\section*{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 \emph{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 \emph{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.''


\section*{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 \cite{bursztyn2020misinformation,del2016spreading,uscinski2020people,kouzy2020coronavirus,brennen2020types}, increased political polarization \cite{pewCOVIDConspiracyPlanned,clinton2020partisan,bakshy2015exposure,conover2011political}, and distrust in scientific findings \cite{swire2020public,pewMediaCovidExaggeration}. 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 \cite{uscinski2020people,bursztyn2020misinformation,swire2020public,bago2020fake,pennycook2019lazy,kouzy2020coronavirus,brennen2020types,van2020using}. 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~\cite{brainard2020agent,kopp2018information,ehsanfar2017incentivizing,del2016spreading,maghool2019coevolution}. 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~\cite{festinger1957theory,uscinski2020people,pennycook2019lazy,pennycook2020fighting,van2018partisan,van2020using,swire2020searching}. We introduce a new class of models, {\em cognitive contagion models,} that adopt a network-based contagion model from social network theory \cite{christakis2013social,centola2007complex}, 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 \cite{pewCovidNews,pewMediaCovidExaggeration,bursztyn2020misinformation} -- 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 \cite{pewTwentyFindings}. 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'' \cite{usaTodayCOVIDNotReal}.

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~\cite{porot2020science,brady2020mad,van2018partisan}. 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~\cite{wiley2019} has recently revealed that some polarized, partisan beliefs and conspiracies have been \emph{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 {\em public opinion:} the theoretical mechanisms by which populations come to certain beliefs, which are notoriously difficult to verify empirically ~\cite{lippmann1946public,bernays2005propaganda}. However, with the advent of social network research, scholars are moving toward that goal, studying how information cascades through groups \cite{kramer2014experimental,fowler2010cooperative,adamic2016information,del2016spreading,christakis2013social}, and the wide-scale adoption of certain beliefs \cite{kearney2020twitter}. 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 \emph{Agent-Based Models} (ABMs), called \emph{Agent Based Social System} (ABSS) \cite{pfeffer2017simulating} -- more specifically, \emph{social contagion} models \cite{centola2007complex}. ABM is a powerful modeling paradigm that has both successes and future potential in a variety of areas, including animal behavior \cite{reynolds1987flocks, scheutz2009artificial,scheutz2006swages,ferreira2018accidental,ferreira2018modeling}, social sciences \cite{schelling1971dynamic}, and, notably, opinion dynamics \cite{goffman1964generalization,centola2005emperor,centola2007complex,centola2007cascade}.

The applications of ABMs, even within the field of opinion dynamics, are very diverse \cite{zhang2019empirically}, but all ABMs must balance complexity of the agent and environment models, and strength of results \cite{jackson2017agent}: 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 \cite{bonabeau2002agent} (for a strong example of this principle, see Schelling's model of segregation \cite{schelling1971dynamic}). Amidst a vast literature of what Zhang \& Vorobeychik \cite{zhang2019empirically} -- 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 \cite{granovetter1978threshold,christakis2013social,centola2007complex}.

By combining the simplicity of the social contagion paradigm with some of the cognitive literature regarding misinformation, we propose our first major contribution: a \emph{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 \cite{festinger1957theory}, and exposure \cite{begg1992dissociation,swire2020public}, capturing what we call \emph{defensive cognitive contagion} (DCC).

We conduct an analysis of the cognitive contagion model within our second contribution, what we call a \emph{public opinion diffusion} (POD) model. We situate agents following cognitive contagion rules in a model that includes \emph{institutional} agents, addressing the crucial ontological fallacy that only individuals play a part in these systems \cite{epstein2012agent}. Media companies have played crucial roles in COVID misinformation \cite{bursztyn2020misinformation,pewMediaCovidExaggeration}, 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 \cite{pewCOVIDConspiracyPlanned,clinton2020partisan,bakshy2015exposure,conover2011political}, and hardly changed throughout 2020 \cite{pewTwentyFindings}. These preliminary results hint at possible interventions and offer plenty of opportunities for future studies to be conducted using these methods.

