u},b)=\frac{1}{\gamma+\alpha|b_{u}-b|}$ (6)

In this equation, $\gamma$ becomes a parameter to add bias towards being
reluctant to update beliefs, and $\alpha$ similarly decreases the probability
of update as it increases. This turns out to be a useful formulation, because
if we set $\gamma$ and $\alpha$ to be very low, then the agent becomes
relatively more “gullible.” Conversely, setting $\gamma$ and $\alpha$ to be
high would make the agent “stubborn.” We hypothesize that the “stubborn” agent
will be most desirable for our purposes of modeling cognitive dissonance.

To compare parameterizations of the inverse linear function, we contrast, on
an Erdős-Rényi random graph, $G(N,\rho)=(V,E)$ with $N=250,\rho=0.05$, in our
POD model setup described above, a relatively “gullible” agent function
($\gamma=1,\alpha=0$) to a “normal” function ($\gamma=1,\alpha=1$), and to a
“stubborn” ($\gamma=10,\alpha=20$) function. We additionally display results
for only the _split_ message set, though results for others can be found in
the supplemental materials (S3-S8). Results are displayed in Fig. 6.

Fig 6: The _split_ message set on an ER random graph, $N=250,\rho=0.05$, for
agents updating their beliefs based on the inverse linear cognitive contagion
function in Eq. (6). Graphs display percent of agents who believe $B$ with
strength $b$ over time. The left graph shows agents parameterized to be
“gullible” ($\gamma=1,\alpha=0$); the middle shows “normal” agents
($\gamma=1,\alpha=1$), and the right, “stubborn” agents
($\gamma=10,\alpha=20$).

As expected, the results show that the “gullible” agents simply believe
everything. The “normal” agents take a bit longer to all update their belief
strengths to that of the messages broadcast, but eventually do. Importantly,
when all agents’ belief strength for $B$ is $b=6$ after the first 50 time
steps, they all quickly switch over to $b=0$, which does not fit the cognitive
dissonance effect we are trying to model. The “stubborn” agent case is the
closest to what we are seeking. Only the agents who are already closest to
$b=6$ believe the first message over time, with some effect on agents with
more distant beliefs. After the messages switch polarity halfway through, the
agents whose strength of belief is $b=6$ do drop significantly, but less so
than in the other conditions. This effect seems closer to the dissonance mixed
with exposure effects that we desire: beliefs are less likely to change as
they are far away, but there is a chance to change with many messages over
time.

#### Threshold Functions

Next, we evaluate the behavior of threshold functions and compare to our
desired effect. Threshold update functions are used in opinion diffusion ABMs,
most notably in the HK bounded confidence model [67, 68, 69]. We anticipate
that these functions will capture the desired dissonance effect. The function
can be parameterized as we already motivated in Eq. (4), with $\gamma$ serving
as the threshold.

Using the same formulations as above, with “gullible” ($\gamma=6$), “normal”
($\gamma=3$), and “stubborn” ($\gamma=1$) agents, we find results on the same
graph structure as displayed in Fig. 7.

Fig 7: The _split_ message set on an ER random graph, $N=250,\rho=0.05$, for
agents updating their beliefs based on the threshold cognitive contagion
function in Eq. (4). Graphs display percent of agents who believe $B$ with
strength $b$ over time. The left graph shows agents parameterized to be
“gullible” ($\gamma=6$); the middle shows “normal” agents ($\gamma=3$); and
the right, “stubborn” agents ($\gamma=1$).

These results confirm our anticipations, and perfectly capture the effect of
only updating if incoming messages are within a certain distance of existing
beliefs. However, the threshold function leaves no possibility of update to
capture the repetition effects of mere-exposure or illusory truth. The
probability of updating is either 0 or 1, which loses much of the nuance of
the actual phenomena.

