# A primer on data-driven modeling of complex social systems

Alexandria Volkening  Department of Mathematics, Purdue University, West
Lafayette, IN 47907 USA<EMAIL_ADDRESS>
###### Abstract.

Traffic jams on roadways, echo chambers on social media, crowds of moving
pedestrians, and opinion dynamics during elections are all complex social
systems. These applications may seem disparate, but some of the questions that
they motivate are similar from a mathematical perspective. Across these
examples, researchers seek to uncover how individual agents—whether drivers,
Twitter accounts, pedestrians, or voters—are interacting. By better
understanding these interactions, mathematical modelers can make predictions
about the group-level features that will emerge when agents alter their
behavior. In this tutorial, which is based on the lecture that I gave at the
2021 American Mathematical Society Short Course, I introduce some of the
terms, methods, and choices that arise when building such data-driven models.
I discuss the differences between models that are statistical or mathematical,
static or dynamic, spatial or non-spatial, discrete or continuous, and
phenomenological or mechanistic. For concreteness, I also describe models of
two complex systems, election dynamics and pedestrian-crowd movement, in more
detail. With a conceptual approach, I broadly highlight some of the challenges
that arise when building and calibrating models, choosing complexity, and
working with quantitative and qualitative data.

###### Key words and phrases:

Complex social systems, complex systems, mathematical modeling, data-driven
modeling, election forecasting, pedestrian movement

In putting together my lecture for the 2021 AMS Short Course, I acknowledge
Heather Zinn Brooks, Jonathan Desponds, Simon Freedman, Brian Hsu, Kara Maki,
Niall Mangan, and Bridget Torsey for helpful examples in their earlier talks
or pointers to references. Special thanks to Jeffrey Humpherys, Rachel Levy,
and Thomas Witelski for their minitutorial [HLW16] on modeling courses at the
2016 SIAM Annual Meeting; I drew from their minitutorial for several of the
concepts in Section 3. Thanks to my Short Course co-organizers Heather, Mason,
and Michelle for their encouragement and for being a terrific team, and to
Mason for introducing me to the term “data-driven modeling of complex systems”
in the first place.

> _A complex system might be defined as a system
>  for which no single model is appropriate._
>  and

> _As Picasso said of art, a good model
>  “is a lie that helps us see the truth.”_
>  (Lee A. Segel and Leah Edelstein-Keshet [SEK13])

##  1\. Introduction

Traffic jams on roads [SFK+08, SCDM+18, BD11, JHZ+14, BHN+95], pedestrian
crowds [BCD18, HM95], swarming locusts [AA15, BCME+20], animal aggregations
[DAB+20, CKJ+02, PEK99, LLEK10, BCC+08, KTI+11], collections of cells [BEK20,
Vol20b, GBKM20, GG93], and echo chambers [SCP+21, EF18, CDFMG+21, CFPSS19] are
examples of complex systems. In each of these cases, rich, group-level
dynamics emerge from the interactions of smaller components—e.g., drivers,
people, locusts, animals, or cells—with one another and with their environment
[DBC+19, Bro22]. The interdisciplinary field of _complex systems_ [New11]
centers on the questions that arise from these emergent dynamics.
Complementing experimental approaches to complex systems, mathematicians
develop methods in dynamical systems, topology, network science, numerical
analysis, probability, partial differential equations, and many other areas.
Here I focus on data-driven mathematical modeling, mainly for complex social
systems. My goal for this tutorial chapter is to help provide a starting point
for folks who are new to this area, and I reflect on some broad questions and
choices that emerge when combining models and data. Figure 1 highlights
several complex social systems, ranging from traffic flow [SFK+08, BD11, NS92]
to Brexit voter dynamics [SHP16]; I also recommend the supplementary material
of [SSS17, SFK+08] and the websites [Loc, PMS17] for related animations.
Across these applications, one interesting feature is the common challenges
that they raise from a modeling perspective. For example, in each of the
images in Figure 1, a researcher may want to characterize alignment. This can
be physical alignment, with pedestrians, locusts, or drivers adjusting how
they move in response to other individuals or obstacles in their environment.
A different type of alignment is present in Figure 1(d)–(e): people are
forming opinions and may be influenced to align with (or against) the beliefs
of others. Another thread in complex systems is heterogeneity [MP07]: each
person, animal, or social-media account in Figure 1 is unique. Guided by the
data available, each modeler must choose how much detail to include. Should we
model voters as having a binary opinion (e.g., “for Brexit” or “against
Brexit”) or allow opinions to live on a spectrum? Changes in behavior are also
present: for example, in evacuation conditions, an emotional contagion can
propagate through a crowd, changing how pedestrians act [BDM+09, BHK+11,
TFB+11, BRSW15].

Figure 1. Examples of complex social systems. (a) Lanes can emerge from the
interactions of pedestrians moving to the right (in black) and left (in red)
in a corridor [SSS17, ZKSS12]; see Section 5.2. (b) Locusts form bands as they
move over the ground, destroying crops [Cre16]. (c) Drivers react to one
another and external signals. In an experiment on a circular road [SFK+08],
Sugiyama _et al_. instructed originally equidistant drivers to drive normally.
Despite the lack of external signals, a phantom traffic jam formed. This jam
or pulse of high density traveled backward relative to the direction that the
cars moved; see the supplementary material of [SFK+08] for an animation. (d)
Election outcomes [SHP16, MBN+16] and (e) echo chambers [CFPSS19] may emerge
from conversations, news coverage, interactions on social media, or other
factors. Image (a) adapted (cropped) from [SSS17] and licensed under CC-BY 4.0
(https://creativecommons.org/licenses/by/4.0/); image (b) reproduced from
[Cre16] with permission from Elsevier, Copyright (2016) Elsevier Inc.; image
(c) reproduced from [epS11] and licensed under CC-BY 2.0
(https://creativecommons.org/licenses/by/2.0/); image (d) reproduced from
[MBN+16] and licensed under CC BY-SA 3.0
(https://creativecommons.org/licenses/by-sa/3.0/deed.en); image (e) reproduced
from [CFPSS19] and licensed under CC-BY 4.0
(http://creativecommons.org/licenses/by/4.0/); I added the boxes and cartoons
with detail in (c) and (e).