\section*{Background}





\subsection*{Simple Contagion}

ABMs have been widely used to model \emph{social contagion} effects -- those which describe the process of ideas or beliefs spreading through a population \cite{christakis2013social}. Such models attempt to explain possible processes underlying the spread of innovations \cite{zhang2019empirically,kiesling2012agent,ryan1943diffusion}, or unpopular norms \cite{centola2005emperor}. These models are extensions of earlier work in sociology that theorized how social network structure and simple decisions, such as the threshold effect \cite{granovetter1978threshold}, 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: \emph{simple} and \emph{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 \cite{goffman1964generalization,kramer2014experimental,fowler2010cooperative,christakis2013social}. 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:

\begin{equation}
    P(b_u = b_v\ |\ b_u) = p.
    \label{eq:simpleContagion}
\end{equation}

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. \ref{fig:simple-contagion}.

\begin{figure*}[h]
    \centering
    \includegraphics[width=\textwidth]{images/simple-contagion.png}
    \caption{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.}
    \label{fig:simple-contagion}
\end{figure*}

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:

\begin{equation}
    |N_i(u)| \geq \ceil{\frac{\log (1-\delta)}{\log (1-p)}}
    \label{eq:simpleContagionGuarantee}
\end{equation}

\noindent For example, if $p=0.5$ and $\delta=0.95$, then $|N_i(u)| \geq \ceil{4.32} \geq 5$ -- $u$ 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 \emph{appear} infectious \cite{christakis2013social}. There are other contagion models that have put forward alternative explanations.

\subsection*{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 \cite{centola2007complex}. 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 \cite{schelling1971dynamic}. Formally, given an ego $u$ and set of neighbors $N(u)$, the probability of adopting belief $b$ can be represented as:

\begin{equation}
    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},
    \label{eq:complexContagion}
\end{equation}

\noindent 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. \ref{fig:complex-contagion}.

\begin{figure*}[h]
    \centering
    \includegraphics[width=\textwidth]{images/complex-contagion.png}
    \caption{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.}
    \label{fig:complex-contagion}
\end{figure*}

It may seem tempting to imagine, given the simple contagion variation in Eq. \ref{eq:simpleContagionGuarantee}, 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 \emph{innate infectiousness} of any belief, but rather the infectiousness of \emph{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 \cite{centola2005emperor}. 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 \cite{guilbeault2018complex}.

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 \emph{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.

\section*{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. ~\cite{pewCOVIDConspiracyPlanned,pewCovidNews,pewMediaCovidExaggeration}, and justified by \cite{cox1980optimal}) 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~\cite{festinger1957theory}. 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:

\begin{equation}
    P(b_u = b_v\ |\ b_u) = \begin{cases}
        1, & |b_u - b_v| \leq \gamma,\\
        0, & otherwise
        \end{cases},
    \label{eq:cog-contagion-threshold}
\end{equation}

\noindent 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. (\ref{eq:complexContagion}).

\begin{figure*}[h]
    \centering
    \includegraphics[width=\textwidth]{images/cognitive-contagion.png}
    \caption{An illustration of cognitive contagion with the DSS contagion function described in Eq. \ref{eq:sigmoid-function}. (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.}
    \label{fig:cognitive-contagion}
\end{figure*}

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 \cite{pereira2018identity,bail2018exposure,porot2020science}. 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 \cite{festinger1957theory}.

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 \cite{begg1992dissociation,swire2020searching} and the mere-exposure effect \cite{zajonc1968attitudinal}. 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 \emph{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:

\begin{equation}
    P(b_u = b\ |\ b_u) = \beta(b_u, b).
    \label{eq:cognitiveContagion}
\end{equation}

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.