#### Sigmoid Functions

Finally, we will test a logistic function – specifically a sigmoid function,
as we are attempting to capture probabilities and the range of the sigmoid is
[0,1]. Sigmoid functions are commonly used in neural network models because
they capture “activation” effects arguably akin to action-potentials in
biological neurons, and rein in outputs so they do not explode while learning
[70]. This property is useful for our purposes as well. We formulate our
sigmoid cognitive contagion function as follows:

$\beta(b_{u},b)=\frac{1}{1+e^{\alpha(|b_{u}-b|-\gamma)}}$ (7)

In this equation, $\alpha$ and $\gamma$ control the strictness and threshold,
respectively. As $\alpha$ increases, the function looks more like a binary
threshold function, and restricts any significant probability to center around
$\gamma-1$. $\gamma$ controls the threshold value by translating the function
on the $x$ axis. Though, in a sigmoid function, the value at $x=\gamma$ will
always be 0.5, so if one wishes to guarantee belief update given a threshold
$\tau$, it must be set as $\tau=\gamma+\epsilon$ where
$\epsilon\propto\frac{1}{\alpha}$. We use this strategy to set our $\gamma$
values throughout our experiments.

Given the same experiments as above, the sigmoid function parameterized for
different agent types (“gullible” ($\alpha=1,\gamma=7$), “normal”
($\alpha=2,\gamma=3$), and “stubborn” ($\alpha=4,\gamma=2$)) yields results as
show in Fig. 8.

Fig 8: The _split_ message set on an ER random graph, $N=250,\rho=0.05$, for
agents updating their beliefs based on the sigmoid cognitive contagion
function in Eq. (7). Graphs display percent of agents who believe $B$ with
strength $b$ over time. The left graph shows agents parameterized to be
“gullible” ($\alpha=1,\gamma=7$), the middle shows “normal” agents
($\alpha=2,\gamma=3$), and the right, “stubborn” agents ($\alpha=4,\gamma=2$).

These results seem to display characteristics of both the inverse linear
function and the threshold function. The “gullible” agents, as always, believe
everything but with a softer transition than for the linear or threshold
functions. “Normal” agents are the first indication that we are getting closer
to our desired effect. After an initial widespread uptake of belief strength
$b=6$ in the first half of the simulation, some agents begin to believe $b=0$,
but the population that decreases the most to engender that gain seem to be
agents with belief strength $b=1$. Though, from our model, some agents with
strength $b=6$ must have believed $b=0$ messages, because if they did not, the
messages would never have been shared and made it to $b=1$ agents – the
institutional agents are only connected to $b=6$ agents as $\epsilon=0$.

Finally, the “stubborn” agents seem to best capture our desired effect.
Initially, $b=6,b=5,$ and $b=4$ agents are the only who update beliefs. There
is also a small effect where some $b=3$ agents update to $b=6$, capturing the
exposure effects combined with a dissonance effect. Importantly, when messages
switch to $b=0$, none of the $b=6$ agents update their beliefs. The exposure
effects would not work as the dissonance effect would take primacy.

Given these initial experiments, it seems most reasonable to choose the
“stubborn” sigmoid cognitive contagion function as that which best captures
our desired effects. We will use this _defensive cognitive contagion_ (DCC)
function in the rest of our experiments as we compare the effects of cognitive
contagion to those of simple and complex contagion.

## Comparing Contagion Models

Now that we have selected a cognitive contagion function that best captures
the effects we wish to model, it is necessary to compare results of this
function to those from simple and complex contagion. As a reminder, for simple
contagion, we are using an infection probability of $p=0.15$, and for complex
contagion, a ratio of $\alpha=0.35$. We will investigate the way that these
different contagion models manifest effects on different network structures.
In addition to significant effects based on the choice of the $\beta$
function, we hypothesize that effects will also significantly differ based on
the structure of the network. The structure will determine which ideas reach
agents, and which do not, and thus should affect the final outcome of belief
distribution over the network.

We will test each contagion model on five types of networks: the Erdős-Rényi
(ER) random graph [63], the Watts-Strogatz (WS) small world network [64], the
Barabási-Albert (BA) preferential attachment network [65], and the
Multiplicative Attribute Graph (MAG) [66]. Each network has distinct
properties that will affect how the contagions play out. Additionally, we will
test each message set for each network type to explore the effects of
different influence strategies.

### Contagion on ER Random Networks

We will begin with a contagion on an Erdős-Rényi [63] random network
$G(N,\rho)=(V,E)$ where $\rho=0.05$ and $N=500$. Note that $\rho$ is not the
chance of simple contagion, $p$, but the chance that two agents connect in the
random graph. This graph type was chosen as a baseline to compare others to,
as is standard in network science. Results of the simple and complex
contagions are shown below in Fig. 9.