Higher-order interactions are widespread in complex systems: peer influence
and social reinforcement from multiple friends may cause someone to change
their opinion or adopt a new technology, when an isolated or pairwise
interaction might not [OT12, BR06, IPBL19, GBC18, Sch73]. In a related vein,
the presence of short- and long-range interactions in complex systems leads to
rich dynamics. In Figure 1(c), drivers are interacting locally, basing their
acceleration on the cars near them. The addition of autonomous vehicles allows
for long-range dynamics. Stern _et al._ [SCDM+18] have shown that judiciously
modulating the speed of one autonomous vehicle can result in the disruption of
phantom traffic jams and improved fuel usage in some experiments. (Phantom
traffic jams are jams that appear to emerge from drivers, rather than through
external forces [JHZ+14, SCDM+18].) Modeling complex social systems stems from
and leads to questions that are of societal and mathematical interest. From an
applied perspective, in the case of traffic flow, we might want to shed light
on what driver behaviors cause jams or suggest how to use external
controls—e.g., time-dependent gating at ramps—to improve traffic. Models can
also provide insight into how echo chambers form or suggest interventions to
help dissipate divisions. These goals fall into the framework of seeking to
understand normal and altered agent interactions, and to predict resulting
group-level features. From a mathematical perspective, modeling complex
systems can be a starting point to drive the development of new methods and
inspire researchers to combine subfields in novel ways. Motivated by the
breadth of complex social systems, my tutorial lecture “Data-driven modeling”
kicked off the 2021 American Mathematical Society (AMS) Short Course on
Mathematical and Computational Methods for Complex Social Systems, and this
chapter is an offshoot of that talk. Many of the figures in this tutorial are
related to slides in my presentation; these slides and my talk recording are
available at [Vol21]. Following the structure of my Short Course presentation,
this chapter has three main parts and takes a conceptual approach throughout.
First, in Section 2, I highlight some resources, including those that I drew
on when preparing my lecture. Second, in Sections 3–4, I overview mathematical
modeling and discuss some of the approaches, challenges, and choices that can
arise when working with data. Third, in Section 5, I discuss two case
studies—election forecasting and pedestrian movement—in more depth.
Mathematical modeling is a big field, and data-driven modeling can be defined
in different ways. The array of approaches that modelers can choose from is a
strength, since different perspectives contribute in complementary ways to our
understanding of complex systems. As a central theme, I want to acknowledge
these choices and use the quotations from Segel and Edelstein-Keshet [SEK13]
at the start of this chapter as a guide. The abundance of modeling approaches
to complex systems, coupled with their multidisciplinary nature, also means
that communication is more challenging; researchers may not mean the same
thing when they say the same term. With this in mind, I discuss some of the
things that I—from my perspective as an applied mathematician and math
biologist—consider when I think about modeling complex systems. There are
many, many perspectives on modeling, and this tutorial represents one,
informed by the references herein.

##  2\. Some Resources on Modeling

I point out some resources below, including the materials that I drew on for
my Short Course lecture [Vol21].

###  2.1. Free Online Resources

The websites [Bro22, DBC+19] provide dynamic examples of research in complex
systems and are an excellent place to gain intuition and explore this field.
The Society for Industrial and Applied Mathematics (SIAM) hosts two modeling
handbooks [BKGL18, BFG14]; and SIAM and the Consortium for Mathematics and its
Applications provide guidelines on teaching mathematical modeling [GAI19].
Humpherys, Levy, and Witelski organized a very useful minitutorial discussing
graduate and undergraduate education in modeling at the 2016 SIAM Annual
Meeting; both their slides and a recording of their presentation are available
online [HLW16]. Kutz and Brunton have posted a rich collection of videos [Bru,
Kut] on YouTube, discussing topics including data-driven model discovery. For
a demonstration of how to go from a biological paper to making simplifications
to building different models, my tutorial lecture [Vol20a] for a broad
audience may be of interest. Also geared toward a biological audience, the
course “What do Your Data Say?” [MJ] includes a large collection of video
lectures with a statistical, data-driven perspective. To see examples of
research talks related to modeling complex systems, I highlight some of the
BIRS workshop videos [BIR] (this collection from the University of British
Columbia library contains a wider selection of topics than just modeling), as
well as videos in the virtual SIAM Data Science minisymposium “Topological
Techniques and Data-Driven Modeling in Complex Systems” organized by Brooks
and Porter [BP20b].

###  2.2. Books

I found the books [KBBP16, Kut13, BK19] to be especially helpful as I
developed my Short Course lecture, and the book [SEK13] by Segel and
Edelstein-Keshet provides the quotations that open this chapter. Additional
books related to complex systems and modeling include [Mit09, THK18, Boc10,
MP07].

###  2.3. Publicly Available Data

Accessing data can be a challenge in complex-systems research. As a starting
point, I highlight some publicly available data for a few specific
applications here. For studies on elections and political opinions in the
United States, I recommend the breadth of polling data aggregated by
FiveThirtyEight [BBG+22]. HuffPost Pollster also curates a broad collection of
public polls, with a search bar for finding data [Huf22a, Huf22b]. At a finer
scale, the 2016 presidential election results in California are available at
the precinct level from the _Los Angeles Times_ [SFK16]. Ciocanel, Topaz, and
other researchers through the Institute for the Quantitative Study of
Inclusion, Diversity, and Equity (QSIDE) [QSI] developed a large-scale
database (called JUSTFAIR, for Judicial System Transparency through Federal
Archive Inferred Records) holding over $500,000$ federal district court
records [CTS+20]. Data from the social-media platform Twitter, as well as
tutorials, are available from sources including [AAA+21, Sto, KS20, Twi].

##  3\. Some Perspectives on Data and Models

Because terminology can vary across fields, I survey some terms for describing
models (Section 3.1) and data (Section 3.2), and then define data-driven
modeling for the purposes of this chapter (Section 3.3). If you are coming to
this tutorial with an applied question that you want to address, I encourage
you to keep your complex system in mind as you read—what are the parameters in
your system, what data could you use to constrain your model, and at what
scale do you want to make predictions or describe the system? If you are a
mathematician new to modeling, what mathematical challenges does thinking from
the perspective of complex systems raise? If you are from a different
disciplinary background than mine, how does what we mean by “data-driven
modeling” differ from and complement each other? And, if you happen to be a
modeler who—like me—was introduced to modeling through research, it might be
interesting to reflect on how we teach modeling.