\section*{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 \emph{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 \cite{pewCovidNews,axiosSkepticsGrowing,mediaMattersSixPolls}. Often, ABMs do not model so-called ``levels'' of social systems: ontological distinctions in abstraction (e.g. those between individuals and institutions) \cite{epstein2012agent}. The POD model addresses this by including the \emph{institutional} level in the form of media  agents. A visual description of the model is included in Fig. \ref{fig:pod-model}.

\begin{figure*}[h]
    \centering
    \includegraphics[width=\textwidth]{images/model-diagram.png}
    \caption{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 $M_1(t=0)=(m_0,m_1)$. (\emph{i}) shows the initial sending of $m_0=4$ to subscribers, and (\emph{ii}) shows $s_1$ and $s_3$ believing the message and propagating it to their neighbors. (\emph{iii}) and (\emph{iv}) show the same for $m_1=3$, but only $s_3$ believes $m_1$.}
    \label{fig:pod-model}
\end{figure*}

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 \emph{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 \cite{NetLogo} and Python 3.5. Code is made available on GitHub \footnote{https://github.com/RickNabb/cognitive-contagion}.

\section*{Contagion Experiments}

\subsection*{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 \emph{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 \emph{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 \emph{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 $M_i(t) = (0)$.

\begin{figure*}[h]
    \centering
    \includegraphics[width=\textwidth]{images/message-sets.png}
    \caption{A visual depiction of the different message set conditions used in our experiments: \emph{single} (top), \emph{split} (middle), and \emph{gradual} (bottom), set against a 100-step simulation from $t=0$ to $t=100$.}
    \label{fig:message-sets}
\end{figure*}

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 \emph{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 \emph{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 \emph{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 \emph{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:

\begin{itemize}
    \item The Erd\H{o}s-R\'{e}nyi (ER) random graph \cite{erdHos1960evolution}
    \item The Watts-Strogatz (WS) small world network \cite{watts1998collective}
    \item The Barab\'{a}si-Albert (BA) preferential attachment network \cite{barabasi1999emergence}
    \item The Multiplicative Attribute Graph (MAG) \cite{kim2011modeling}
\end{itemize}

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.

\subsection*{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.

\subsubsection*{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:


\begin{equation}
    P(b_u = b\ |\ b_u) = \beta(b_u, b) = \frac{1}{\gamma+\alpha|b_u-b|}
    \label{eq:linear-function}
\end{equation}

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 Erd\H{o}s-R\'{e}nyi 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 \emph{split} message set, though results for others can be found in the supplemental materials (S3-S8). Results are displayed in Fig. \ref{fig:split-linear-function}.

\begin{figure*}[h]
    \centering
    \includegraphics[width=\textwidth]{images/split-linear-function.png}
    \caption{The \emph{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. (\ref{eq:linear-function}). 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 ($\gamma=10, \alpha=20$).}
    \label{fig:split-linear-function}
\end{figure*}

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.

\subsubsection*{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 \cite{grim2020computational,li2020opinion,hegselmann2002opinion}. We anticipate that these functions will capture the desired dissonance effect. The function can be parameterized as we already motivated in Eq. (\ref{eq:cog-contagion-threshold}), 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. \ref{fig:split-threshold-function}.

\begin{figure*}[h]
    \centering
    \includegraphics[width=\textwidth]{images/split-threshold-function.png}
    \caption{The \emph{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. (\ref{eq:cog-contagion-threshold}). 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 ($\gamma=1$).}
    \label{fig:split-threshold-function}
\end{figure*}

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.

\subsubsection*{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 \cite{ding2018activation}. This property is useful for our purposes as well. We formulate our sigmoid cognitive contagion function as follows:

\begin{equation}
    \beta(b_u, b) = \frac{1}{1+e^{\alpha(|b_u-b|-\gamma)}}
    \label{eq:sigmoid-function}
\end{equation}

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. \ref{fig:split-sigmoid-function}.

\begin{figure*}[h]
    \centering
    \includegraphics[width=\textwidth]{images/split-sigmoid-function.png}
    \caption{The \emph{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. (\ref{eq:sigmoid-function}). 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 ($\alpha=4, \gamma=2$).}
    \label{fig:split-sigmoid-function}
\end{figure*}

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 \emph{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.