Fig 9: Simple (top row) and complex (bottom row) contagion on a random network
with $N=500$, and connection chance $\rho=0.05$. Graphs show the percent of
agents who believe $B$ with strength $b$ over time.

Results from the simple contagion experiments show that in each message set
condition, the belief strength being spread pervaded the entire network in all
cases. Moreover, the strength of beliefs broadcast were adopted by the
population very quickly. In the case of complex contagion, the initial
distributions of belief strengths did not change in any message set condition.
This is likely due to the fact that in an ER random graph, nodes are likely to
have a uniform distribution of neighbors (which, for seven values of $B$,
would yield a 0.143 ratio for each $b$), so the complex contagion threshold of
$0.35$ could not be reached given any message’s $b$ value.

Cognitive contagion on the ER graphs yielded markedly different results. As
depicted in Fig. 10, both the _single_ and _split_ message sets were only able
to sway agents that started with $b=6$, $b=5$, or $b=4$, with what appears to
be a few $b=3$ agents persuaded. Importantly, no agents were swayed after the
messaging change in the _split_ condition. The _gradual_ message set is the
only that was able to sway all agents over to $b=0$.

Fig 10: DCC on a random network with $N=500$, and connection chance
$\rho=0.05$. Graphs show the percent of agents who believe $B$ with strength
$b$ over time.

### Contagion on WS Small World Networks

Our second set of experiments were conducted on Watts-Strogatz [64] small
world networks, $G(N,k,\rho)=(V,E)$, where $N=500,k=5$, and $\rho=0.5$. In
this formulation, $k$ is the number of initial neighbors any node is connected
to, and $\rho$ is the chance of rewiring any edge. We chose this graph
topology because small world networks exhibit some attributes of real-world
social networks (low diameter and triadic closure). Results of simple and
complex contagions on these WS graphs are shown in Fig. 11.

Fig 11: Simple (top row) and complex (bottom row) contagion on a Watts-
Strogatz small world network with $N=500$, initial neighbors $k=5$, and
rewiring chance $\rho=0.5$. Graphs show the percent of agents who believe $B$
with strength $b$ over time. Asterisks (*) denote these contagions had
significant variance over simulation iterations.

Simple contagion results showed largely the same pattern as in the ER random
graph, but a significantly slower spread through the population. This is
likely due to the fact that the WS graphs had a total of 2,500 edges, where
the rounded expected number of edges in the ER graphs was 6,238.
Interestingly, complex contagion was successful on the WS graphs, but with a
high amount of variance over simulations (variance shown in S1). With only 5
neighbors, the barrier to fulfilling the $0.35$ threshold for belief is much
easier - requiring only 2 neighbors who match the message’s $b$ value. This
likely led to contagion whereas the threshold given an expected neighbor count
in the ER graph of 25 is much harder to satisfy with 9 neighbors.

Results from cognitive contagion experiments closely match those from the ER
random graph experiments. These are displayed in Fig. 12. The population
displayed the same patterns given a significantly different graph structure,
which begs explanation. We will discuss this further below.

Fig 12: DCC on a Watts-Strogatz small world network with $N=500$, initial
neighbors $k=5$, and rewiring chance $\rho=0.5$. Graphs show the percent of
agents who believe $B$ with strength $b$ over time.

### Contagion on BA Preferential Attachment Networks

We continued our experiments by testing Barabási-Albert [65] preferential
attachment networks, $G(N,m)=(V,E)$, where $N=500$ and $m=3$. $m$ represents
the number of edges added with each newly added node. This network type was
also chosen because of its properties that closely resemble real-world social
networks (low diameter, power law degree distribution). Results are shown in
Fig. 13.

Fig 13: Simple (top row) and complex (bottom row) contagion on a Barabási-
Albert preferential attachment network with $N=500$, and added edges $m=3$.
Graphs show the percent of agents who believe $B$ with strength $b$ over time.
Asterisks (*) denote these contagions had significant variance over simulation
iterations.