###  3.1. Types of Models

The term “model” means different things in different fields. In the life
sciences, “model” may refer to a model organism (e.g., zebrafish, fruit flies,
or worms) [HLW16] or a schematic hypothesizing the relationship between
things. In mathematics, we may think of models that take the form of
differential equations or stochastic rules, for example. Mathematical models
are described using many terms, and I include a few in Figure 2. Figure 2 also
highlights some of the initial choices that modelers face, often constrained
by their data. Importantly, the distinctions in Figure 2 are not sharp: models
often fall on a spectrum and this can depend heavily on the perspective that
one takes.

Figure 2. Example modeling perspectives [Vol21]. Many models fall somewhere in
the middle of each of these scales. For example, a model may have both
stochastic components (e.g., stochastic rules for when new pedestrians enter
the corridor in Figure 1(a)) and deterministic components (e.g., differential
equations for pedestrian movement). Models can be discrete or continuous in
many ways: they can be discrete in terms of types of opinions (e.g.,
Republican or Democratic voting opinions in the United States), physical
space, or time, for example. The distinction between phenomenological and
mechanistic models is difficult, and folks have different opinions on what
this means, as I discuss in Section 3.1.

Models can be described as deterministic or stochastic; stochastic models
include variability. Depending on their goals and data, researchers must
choose whether to build models that are static (time-independent) or dynamic.
Similarly, scientists are faced with the choice of building models that are
spatial or non-spatial. Do we need to understand where individual cars are
located on a road, or is it sufficient to know how the number of cars evolves?
Multiscale approaches are also possible, and I provide an example for the case
of pedestrian movement in Section 5.2. We can think of models as being
discrete or continuous in time or in space (e.g., so-called “on-lattice” or
“off-lattice” microscopic models; see Section 5.2), but models can also be
discrete in terms of types of agents; for example, do we assume voters live on
an ideological spectrum or assign them a binary opinion? Whereas the choice of
making a model discrete or continuous in space and time is often a choice of
mathematical and computational implementation, the choice of modeling agents
as having discrete or continuous features can be particularly meaningful from
the perspective of the application. Understanding how choices of
implementation impact model predictions is an important area of research
(e.g., [KBF17]), as is uncovering how different modeling approaches—such as
microscopic and macroscopic (see Section 5.2)—are related (e.g., [BT11, CP21,
BV05]). Some researchers distinguish between mathematical and statistical
models, and others see statistical models as a type of mathematical model. A
related categorization is phenomenological or mechanistic. These are difficult
distinctions, and, in my opinion, scientists use the terms “phenomenological”
and “mechanistic” in different ways. Mechanistic models of complex systems get
at the mechanism underlying agent behavior. For example, the drivers in Figure
1(c) want to avoid running into one another, and we could model this by
specifying repulsive forces between cars. This model can be seen as
phenomenological since it describes the affect (e.g., drivers avoid one
another) without getting at the mechanism of how the repulsion occurs. If we
modeled the physics of the vehicles, the vision cone of individual drivers,
and each driver’s internal decision process, this would be more mechanistic.
However, what are the variables in a model of how people make decisions? This
is in some sense a phenomenological model as well, raising further questions
that involve neuroscience. I suggest that the meanings of “mechanistic” and
“phenomenological” depend on the question that we want to answer and the
perspective from which we are studying an application. In my opinion, many
models are mechanistic at one scale, and phenomenological as soon as we step
deeper into the complex system.

###  3.2. Types of Data

The methods that modelers use to build predictive models that balance model
and data complexity look different depending on the form of their data.
However, the core concepts are the same when building and validating data-
driven models if we look more closely, and, for this reason, I overview some
types of data here. For example, data may be quantitative (e.g., the speed of
the $i$th car in Figure 1(c)) or qualitative (e.g., the presence of lanes
emerging from pedestrian behavior in Figure 1(a)). See Section 5.1 for an
example of modeling with quantitative data, and Section 5.2 for a discussion
of the challenges that qualitative data introduce to the modeling process.
Textual data also emerge from many complex social systems (e.g., [AAA+21,
MCM+22]). Sometimes we find ourselves with so much data that we cannot open
the files, and other times there is nearly no data. In the first case, the
“black-box” modeling approaches that I discuss in Section 3.3 may be useful;
for example, if we are working with a huge set of tweets, we could complete
some data analysis to identify meaningful categories of accounts. If we are
working with large sets of qualitative data (e.g., many images), this may
motivate the development of new computational and mathematical approaches for
extracting quantitative information from our data. On the other hand, if we
have nearly no data, it can be challenging to know where to start. In this
case, it is a matter of making many simplifications (and being actively aware
of the choices that we make in this process), so that the number of
assumptions that we build into our model is balanced with the small amount of
data available. On a related note to amount, data for some complex systems
describe rare events. For example, a model may be fit to measurements of
average traffic flow, but how do we account for events that are relative
outliers, like car accidents? In the case of election forecasting, we might
judge a model as wrong if, despite giving Candidate $A$ a $75$% chance of
winning and Candidate $B$ a $25$% chance of winning, Candidate $B$ wins. The
reality is that we do not have enough information to determine whether the
model is good or bad. Forecasts are more meaningfully judged in aggregate
across many elections, but limited polling data are available. Like models,
data can also be time-independent (e.g., a Twitter followership network at one
snapshot in time) or dynamic (e.g., the timeline of tweets from a given
account) and spatial or non-spatial. The initial form of data is often messy,
and in some cases a large portion of the time that researchers spend modeling
complex systems is focused on cleaning [BKGL18], gathering, and tracking down
the oddities in their data. All forms of data can have bias and require human
choices, particularly in the case of complex social systems. I point the
reader to the chapter [Por22] by Porter and references therein in this volume
for a discussion of data ethics. Importantly, just because data exist does not
mean they should be used, and as the author mentions in [Por22], determining
when to use or not use data is a critical step in research on complex social
systems. Modelers need to be actively aware of the choices that they make when
handling data, and of the presence of any choices made prior to the time that
they gained access to the data. For example, if we are interested in
understanding the online conversation about a recent event, we might start by
downloading a large set of tweets using hashtags associated with that event.
There are multiple choices wrapped up in this process, and I name a few here
[CY16, MPLC14, Tuf14, TECP20]. First, we chose one of many social-media
platforms, so our analysis will be specific to the groups that use Twitter
[Tuf14]. Second, we had to select what hashtags to search for and how we would
identify tweets “associated with” our recent event [MPLC14, TECP20]. Third,
while the Twitter API provides a rich sample of tweets, it is not fully clear
how this selection is made [MPLC14]. All of these choices will affect the
results of our model.