\section*{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 Erd\H{o}s-R\'{e}nyi (ER) random graph \cite{erdHos1960evolution}, the Watts-Strogatz (WS) small world network \cite{watts1998collective}, the Barab\'{a}si-Albert (BA) preferential attachment network \cite{barabasi1999emergence}, and the Multiplicative Attribute Graph (MAG) \cite{kim2011modeling}. 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.

\subsection*{Contagion on ER Random Networks}

We will begin with a contagion on an Erd\H{o}s-R\'{e}nyi \cite{erdHos1960evolution} 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. \ref{fig:er-simple-complex}.

\begin{figure*}[h]
    \centering
    \includegraphics[width=\textwidth]{images/er-simple-complex.png}
    \caption{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.}
    \label{fig:er-simple-complex}
\end{figure*}

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. \ref{fig:er-cognitive}, both the \emph{single} and \emph{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 \emph{split} condition. The \emph{gradual} message set is the only that was able to sway all agents over to $b=0$.

\begin{figure*}[h]
    \centering
    \includegraphics[width=\textwidth]{images/er-cognitive.png}
    \caption{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.}
    \label{fig:er-cognitive}
\end{figure*}

\subsection*{Contagion on WS Small World Networks}

Our second set of experiments were conducted on Watts-Strogatz \cite{watts1998collective} 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. \ref{fig:ws-simple-complex}.

\begin{figure*}[h]
    \centering
    \includegraphics[width=\textwidth]{images/ws-simple-complex.png}
    \caption{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.}
    \label{fig:ws-simple-complex}
\end{figure*}

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. \ref{fig:ws-cognitive}. The population displayed the same patterns given a significantly different graph structure, which begs explanation. We will discuss this further below.

\begin{figure*}[h]
    \centering
    \includegraphics[width=\textwidth]{images/ws-cognitive.png}
    \caption{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.}
    \label{fig:ws-cognitive}
\end{figure*}

\subsection*{Contagion on BA Preferential Attachment Networks}

We continued our experiments by testing Barab\'{a}si-Albert \cite{barabasi1999emergence} 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. \ref{fig:ba-simple-complex}. 

\begin{figure*}[h]
    \centering
    \includegraphics[width=\textwidth]{images/ba-simple-complex.png}
    \caption{Simple (top row) and complex (bottom row) contagion on a Barab\'{a}si-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.}
    \label{fig:ba-simple-complex}
\end{figure*}

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 \cite{ebrahimi2017complex}. Also interestingly, complex contagion appears to have slight effects on these graphs, with the \emph{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. \ref{fig:ba-cognitive}.

\begin{figure*}[h]
    \centering
    \includegraphics[width=\textwidth]{images/ba-cognitive.png}
    \caption{DCC on a Barab\'{a}si-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.}
    \label{fig:ba-cognitive}
\end{figure*}

\subsection*{Contagion on MAG Networks}

Finally, we tested contagion on the Multiplicative Attribute Graph \cite{kim2011modeling} 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:

\begin{align}
    \Theta_b &= (\theta_{ij}) \in \mathbb{R}^{m \times n} = \frac{1}{1+50(j-i)^2}\\
    &= \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}
    \label{eq:mag-theta}
\end{align}

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. (\ref{eq:homophilyMeasure}):

\begin{equation}
    h(G = (V,E)) = \frac{\sum\limits_{v \in V} \sum\limits_{u \in N(v)} |b_u - b_v|}{2|V|^2},
    \label{eq:homophilyMeasure}
\end{equation}

\noindent 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. \ref{fig:mag-simple-complex}.