Again, simple contagion results are similar to those of the WS graphs: slower
contagion than in ER random graphs, but faster than in WS graphs. This is
interesting because the number of edges in each BA graph would be 1,500 -
clearly less than our WS graphs - yet spread was faster. In this case, spread
is facilitated by the power law distribution of node degree – even a few nodes
with high degree believing the message can have an outsize effect in spreading
quickly to the outskirts of the network [71]. Also interestingly, complex
contagion appears to have slight effects on these graphs, with the _gradual_
message set having most effect. Complex contagion results also showed
significant variance (shown in S2). Perhaps the power law degree distribution
also created nodes able to be influenced with fewer edges than in the ER
graph. But since node degree was more distributed than in a WS graph, it
logically follows that complex contagion was not as ubiquitous for all nodes.

Again, results from cognitive contagion experiments were incredibly similar to
those of the ER random and WS graphs – most similar to results from the WS
graph. Results are shown in Fig. 14.

Fig 14: DCC on a Barabási-Albert preferential attachment network with $N=500$,
and added edges $m=3$. Graphs show the percent of agents who believe $B$ with
strength $b$ over time.

### Contagion on MAG Networks

Finally, we tested contagion on the Multiplicative Attribute Graph [66] with
an affinity matrix $\Theta_{b}$ that yielded a graph with very high homophily.
We chose this graph topology because real world social networks are homophilic
– people who have similar interests tend to connect. Testing a highly
homophilic graph (higher than in a real social network) can allow us to test
the extreme case of communities in silos based on their belief strength. The
affinity matrix was constructed as follows:

$\displaystyle\Theta_{b}$ $\displaystyle=(\theta_{ij})\in\mathbb{R}^{m\times
n}=\frac{1}{1+50(j-i)^{2}}$ (8)
$\displaystyle=\begin{bmatrix}0.167&0.018&0.005&0.002&0.001&0.0008&0.0006\\\
0.018&0.167&0.018&0.005&0.002&0.001&0.0008\\\
0.005&0.018&0.167&0.018&0.005&0.002&0.001\\\
0.002&0.005&0.018&0.167&0.018&0.005&0.002\\\
0.001&0.002&0.005&0.018&0.167&0.018&0.005\\\
0.0008&0.001&0.002&0.005&0.018&0.167&0.018\\\
0.0006&0.0008&0.001&0.002&0.005&0.018&0.167\\\ \end{bmatrix}$ (9)

To measure homophily, we used a simple measure of the global average neighbor
distance given the $b$ value of each node, and compared against a random ER
graph. The measure is detailed in Eq. (10):

$h(G=(V,E))=\frac{\sum\limits_{v\in V}\sum\limits_{u\in
N(v)}|b_{u}-b_{v}|}{2|V|^{2}},$ (10)

where $N(v)$ is a function that returns neighbors $u\in V$ of $v$. Over ten ER
random graphs with $N=500$ and $\rho=0.05$, the mean average neighbor distance
was 2.30 with a mean variance of 0.349. Over ten MAG graphs generated with
$\Theta_{b}$, the mean average neighbor distance was 0.31 with a mean variance
of 0.02. Results from simple and complex contagion on these homophilic MAG
graphs are shown in Fig. 15.

Fig 15: Simple (top row) and complex (bottom row) contagion on a homophilic
MAG network with $N=500$, and $\Theta_{b}$ detailed in Eq. (9). Graphs show
the percent of agents who believe $B$ with strength $b$ over time.

As it turns out, a high degree of homophily did not appear to make a
significant difference to any contagion patterns. The patterns generated from
simple and complex contagion appear incredibly similar to those from the ER
random graphs. The same is true for cognitive contagion results, which are
depicted in Fig. 16. This notion makes sense for simple contagion, because as
long as the expected number of neighbors for each node is similar to that of
our ER graphs, the belief strengths of neighbors make no difference. For
complex contagion, it also makes sense that no agents seemed to update their
belief strength because they are surrounded almost entirely by other agents of
the same belief strength – meeting the 0.35 threshold for any message would be
infeasible. But the result for cognitive contagion may be surprising, as it
still resembles the same pattern seen in each prior experiment with only
slight variation.