###  3.3. Perspectives on Modeling with Data

Figure 3. Shades of modeling with data [HLW16]. Black-box, gray-box, and
white-box models depend on data to varying degrees and have different
relationships with parameters. Black-box modeling approaches rely on data and
often have internal parameters, while white-box models are largely dictated by
first principles and have measurable parameters (e.g., conductivity of a
material). Gray-box modeling involves visible, interpretable parameters that
are fit, specified, or measured using data. All of these approaches require
domain expertise. As some examples, I highlight principal component analysis
(PCA) applied to media followership on Twitter [TECP20], and recognizing
handwritten numbers [LBBH98] (black-box modeling); deterministic models of
traffic flow and game-theoretic models of opinion dynamics on networks [EF18]
(gray-box modeling); and fluid flow past a sphere (white-box modeling).
PCA–Twitter image and opinion-network images reproduced from [TECP20] and
[EF18], respectively, and licensed under CC-BY 4.0
(https://creativecommons.org/licenses/by/4.0/); image-classification image
reproduced from [LBBH98] with permission, Copyright (1998) IEEE; traffic image
reproduced from [epS11] and licensed under CC-BY 2.0
(https://creativecommons.org/licenses/by/2.0/).

In their 2016 SIAM Annual Meeting minitutorial [HLW16], Humpherys, Levy, and
Witelski discussed a useful classification of models based on “shades of model
uncertainty”. As I highlight in Figure 3, black-box, gray-box, and white-box
models have different levels of dependency on data [HLW16], and their
parameters mean different things. According to the classification system in
[HLW16], black-box models are based heavily on data and can be thought of as
maps between inputs and outputs; these models include regression,
classification, and machine learning. For example, Tien _et al._ applied
[TECP20] principal component analysis to Twitter data (the input) to
distinguish groups of accounts (the output) based on their media followership.
The parameters in black-box models may be internal or hidden, and it is the
model output—rather than the model structure—that is often of most interest.
On the other hand, white-box models are based on first principles; these
include equations from physics, such as those describing fluid dynamics or
optics [HLW16]. The parameters in white-box models are measurable, and
examples are viscosity and conductivity. Gray-box models depend on a
combination of data, first principles, and domain expertise. For example, an
ordinary differential equation (ODE) model for driver movement could include
equations for velocity and acceleration that are based on phenomenological
descriptions of repulsion and attraction between cars (i.e., domain expertise)
and measurements of speeds (i.e., data). The distinctions in Figure 3, like
the distinctions in Figure 2, are not perfect. For example, equation-learning
and model-selection approaches (e.g., [MKBP17, BPK16, NBSF21, KBT+22, MBPK16])
might be thought of as “dark gray”. It is also important to keep in mind that
there are choices present and domain expertise needed across the spectrum in
Figure 3. This is especially true when working with data from complex social
systems, since even the data that are selected for training black-box models
rely on a modeler’s choice to use those data [Por22]. For the purposes of this
tutorial, I thus think of data-driven modeling as being mathematical modeling
that is driven by data, motivated by a given question, and combined with
domain expertise. This encompasses developing predictive, mechanistic models
based on data; equation learning and model selection111Equation learning and
model selection—sometimes referred to as “data-driven modeling”—are outside
the scope of this survey. See, for example, [MKBP17, BPK16, NBSF21, KBT+22,
MBPK16] for more discussion of these topics.; machine learning, regression, or
classification to understand data; and using models to raise questions and
drive further data collection. Both black-box and gray-box models fit this
description, but I predominantly focus on gray-box models in this survey,
though again I stress that the distinctions are not sharp.

##  4\. Challenges, Choices, and Creativity in Data-Driven Modeling

Data-driven modeling involves creativity and choices, informed by the
modeler’s driving question, data, and domain expertise. In Section 4.1, I
provide an example modeling process and highlight some of the places where
modelers make choices. In Section 4.2, I then discuss challenges related to
data and model calibration. See Sections 5.1 and 5.2 for illustrations of
these topics for two specific applications. I take a conceptual approach
throughout.

###  4.1. Building Data-Driven Mathematical Models

As an example data-driven, gray-box modeling process, we might follow the
steps below [GAI19, BFG14, BKGL18]:

  1. (1)

formulate our broad motivation and specific goals

     * •

get to know the application area or talk to domain experts

     * •

search for data (qualitative or quantitative) and prior work

     * •

identify hypotheses to be tested or proposed and questions to be “answered” or
raised

  2. (2)

come up with a plan for building and evaluating our model

     * •

determine baseline assumptions and simplify where possible

     * •

identify our variables, parameter names, timescales, and units

     * •

specify the values of measurable parameters and determine what parameters need
to be fit

     * •

handle formatting, cleaning, and quantifying our data as needed

     * •

break our data into sets for fitting parameters, testing, and predicting

  3. (3)