\begin{figure*}[h]
    \centering
    \includegraphics[width=\textwidth]{images/mag-simple-complex.png}
    \caption{Simple (top row) and complex (bottom row) contagion on a homophilic MAG network with $N=500$, and $\Theta_b$ detailed in Eq. (\ref{eq:mag-theta}). Graphs show the percent of agents who believe $B$ with strength $b$ over time.}
    \label{fig:mag-simple-complex}
\end{figure*}


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. \ref{fig:mag-cognitive}. 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.

\begin{figure*}[h]
    \centering
    \includegraphics[width=\textwidth]{images/mag-cognitive.png}
    \caption{DCC on a homophilic MAG network with $N=500$, and $\Theta_b$ detailed in Eq. (\ref{eq:mag-theta}). Graphs show the percent of agents who believe $B$ with strength $b$ over time.}
    \label{fig:mag-cognitive}
\end{figure*}

\section*{Analysis of Results}

Results across different graph toplogies further support our motivations for introducing cognitive contagion models. Across Erd\H{o}s-R\'{e}nyi random graphs \cite{erdHos1960evolution}, Watts-Strogatz small world networks \cite{watts1998collective}, Barab\'{a}si-Albert preferential attachment networks \cite{barabasi1999emergence}, and highly homophilic Multiplicative Attribute Graphs \cite{kim2011modeling}, 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:

\begin{enumerate}[label=(\arabic*)]
    \item Whether or not any given agent is exposed to some message;
    \item How many times an agent is exposed to similar messages; and
    \item The difference between agent beliefs and that message.
\end{enumerate}

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 \cite{del2016spreading,van2018partisan,porot2020science,bail2018exposure}. 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 \cite{begg1992dissociation,swire2020searching} and mere-exposure effects \cite{zajonc1968attitudinal}.

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,

\begin{enumerate}
    \item[$(1)$] becomes the probability of $m_i$ being received and believed by any neighbor $N(u)$ of $u$;
\end{enumerate}
    
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:

\begin{equation}
    \mathcal{P}(m_j, w) = \prod\limits_{v \in w}{\beta(v, m_j)}.
    \label{eq:analysis-path}
\end{equation}

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,

\begin{enumerate}
    \item[$(2)$] becomes $|N_\beta(i,u,m_j)|$, the number of neighbors of $u$ who are likely to believe $m_j$ coming from $i$.
\end{enumerate}

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:

\begin{equation}
    N_\beta(i,u,m_j) = \{\ v \ |\ v \in N(u) \land \mathcal{P}(w_{iu},m_j) \geq 1 - \delta\ \},
    \label{eq:analysis-believing-neighbors}
\end{equation}

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:


\begin{equation}
    \mathcal{W}^*_{iu} = \max_{w}\prod{\mathcal{P}(m_j,w)}\ \{\ w_{iu}\ |\ \bigcup_{w_{iu}}(\cdot) = \O \ \}
    \label{eq:analysis-non-overlapping-paths}
\end{equation}

\noindent 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 \ref{table:probabilities}, 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.

\begin{table}[h]
    \begin{center}
        \begin{tabular}{ l | c | c | c | c | c | c | c }
        \textbf{$b_u/b_v$} & 0 & 1 & 2 & 3 & 4 & 5 & 6\\
        \hline
        0 & 0.999 & 0.982 & 0.500 & 0.018 & $<$0.001 & $<$0.001 & $<$0.001\\
        \hline
        1 & 0.982 & 0.999 & 0.982 & 0.500 & 0.018 & $<$0.001 & $<$0.001\\
        \hline
        2 & 0.500 & 0.982 & 0.999 & 0.982 & 0.500 & 0.018 & $<$0.001\\
        \hline
        3 & 0.018 & 0.500 & 0.982 & 0.999 & 0.982 & 0.500 & 0.018\\
        \hline
        4 & $<$0.001 & 0.018 & 0.500 & 0.982 & 0.999 & 0.982 & 0.500\\
        \hline
        5 & $<$0.001 & $<$0.001 & 0.018 & 0.500 & 0.982 & 0.999 & 0.982\\
        \hline
        6 & $<$0.001 & $<$0.001 & $<$0.001 & 0.018 & 0.500 & 0.982 & 0.999\\
        \hline
        \end{tabular}
    \end{center}
    \caption{Probabilities given by $\beta(b_u,b_v)$ for the DCC function, described in Eq. (\ref{eq:cognitiveContagion}).}
    \label{table:probabilities}
\end{table}