Fig 16: DCC on a homophilic MAG network with $N=500$, and $\Theta_{b}$
detailed in Eq. (9). Graphs show the percent of agents who believe $B$ with
strength $b$ over time.

## Analysis of Results

Results across different graph toplogies further support our motivations for
introducing cognitive contagion models. Across Erdős-Rényi random graphs [63],
Watts-Strogatz small world networks [64], Barabási-Albert preferential
attachment networks [65], and highly homophilic Multiplicative Attribute
Graphs [66], simple and complex contagions had varying results. Due to highly
varied graph structure, it makes sense that contagion models which are very
reliant on structure yield differing results.

Our results from the DCC cognitive contagion model do not show such variation.
In fact, across all graph types tested, it appears that the cognitive
contagion results do not significantly differ. These results are encouraging
because the entire motivation for our model takes out the structural
dependencies of complex contagion, and replaces simple contagion’s random
chance of spread with one motivated by what agents already believe. Thus the
key factors in our cognitive contagion model are:

  1. (1)

Whether or not any given agent is exposed to some message;

  2. (2)

How many times an agent is exposed to similar messages; and

  3. (3)

The difference between agent beliefs and that message.

These results qualitatively match what has been observed in misinformation
literature. Even when exposed to factual or scientific evidence (e.g. that
wearing masks would mitigate the COVID-19 pandemic), people who are already
skeptical of mask-wearing are not able to be swayed. They often instead
rationalize their existing beliefs [2, 21, 28, 58]. Additionally, mass-
exposure to a given message still has a chance to sway agents in our model –
proportionally to how distant that message is to agent beliefs. This captures
the illusory truth [51, 22] and mere-exposure effects [59].

We can also quantify this result. For any message $m_{j}$ from a list
$M_{i}:t\to(m_{0},m_{1},...,m_{j}),m_{j}\in\mathcal{M},j\geq 0$ sent by an
institution $i$ at time step $t$, there will be a probability that any agent
$u$ will believe the message, which depends on the factors listed above. In
terms of our model,

  1. $(1)$

becomes the probability of $m_{i}$ being received and believed by any neighbor
$N(u)$ of $u$;

To travel from institution $i$ to agent $u$, a message must follow a directed
path through the graph, $w_{iu}=(v_{1},v_{2},...,v_{n})$ where $v_{1}=i$ and
$v_{n}=u$. The probability of a message being passed down the entire path can
be expressed as:

$\mathcal{P}(m_{j},w)=\prod\limits_{v\in w}{\beta(v,m_{j})}.$ (11)

To then properly represent (1), we can limit $w_{iu}$ to end at neighbors of
$u$.

The next step requires us to determine how many neighbors $N(u)$ of $u$
believe $m_{j}$ – as they would then subsequently propagate the message to
$u$. Therefore,

  1. $(2)$

becomes $|N_{\beta}(i,u,m_{j})|$, the number of neighbors of $u$ who are
likely to believe $m_{j}$ coming from $i$.

However, we do not know ahead of time which neighbors will actually believe
$m_{j}$ as the model is stochastic. We can argue that a probability within
some $\delta$ will suffice. We can represent $N_{\beta}(i,u,m_{j})$ as:

$N_{\beta}(i,u,m_{j})=\\{\ v\ |\ v\in N(u)\land\mathcal{P}(w_{iu},m_{j})\geq
1-\delta\ \\},$ (12)

Because in the POD model, agents can only believe and share $m_{j}$ once, to
determine how many neighbors of $u$ would share the message, we must choose a
set of non-overlapping paths $\mathcal{W}^{*}$ from $i$ to neighbors of $u$ –
moreover, one that maximizes total path probabilities:

$\mathcal{W}^{*}_{iu}=\max_{w}\prod{\mathcal{P}(m_{j},w)}\ \\{\ w_{iu}\ |\
\bigcup_{w_{iu}}(\cdot)=\O\ \\}$ (13)

The algorithmic formulation of such a process would best be captured in future
work.