simulate, analyze, and use our model

     * •

identify remaining parameter values using data for fitting

     * •

validate our model on test data

     * •

perform a sensitivity analysis or bifurcation analysis if possible

     * •

use our model to gain intuition, raise questions, and make predictions

     * •

communicate results to an interdisciplinary audience

     * •

iterate to improve

These steps are not necessarily linear and data-driven modeling is iterative
[GAI19, BFG14, BKGL18]. The starting point may be data, domain expertise, or
questions, and Step ($1$) involves research to begin filling in gaps in our
knowledge of these three areas and to formalize our goals. I often review
literature in Step ($1$) with Step ($2$) in mind, tagging papers with
quantitative data that I can use later for parameter fitting and noting
studies that show alternative experimental conditions that could be used for
model testing. Steps ($2$) and ($3$) then treat complementary parts of model
building. In Step ($2$), we select our overall approach and the variables,
parameters, group dynamics, and agent behaviors in which we are most
interested. This means making choices related to the concepts in Figures 2 and
3: for example, if we are studying traffic flow on a stretch of roadway, will
we track the number $N(t)$ of cars on the road in time, or the position
$\textbf{x}_{i}(t)$ and velocity $\textbf{v}_{i}(t)$ of each vehicle $i$ in
time? If we are accounting for driver differences, will we assume that each
driver’s phenomenological “level of cautiousness” is time-dependent or static?
It is important to make these choices in a way that accounts for the
complexity of the problem and our data, so Step ($2$) involves making a plan
for how we will use data to develop (or train, or fit) our model and later
test (or validate) our model, as I discuss in Section 4.2. At the end of Step
($2$), our model is written down (e.g., as a system of differential equations
on paper or as a set of stochastic rules in code). In Step ($3$), we turn to
filling in parameter names with parameter values, setting initial conditions,
and determining our boundary conditions, as needed. Step ($3$) involves
validating our model to test its predictive value and performing various
analyses to check how sensitive our model is to uncertainty in parameter
values, initial conditions, boundary conditions, or data. Depending on the
form of our model, we may be able to perform a bifurcation analysis to
understand how changes in parameters influence our results. We may also
brainstorm alternative ways of judging model output and comparing this with
data, since how we choose to describe model output can impact how we interpret
our results. At the end of Step ($3$), we use our model to gain understanding
and, if possible, suggest new experiments, resulting in model-driven data
collection. More broadly, Step ($1$) is where we realize that a model can help
us accomplish our goals, Step ($2$) is the place where we build the model
structure, and Step ($3$) is where we test and prod this structure. Data enter
the picture in Step ($1$) as motivation. In Steps ($2$) and ($3$), we work
closely with data to build, test, and use our model in a way that balances
model and data complexity to accomplish our goals. In the remainder of this
tutorial, I focus primarily on the later parts of Step ($2$) and broadly
discuss the early parts of Step ($3$). To learn more about some of the
analyses and computational approaches possible in Step ($3$), I suggest the
books [Smi14, Str15, Kut13, SEK13].

###  4.2. Balancing Model and Data Complexity

While data-driven models take many forms and scientists use a range of methods
to understand them, the overarching theme of balancing model and data
complexity is present throughout. Depending on our goals and data, what
modeling approach do we choose? How do we build a data-driven, _data-
appropriate_ model? If we have access to a wealth of domain expertise and a
rich set of data, it may make sense to build a complex model, since, in this
case, the majority of the model will be purely descriptive, framing known
agent interactions in a mathematical way. The new hypothesis that we are
testing, along with its few parameters, enters the picture as our assumption.
On the other hand, if we are leading the way to model a poorly understood
complex system, our model needs to be very simple, again so that the
assumptions and hypotheses that we introduce match the amount of data
available.

Figure 4. Balancing model and data complexity. (a) Underfit models miss
meaningful features in data, (b) overfit models include too many assumptions,
and (c) parsimonious models balance model and data complexity. In (a)–(c),
blue points denote data that we use to develop our model and fit parameters,
and red points denote our testing set. (d) Underfit models agree poorly with
both our training and testing data, while overfit models represent our
training data well and our testing data poorly [BK19, MKBP17]. Parsimonious
models perform well on both sets of data. (e) Creating a plan for model
training/development and testing is key to data-driven modeling. This involves
breaking data into sets for training and testing, a process that depends on
our complex system. For example, if our goal is to understand how cells
interact to form patterns in fish skin, this could mean breaking our
(qualitative) data into images of fish that are well understood (and involve
setting parameters in a model to zero in a clear way), more images of fish
that are well understood (and involve changing the values of nonzero
parameters in a clear way), and images of poorly understood fish (and involve
changing parameters in unknown ways) [VS18, Vol17]. The first set is used for
model development, the second for testing, and the third as a place where we
can make predictions [VS18, Vol17]. Image (d) is based partly on [BK19,
MKBP17]; first row, third row, and left image in second row of (e) adapted
from [SNV15] with permission from Elsevier, Copyright (2015) Elsevier Ltd.;
remaining images in second row of (e) reproduced from [PT03] with permission
of The Company of Biologists, Ltd.

In either case, it is helpful to break our data up into sets for model
training (or development) and testing. Training/development data is the data
that we use to build our model, specifying parameter values as well as the
form of model rules and terms as appropriate. After this, we take a step back
and test whether or not our model behaves well on the data that we
withheld—our testing data. If the model does well on the testing set, we can
use our model to predict future dynamics or shed light on poorly understood
dynamics. If the model does not do well on the testing set, we need to return
to model development. As a guiding principle, the more parameters and
assumptions that we build into a model, the more that it needs to be able to
reproduce in order to have predictive value222It is worth noting that data-
driven models can be used for many purposes, including describing, explaining,
or predicting, and Shmueli discusses these goals from a statistical
perspective in [Shm10].. Figure 4(c)–(d) highlights two concepts that are
related to balancing model and data complexity: underfitting and overfitting.
For illustrative purposes, I consider the example of population growth of some
organism in time, given some (imperfect) measurements of the number of
organisms at discrete time points. At one extreme, I could assume a linear
relationship between population size and time, fitting a line to the data.
This involves few parameters, and the difference between the model and
training data is high. At the other extreme, I could draw a curve that goes
through every single data point [BKGL18]—this would mean introducing many
parameters. In terms of these models’ ability to approximate population size
at some new time in our testing set, neither will do well [MKBP17]. A better
model lies somewhere in between these two extremes. What we are after is a
“parsimonious” model [MKBP17, BK19]: a model that it is supported by our data
and no more complex than it needs to be. Building predictive, data-appropriate
models that avoid overfitting and have strong predictive value looks different
based on the problem and relies on domain expertise. (See e.g., [Smi14, BK19]
for a more detailed discussion of methods—I focus on broad concepts here.) If
our goal is to understand social-media engagement in time, for example, we
might build a gray-box model driven by some data $\\{w_{i}\\}_{i=1,....,T}$,
where $w_{i}$ is the number of accounts on a social-media platform on day
$t=i$. As one approach, we could split the data into a training set
$\\{w_{i}\\}_{i=1,....,\tilde{T}}$ and a testing set
$\\{w_{i}\\}_{i=\tilde{T}+1,....,T}$ with $\tilde{T}<T$. We could develop our
model and specify its parameters using the training set and then run our model
until $t=T$ to evaluate how well it does on the testing set. If our model does
well in testing, we could use it to predict future social-media engagement.
When working with qualitative data, the process of balancing model and data
complexity looks different, but it is the same at its core. In Figure 4(e), I
highlight the complex biological system that most of my work is on: pattern
formation in zebrafish skin [VS15, VS18]. Wild-type and mutant zebrafish
feature different patterns, which form through the interactions of pigment
cells [SNV15, PT03]. Although there are some quantitative data (e.g., cell
speeds), most take the form of images of fish. To build the model [VS18], we
broke these qualitative data into three sets. The first set of images contains
patterns that correspond to setting specific parameters to zero in a
mathematical model (e.g., setting the birth rate of black cells to zero). The
second set holds some fish patterns that are relatively well understood; in
this case, we know simulating them means changing parameters in a clear way
(e.g., slowing domain growth). The final set contains mutant patterns that are
poorly understood, patterns that form due to cell interactions that are
altered in unknown ways. The first set serves as a natural model
development/training set, and once we identified a model that could reproduce
these fish patterns, the next step was to step back and break it down,
checking if there were any ways that we could simplify the model and still
maintain consistency [VS18]. “Minimal” model in hand, we used the second set
of images for testing, asking whether or not the model could reproduce data
that we did not build into it. And, finally, the tested model now serves as a
predictive tool to understand the fish in the third set: at this stage, we
change parameters in the model with the goal of identifying altered cell
interactions that may lead to mutant patterns [Vol17]. In order to further
improve predictive value and avoid overfitting, there are a wealth of other
approaches modelers can take. We can test how uncertainty in our parameters,
boundary conditions, or initial conditions affect our results, and we can
explore whether other modeling approaches lead to the same conclusions. We can
set parameters in our models to zero or remove rules, checking to see if our
models can be made simpler without losing agreement with the training set. We
can also ask questions about whether the methods that we use to judge our
models influence our results: what alternative methods for measuring agreement
between model output and data can we test? Throughout this process, the goal
is to critically investigate our modeling assumptions as we build a
parsimonious—or minimally complex—model based in our data.