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 \ref{table:probabilities} 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 \emph{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.

\begin{table}[h]
    \begin{center}
        \begin{tabular}{ l | c | c | c | c | c | c | c }
        ($\tau=1$) & $b_u=0$ & $b_u=1$ & $b_u=2$ & $b_u=3$ & $b_u=4$ & $b_u=5$ & $b_u=6$ \\
        \hline
        ER & 0.91 & 0.9 & 0.97 & 0.95 & 1.0 & 1.0 & 1.0\\
        \hline
        WS & 0.51 & 0.56 & 0.59 & 0.7 & 0.64 & 0.62 & 1.0\\
        \hline
        BA & 0.73 & 0.66 & 0.66 & 0.75 & 0.63 & 0.75 & 1.0\\
        \hline
        MAG & 0.13 & 0.2 & 0.37 & 0.54 & 0.8 & 1.0 & 1.0\\
        \hline
        \hline
        ($\tau=2$) & $b_u=0$ & $b_u=1$ & $b_u=2$ & $b_u=3$ & $b_u=4$ & $b_u=5$ & $b_u=6$ \\
        \hline
        ER & 0.98 & 0.99 & 0.97 & 1.0 & 1.0 & 1.0 & 1.0\\
        \hline
        WS & 0.65 & 0.8 & 0.76 & 0.72 & 0.76 & 0.81 & 1.0\\
        \hline
        BA & 0.84 & 0.89 & 0.84 & 0.82 & 0.89 & 0.88 & 1.0\\
        \hline
        MAG & 0.29 & 0.29 & 0.45 & 0.84 & 0.99 & 1.0 & 1.0\\
        \end{tabular}
    \end{center}
    \caption{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 $b_{mj}=6$.}
    \label{table:paths}
\end{table}

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 \emph{all} belief strengths. Particularly for Erd\H{o}s-R\'{e}nyi random graphs, both path types are almost always present. Both Watts-Strogatz and Barab\'{a}si-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 \ref{table:probabilities}, 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.

\section*{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 \emph{single} and \emph{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 \emph{single} and \emph{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 \cite{porot2020science,swire2020searching}. The only message set condition that was able to sway the entire population in the cognitive contagion condition was the \emph{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 \cite{pewCOVIDConspiracyPlanned,clinton2020partisan,bakshy2015exposure,conover2011political,pewTwentyFindings}, 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 \cite{brennen2020types}, (3) would prevent staunch conspiracy believers from changing beliefs if exposed to a contradictory message. Some analyses attempt to focus on the network structure \cite{del2016spreading,brainard2020agent,ross2019social} -- 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 \emph{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 \cite{matz2017psychological,wiley2019,thaler2009nudge}. 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.

\subsection*{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 \cite{pereira2018identity,van2020using,van2018partisan}, trust in message sources \cite{swire2020public}, more nuanced belief structures \cite{jost2003political,jost2009political,jost2012political,porot2020science}, or effects of emotion on social contagion \cite{brady2020mad,van2020using}. 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 \cite{brainard2020agent}, social media messages \cite{nasrinpour2016agent,ross2019social}, and value-laden topics \cite{shugars2020good}, 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.

\section*{Conclusion}


This paper lays out two major contributions: a \emph{cognitive contagion} model for identity-related belief spread, and a \emph{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 \emph{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.

\section*{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. 