Regardless of methods, it stands that $\mathcal{P}(m_{j},w)$ is crucial in
determining whether any agent $u$ will have a chance of receiving a message
and believing it. This result can help explain why simple and complex
contagion showed such variation across graph types, but cognitive contagion
did not. Given the probabilities of adopting belief strengths $b_{u}$ given a
prior belief of $b_{v}$ for DCC shown in Table 1, it becomes clear that if any
$\beta(v,m_{j})$ is given a belief strength difference of 3 or higher, then
the entire chain’s probability $\mathcal{P}(m_{j},w)$ will collapse to very
close to 0.

$b_{u}/b_{v}$ | 0 | 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---|---|---
0 | 0.999 | 0.982 | 0.500 | 0.018 |  $<$0.001 |  $<$0.001 |  $<$0.001
1 | 0.982 | 0.999 | 0.982 | 0.500 | 0.018 |  $<$0.001 |  $<$0.001
2 | 0.500 | 0.982 | 0.999 | 0.982 | 0.500 | 0.018 |  $<$0.001
3 | 0.018 | 0.500 | 0.982 | 0.999 | 0.982 | 0.500 | 0.018
4 |  $<$0.001 | 0.018 | 0.500 | 0.982 | 0.999 | 0.982 | 0.500
5 |  $<$0.001 |  $<$0.001 | 0.018 | 0.500 | 0.982 | 0.999 | 0.982
6 |  $<$0.001 |  $<$0.001 |  $<$0.001 | 0.018 | 0.500 | 0.982 | 0.999
Table 1: Probabilities given by $\beta(b_{u},b_{v})$ for the DCC function,
described in Eq. (5).

To satisfy (1), a path of agents with belief strengths at most distance 1 away
from the message would reliably transmit it with high probability. If any
agents in the path have a belief of distance 2 from the message, the
transmission probability would decrease; halving the probability with each
agent of distance 2. Compare this with simple contagion, where every agent has
a flat 0.15 probability of sharing, and the path probability converges close
to 0 in only two steps; or complex contagion where if any agent in the path
does not meet the threshold of 0.35, the path probability immediately
collapses to 0.

Taking this criteria for (1) into account, Table 1 also makes clear that to
satisfy (2) and (3), the chain may only need to end with one agent – i.e.
$|\mathcal{W}^{*}_{iu}|=1$. If the message belief strength is distance 0 or 1
from $u$, then there is already a near guaranteed chance that $u$ will believe
the message after receiving it only once. Conversely, in any quantitative
analysis of which agents may believe a message, we can exclude with high
confidence all agents with a belief strength difference of 3 or higher from
consideration, as their chances of believing the message even if it made it to
them would be near zero.

Therefore, to quantitatively demonstrate why the DCC results were so stable
across random graph types, we can show, for a randomly selected agent $u$ with
belief strength $b_{u}$, the percentage of 100 random generations of the graph
which yield at least one path of agents $v$ entirely with $b_{v}$ distance at
most $\tau$ away from a message with belief strength $b_{mj}$. Moreover, we
can show this for all potential values of $b_{u}$, keeping the belief strength
of the message constant, $b_{mj}$, at 6 (quantifying the _single_ message
condition). Importantly, this path does not include $u$ because for $b_{u}=3$
and below, the distance of $b_{u}$ to $b_{mj}$ would always be too high; thus,
our paths lead to neighbors of $u$. These paths were found by assigning edge
weights equal to the distance between the message and the belief strength of
the source node in the directed pair, and running Dijkstra’s weighted shortest
path algorithm with $i$ as the source and $u$ as the target.