##  5\. Illustrative Case Studies

In the remainder of this tutorial, I turn to two case studies of complex
social systems: opinion dynamics during elections (Section 5.1) and pedestrian
movement in crowds (Section 5.2). These examples illustrate some of the types
of models and data from Section 3 in the broader framework of the challenges
and choices introduced in Section 4. I highlight the benefits and drawbacks of
different modeling choices, with the quotations from Segel and Edelstein-
Keshet [SEK13] at the beginning of this chapter as a guide.

###  5.1. Forecasting Elections

Political opinion dynamics are a complex social system, and here I focus on
the goal of forecasting elections in the United States. Election forecasting
is highly interdisciplinary, drawing on probability, geometry, dynamical
systems, topology, and statistics, as well as political science, history,
economics, computer science, and sociology more broadly. It naturally involves
communication and public science, and different forms of data (Section 5.1.1).
Framed by this interdisciplinarity, I illustrate a statistical, static
modeling approach to elections in Section 5.1.2 and a dynamic, mathematical
model in Section 5.1.2. Many other models and methods for incorporating data
into forecasts exist beyond the scope of this survey (e.g. data-assimilation
techniques [LSZ15]). Election forecasting raises questions at many different
scales; for example, using a compartmental model, Restrepo _et al._ [RRH09]
investigated how polling data affect whether potential voters decide to vote,
and Biondo _et al_. [BPR18] developed an agent-based model to better
understand how surveys influence opinions. Election forecasting is related to
the broader field of opinion dynamics [CFL09, PG16], which includes the
formation and dynamics of echo chambers (e.g., [SCP+21, EF18, CDFMG+21]) and
polarization (e.g., [SMA20, YAKM20]). There are many approaches to opinion
formation, such as voter models [FGSR+14, BdA17] and threshold models [LYY18].

Figure 5. Example approaches to forecasting U.S. elections. (a) The
president’s June approval rating and the percentage of the national vote for
their party tend to be related [Abr08]. (b) Abramowitz’s [Abr08] model is
driven by fundamental data. (c) This statistical model is deterministic and
continuous in the sense that, once the parameter values are set, the result is
one prediction of the national vote for the incumbent party (a continuous
number between $0$ and $100$). (d) In a network model, one could investigate
the interactions between undecided (purple), Republican (red), and Democratic
(blue) voters, as I illustrate in this cartoon. Because networks [PG16, Str01,
New10] are the focus of another chapter in this volume, I do not cover them; I
suggest the lectures by Brooks and DeFord in our Short Course [Bro21, DeF21].
(e) Compartmental modeling [Het00, DH00, BCC12] involves grouping individuals
into categories and investigating how folks change compartments. (f) The
compartmental model [VLPR20] is a stochastic mathematical approach to
forecasting elections. It is spatial in the sense that it produces state-level
forecasts, and non-spatial in the sense that it does not track the locations
of individual voters. Image (a) reproduced from [Abr08] with permission,
published by Cambridge University Press, and Copyright (2008) The American
Political Science Association; images (d)–(e) adapted from [VLPR20].

Because elections receive attention so widely and forecasts have the potential
to impact turnout, the example of election forecasting highlights a place in
complex-systems research where carefully presenting the results of data-driven
models is especially important. Communicating probabilistic forecasts in a
tangible, interpretable way itself leads to questions, and I suggest [GHWM20,
FPS+21] for further discussion about visualizing and communicating
uncertainty. Election forecasting also presents interesting challenges when it
comes to evaluating model success and forecast accuracy [GHWM20], as I
mentioned in Section 3.2.

####  5.1.1. Election Data

In terms of Step (1) in Section 4.1, as a starting point, the data used to
build election-forecasting models include historical results, approval
ratings, economic indicators, information about incumbency, and polls [HR14,
Sil12, Abr08]. Analysts often separate these data into two types: polls and
“fundamental data” (or “fundamentals”). Fundamental data are the data from
which voters may form their opinions and determine how they will vote [GK93];
for example, economic data fall into the fundamentals category. Regardless of
the type, all data come with challenges: data may not go back in time as far
as a we would hope or may not be as fine-scale as we would like (e.g., data at
the national or state level, rather than the district level). For forecasts
that depend on historical data, one assumption is that the past and the future
will behave similarly. Modeling with fundamental data allows forecasters to
produce early predictions, prior to when accurate polls may be available
[HR14, Lin13]. However, opinions are dynamic—both across years and within the
same election year—and past elections may not be representative of how voters
will behave in the future. On the other hand, it is not always clear whether
shifts in polls in a given year represent real shifts in opinion or just
differences in pollster methods [GK93, WE02, Jac05]. Moreover, polling data
are often bias [Jac05] and adjusted in proprietary ways; for example,
pollsters make decisions such as how to define “likely voters”. Polling data
can be spotty, with some states being polled more frequently than others
[Lin13]. Adding another layer of complexity, pollster herding is a phenomenon
in which polling organizations adjust their results when their data do not
align with other polls [Sil14, CR13, GHWM20].