($\tau=1$) | $b_{u}=0$ | $b_{u}=1$ | $b_{u}=2$ | $b_{u}=3$ | $b_{u}=4$ | $b_{u}=5$ | $b_{u}=6$
---|---|---|---|---|---|---|---
ER | 0.91 | 0.9 | 0.97 | 0.95 | 1.0 | 1.0 | 1.0
WS | 0.51 | 0.56 | 0.59 | 0.7 | 0.64 | 0.62 | 1.0
BA | 0.73 | 0.66 | 0.66 | 0.75 | 0.63 | 0.75 | 1.0
MAG | 0.13 | 0.2 | 0.37 | 0.54 | 0.8 | 1.0 | 1.0
($\tau=2$) | $b_{u}=0$ | $b_{u}=1$ | $b_{u}=2$ | $b_{u}=3$ | $b_{u}=4$ | $b_{u}=5$ | $b_{u}=6$
ER | 0.98 | 0.99 | 0.97 | 1.0 | 1.0 | 1.0 | 1.0
WS | 0.65 | 0.8 | 0.76 | 0.72 | 0.76 | 0.81 | 1.0
BA | 0.84 | 0.89 | 0.84 | 0.82 | 0.89 | 0.88 | 1.0
MAG | 0.29 | 0.29 | 0.45 | 0.84 | 0.99 | 1.0 | 1.0
Table 2: Proportion of 100 random graphs with at least one path leading from
the institutional agent $i$ to a randomly selected node $u$ with belief
strength $b_{u}$, where each agent $v$ in the path has belief strength
$|b_{v}-b_{mj}|\leq\tau$, and $b_{mj}=6$.

When $\tau$ is 1 the path yields an almost guaranteed probability of the
message reaching $u$. When $\tau$ equals 2, the path can yield a range of
chances to reach $u$ – depending on how many distances of 2 there are in the
path, each inserting a probability of 0.5 into the total product. However,
compared to the path probability in simple or complex contagion – where the
former is a path entirely of probabilities of 0.15, and the latter requiring
meeting a threshold of 0.35 for each agent in the path to even have a non-zero
chance – both path types yield significantly higher probabilities of messages
reaching target agents.

Moreover, the analysis shows that across graph types, where there are high
proportions of both path types present, there is a high likelihood for
messages to reach agents of _all_ belief strengths. Particularly for Erdős-
Rényi random graphs, both path types are almost always present. Both Watts-
Strogatz and Barabási-Albert networks show lower, but still high proportions
of both path types being present. This likely accounts for the slight
variations in cognitive contagion results displayed in graphs above.
Predictably, homophilic MAG graphs show a decreasing likelihood of both path
types as the distance between $b_{u}$ and $b_{mj}$ increases. If any of these
paths yielded a message reaching agent $u$, then combined with the
probabilities in Table 1, we see that target agents with belief strength
distance 2 or less from the message will likely believe it, and update
accordingly. This is exactly what we see qualitatively in the above results,
as regardless of graph type, agents with belief strength 4 or higher quickly
adopt a stronger belief of 6 from the message, and agents with belief strength
of 3 eventually update after enough messages reach them.

## Discussion

From our experiments, it is clear that the three types of social contagion
affect populations differently given the same initial conditions and beliefs
to spread. We were able to show, as predicted, that simple and complex
contagion methods change agent belief strengths in a manner that does not
depend on what they were believing previously. In the _single_ and _split_
message set conditions, many agents’ belief strengths were able to be swayed
from value-to-value regardless of their initial belief. Thus, these contagion
models do not capture the cognitive phenomena that motivated our experiments.

On the other hand, our simple cognitive contagion model also performed as
expected. The results were fairly robust across several graph topologies. In
the _single_ and _split_ message set conditions, most agents following the DCC
function did not change their belief strength over time. This fits the
underlying social theory because the messages were too far from what agents
initially believed, so not updating their beliefs accurately models the
defensive or entrenching effects observed when people are exposed to identity-
related beliefs that they do not agree with [28, 22]. The only message set
condition that was able to sway the entire population in the cognitive
contagion condition was the _gradual_ set.

The POD model with DCC also appears to capture population-level trends in
opinion data that originally motivated our study. Results match the partisan
polarization phenomena being observed [6, 7, 8, 9, 26], as agents who only
update their belief strengths in this manner are highly unlikely to be swayed
by belief strengths that are too far from theirs. Once swayed in one direction
or another, our agents could not adopt significantly differing belief
strengths without being nudged along.

Given these similarities in results, even our very simple model may be able to
lend insight into potential ways to deal with spread of conspiracy beliefs –
though we in no way mean for these to be taken as policy recommendations. Our
analysis revealed that the three most important factors in swaying any agent’s
belief were (1) whether or not an agent is exposed to a message, (2) number of
exposures, and (3) the difference between prior agent beliefs and those
expressed in the message. Even if (1) and (2) are met, as in some attempts to
debunk misinformation [5], (3) would prevent staunch conspiracy believers from
changing beliefs if exposed to a contradictory message. Some analyses attempt
to focus on the network structure [2, 15, 72] – i.e. (1) and (2) – without
acknowledging that individual psychology is just as important – as in (3). Our
model, which captures both network effects and individual effects, therefore
gives novel insights into a more holistic intervention. Our analysis of
results showed that for a highly homophilic network (a trait present in real
social networks), certain messages have a slim chance of reaching those with
certain beliefs. Any intervention would need to take this into account, and
imagine what types of messages would be most likely to reach certain
populations.

Moreover, on the individual level, an intervention under our model would have
to gradually nudge the agent away from an undesirable belief strength. This
brings _the individual_ into the debate over interventions. The only message
sets from our experiments that successfully swayed all agents in the
population were those which gradually eased agents from one belief polarity to
another. It should be noted that belief change tactics aimed at individuals’
psychology are already widely considered and used by private and public
institutions to manipulate target population beliefs, but not yet in the case
of domestic misinformation [73, 30, 74]. The ethics of these interventions –
often targeted at anti-radicalization, voter manipulation, or behavior change
for economic gain – are clearly fraught. This begs more analysis of the ethics
of intervention techniques, but such an endeavor is easily outside the scope
of this report.

### Limitations and Future Work

These results are in their early stages. While the motivation for models of
cognitive contagion exists, there are several steps that still must be taken
in order to flesh the model out further. For instance, in the social science
literature that backs misinformation and identity-based reasoning effects,
there is a great deal of evidence backing belief effects that rely on in/out-
group effects [57, 14, 21], trust in message sources [10], more nuanced belief
structures [75, 76, 77, 28], or effects of emotion on social contagion [29,
14]. These theories and findings could motivate more complex agent cognitive
models than either the simple sigmoid distance function, or the singular
proposition, we used in our model.

Further, a more rigorous application of the model to empirical population-
level belief changes would help in verifying the legitimacy of the model’s
results. Using both real network structures, such as snapshots of social media
networks, and real institutional message data, such as tweets or posts from
notable figures, would be steps forward for this goal. While ABMs have been
used to model spread of misinformation [15], social media messages [78, 72],
and value-laden topics [79], it appears that few have verified outcomes
against ground truth. Among other reasons, this is likely due to the fact that
such data seems difficult to obtain.

There is great promise for computational social scientific tools like ABM to
leverage computational power to tackle complex social problems previously
limited to thought experiments and small experiments – such as modeling
misinformation spread. However, if the promise is to be fulfilled, greater
pains must be taken to motivate and empirically ground both individual agent
models, and global network structures and population models. But this work is
a step towards establishing fruitful collaboration between the computational
modeling community, and social scientists, in order to tackle one of the
greatest political challenges of our time.

## Conclusion

This paper lays out two major contributions: a _cognitive contagion_ model for
identity-related belief spread, and a _Public Opinion Diffusion_ (POD) model
in which external, institutional agents (modeling media companies) dictate
influence of internal agent beliefs. The cognitive contagion model, by giving
each individual agent a cognitive model to direct belief update, allows a
level of expressiveness above existing simple and complex contagion models.
After proposing the cognitive contagion model, we compared potential contagion
functions to arrive at one capturing misinformation spread – what we called a
_Defensive Cognitive Contagion_ (DCC) function – which adequately captured the
cognitive dissonance and exposure effects referenced in empirical literature.
This allowed us to run simulation models of networked populations of agents
whose belief strength in a given proposition is influenced by an external
agent. Across several graph topologies, keeping the POD model consistent, we
compared simulation results for simple contagion, complex contagion, and our
DCC function. Analysis of these results revealed that our cognitive contagion
model is much less sensitive to graph topology than either other contagion
method. It showed that the crucial factor in belief change, was not only who
surrounds a given agent, but the content of any message and its relation to
the agent’s prior belief. We concluded by motivating potential interventions
to correct misinformation and conspiracy beliefs that address the individual
and the network holistically, rather that only the network they are embedded
in.

## Acknowledgements

We thank the Tufts Data Intensive Studies Center (DISC) for the seed grant
funding that supported this research. LC was additionally supported by NSF
grant 1934553.

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