####  5.1.2. Example Statistical Approach

In Figure 5(a), I reproduce a plot from [Abr08] of net presidential approval
ratings333This is defined as approval minus disapproval (see [Abr08] for
details), so it can be negative. in June versus the percentage of the vote
that went for the incumbent president’s party in November of the same year.
This motivates a statistical modeling approach to forecasting U.S. elections
that is driven by fundamental, historical data. As an example of such an
approach, I highlight some of the ideas in Abramowitz’s “time-for-change”
model [Abr08, Abr88]:

(5.1) $\displaystyle\begin{bmatrix}v_{1}\\\ v_{2}\\\ v_{3}\\\ .\\\ .\\\
v_{m}\end{bmatrix}=\begin{bmatrix}1&a_{1}&g_{1}&c_{1}\\\
1&a_{2}&g_{2}&c_{2}\\\ 1&a_{3}&g_{3}&c_{3}\\\ .&.&.&.\\\ .&.&.&.\\\
1&a_{m}&g_{m}&c_{m}\\\ \end{bmatrix}\begin{bmatrix}\alpha\\\ \beta\\\
\gamma\\\ \varepsilon\end{bmatrix},$

where $v_{i}$ is the percentage of the national vote that went for the
presidential candidate from the incumbent party in the $i$th election in the
data set; $m$ is the number of years for which data is available; $a_{i}$ is a
measurement of presidential approval before the $i$th election; $g_{i}$
includes information about economic growth in the year leading up to the $i$th
election; and $c_{i}$ is a variable related to incumbency. Once the parameters
$\alpha,\beta,\gamma,$ and $\varepsilon$ are determined from historical data
(e.g., using regression), the time-for-change model [Abr08, Abr88] can predict
an election $m+1$ by computing $v_{m+1}=\alpha+\beta a_{m+1}+\gamma
g_{m+1}+\varepsilon c_{m+1}$. Equation (5.1) has the general form
$\textbf{v}=\textbf{A}\textbf{p}$, where p corresponds to parameters, A
contains fundamental data, and v holds $m$ past election outcomes. If we were
to introduce more types of historical data, the number of parameters $n$ would
grow. With more parameters, we would expect to get a better fit between the
model predictions and past election results. As Figure 4 highlights, however,
this does not necessarily correspond to better predictions of future
elections, since allowing $n$ to become too large can lead to overfitting.
This raises questions about model complexity. How many kinds of fundamental
data should a modeler include? How many terms in the model is the “right”
number of terms? To address these questions, we need to define what a good
model means and choose how to measure error. For example, consider the
function [BK19]:

(5.2) $\displaystyle E(\textbf{p})$
$\displaystyle=\underbrace{||\textbf{A}\textbf{p}-\textbf{v}||_{2}}_{\text{least-
squares term}}+\underbrace{\lambda_{1}||\textbf{p}||_{1}}_{\text{LASSO
term}}+\underbrace{\lambda_{2}||\textbf{p}||_{2}}_{\text{ridge-regression
term}}.$

We can minimize $E(\hat{\textbf{p}})$ to find the parameter values most
consistent with our data:

p
$\displaystyle=\underset{\hat{\textbf{p}}}{\mathrm{argmin}}~{}E(\hat{\textbf{p}}).$

When $\lambda_{1}=\lambda_{2}=0$ in Equation (5.2), $E(\textbf{p})$ is the
least-squares difference between the model’s predictions and the election
outcomes under the parameters p. This method for measuring goodness-of-fit is
sensitive to variability [BK19]. If $\lambda_{1}>0$ and $\lambda_{2}=0$, we
instead implement LASSO regression [Tib96], which selects sparse models and
helps prevent overfitting by forcing some parameters to zero [BK19]. When
$\lambda_{1}>0$ and $\lambda_{2}>0$, Equation (5.2) corresponds to elastic-net
regularization. Importantly, $\lambda_{1}$ and $\lambda_{2}$ provide a means
of calibrating model complexity. We can choose to minimize Equation (5.2) for
different values of the hyper-parameters $\lambda_{1}$ and $\lambda_{2}$,
resulting in different models (in the form of the parameter values p) for each
choice. Information criteria, such as Akaike information criteria (AIC) and
Bayes information criteria (BIC), can come in handy to select the best model
from among these alternatives [MKBP17, Aka98, Aka74, Sch78]. _The Economist_
’s 2020 forecasts [eGH20], for example, depend in part on a statistical model
of the form $\textbf{A}\textbf{p}=\textbf{v}$ with a matrix A that contains
many types of fundamental data. To help prevent overfitting, _The Economist_
[eGH20] team combines leave-$1$-out cross validation [BK19] and elastic-net
regularization with a range of $\lambda_{1}$ and $\lambda_{2}$. Broadly,
leave-$k$-out cross validation is a means of breaking data into training and
validation sets. To implement this method, one removes $k$ samples of the
training data; the removed data then becomes the validation set, and the
remaining data is used for training [BK19]. For example, if $k=1$ in the
presidential election setting and the available data are for the years 2004,
2008, 2012, and 2016, one first removes one year of data (e.g., the 2012
data). The next step is determining the parameter values
$\textbf{p}_{\text{2012}}$ that result from fitting based on the data for the
remaining years (2004, 2008, and 2016, in this example). Repeating this for
the other years leads to four sets of parameter values. One option is to
define the final parameter values p as the mean of these four sets of
parameters. Other approaches to testing and validation include $k$-fold cross
validation [BK19]. In Figure 5(b)–(c), the statistical, phenomenological
approach of this section has benefits and drawbacks, like all models do.
Because it is driven by fundamental data, the time-for-change model [Abr08,
Abr88] is not dependent on noisy polling data; instead, it is able to generate
forecasts as early as approval, economic, and incumbency data are available.
Moreover, this model is simple and has few parameters. On the other hand, the
model [Abr08, Abr88] in Figure 5(b) is static, and it does not add mechanistic
understanding of what causes opinions to change in time during an election
year.

####  5.1.3. Example Dynamical-Systems Approach

As a more mechanistic approach, one example is the mathematical model [VLPR20]
that my collaborators and I developed for forecasting U.S. elections. This
model, driven by polling data [Huf22a, Huf22b, Rea22, BBG+22], has a
compartmental Susceptible–Infected–Susceptible (SIS) model at its core.
Compartmental modeling is a widely used method for describing disease dynamics
(e.g., [KM27, KM32, KM33, Het00, DH00, BCC12]), and it has also been applied
to social contagions (e.g., [BCAKCC06, BGBD+18]). The central concept is that
the population of interest can be grouped into compartments444This general
structure is very flexible: for example, in Figure 6(d), I highlight one way
that compartmental modeling could be used to describe pedestrian dynamics.
Here the compartments are leading pedestrians moving to the right, following
pedestrians moving to the right, leading pedestrians moving to the left, and
following pedestrians moving to the left. If we are mainly interested in
understanding how many leaders and followers are present, this approach could
suffice. We might consider the transition of left-moving leaders to left-
moving followers as dependent on interactions with other leaders.. In the SIS
setting (Figure 5(e)), there are two compartments: susceptible and infected.
Susceptible individuals become infected through interactions with infected
folks (i.e., transmission), and infected individuals recover, becoming
susceptible. If we track the fraction of the population that is susceptible or
infected in time, the result is a gray-box model in the form of differential
equations. In the approach [VLPR20], we adapt the traditional SIS
compartmental model by introducing two “contagions” (Democratic and Republican
voting inclinations) and replacing susceptible individual with undecided or
other voters. For each state or region $i$, we track the fraction of undecided
$S^{i}(t)$, Democratic $I^{i}_{\text{D}}(t)$, and Republican
$I^{i}_{\text{R}}$ voters in time according to the stochastic ordinary
differential equations:

(5.3) $\displaystyle{dI_{\text{D}}^{i}}(t)$
$\displaystyle=\underbrace{-\gamma^{i}_{\text{D}}I^{i}_{\text{D}}}_{\text{Dem.\
recovery}}dt+\underbrace{\sum_{j=1}^{M}\beta^{ij}_{\text{D}}\frac{N^{j}}{N}S^{i}I^{j}_{\text{D}}}_{\text{Dem.\
transmission}}dt+\underbrace{\sigma
dW^{i}_{\text{D}}(t)}_{\text{uncertainty}}\,,$ (5.4)
$\displaystyle{dI_{\text{R}}^{i}}(t)$
$\displaystyle=\underbrace{-\gamma^{i}_{R}I^{i}_{\text{R}}}_{\text{Rep.\
recovery}}dt+\underbrace{\sum_{j=1}^{M}\beta^{ij}_{\text{R}}\frac{N^{j}}{N}S^{i}I^{j}_{\text{R}}}_{\text{Rep.\
transmission}}dt+\underbrace{\sigma
dW^{i}_{\text{R}}(t)}_{\text{uncertainty}}\,,$

where we use that $S^{i}(t)=1-I^{i}_{\text{D}}(t)-I^{i}_{\text{R}}(t)$ to
reduce the number of equations. Here $I_{\text{D}}^{i},I_{\text{R}}^{i},$ and
$S^{i}$ are stochastic processes; $W^{i}_{\text{D}}$ and $W^{i}_{\text{R}}$
are Wiener processes; $M$ is the number of states or regions; $N^{j}$ is the
number of voting-age individuals in state $j$; and $N$ is the total number of
voting-age individuals across our $M$ regions. This model involves the
simplifying assumption that we can bin voters as Democratic, Republican, or
undecided. Bounded-confidence and related models (e.g., [WPCG+14, DNAW00,
HK02, BP20a]) account for opinions existing on a continuous spectrum. The
parameters in Equations (5.3)–(5.4) call for special attention. There are
$2\times M$ parameters
$\\{\gamma_{\text{D}}^{i},\gamma_{\text{R}}^{i}\\}_{i=1,...,M}$ that describe
the rates at which committed voters become undecided. There are also $2\times
M^{2}$ parameters
$\\{\beta^{ij}_{\text{D}},\beta^{ij}_{\text{R}}\\}_{i,j=1,...,M}$ for the
rates at which Democratic (Republican) voters in state $j$ “infect” undecided
voters. To find the values of these parameters, we [VLPR20] relied on polling
data. For the ODEs associated with Equations (5.3)–(5.4) (with $\sigma=0$), we
minimized the least-squares difference between our model output under
parameters p:

$\displaystyle\textbf{X}(t_{k};\textbf{p})$
$\displaystyle=[I^{1}_{\text{R}}(t_{k};\textbf{p}),...,I^{M}_{\text{R}}(t_{k};\textbf{p}),I^{1}_{\text{D}}(t_{k};\textbf{p}),...,I^{M}_{\text{D}}(t_{k};\textbf{p}),S^{1}(t_{k};\textbf{p}),...,S^{M}(t_{k};\textbf{p})],$

and the averaged state- or region-level polling data:

$\displaystyle\textbf{x}(t_{k})$
$\displaystyle=[R^{1}(t_{k}),...,R^{M}(t_{k}),D^{1}(t_{k}),...,I^{M}_{\text{D}}(t_{k}),S^{1}(t_{k}),...,S^{M}(t_{k})],$

where $k=1,...,T$ with $T$ months of polling data considered. The parameter
values are different for each election year and race, depending on the
associated polls. The goal of forecasting elections provides a natural means
of building and testing a model. By using only the polling data (but not the
election results) for past races, we can test Equations (5.3)–(5.4) by
retroactively forecasting past elections [VLPR20]. For the statistical model
in Figure 5(b), one of the challenges is selecting what types of fundamental
data to include in the model, and this comes down to determining what
parameters are zero or nonzero. In contrast, for the mathematical model here,
it is more the format of the differential equations and the assumptions of an
SIS-style model, rather than the values of the parameters, that we want to
evaluate. Because the parameters in Equations (5.3)–(5.4) depend only on the
polls for a given election year, this model can be tested by applying it to
forecast previous elections, one at a time. This step in some sense combines
model training and validation together. In terms of predictions, there is also
a natural—and high-stakes—opportunity: the model can be used fo forecast
upcoming elections. One of the benefits of the continuous, stochastic
mathematical model in Equations (5.3)–(5.4) is that it includes some
mechanistic hypotheses about opinion dynamics. The model [VLPR20] is also
dynamic in time; see Figure 5(f). Once polls becomes available, Equations
(5.3)–(5.4) can forecast a new U.S. election with parameters that are specific
to that election. However, opinion dynamics are not the same as biological
disease transmission. Instead, we might think of the transmission terms in
Equations (5.3)–(5.4) as capturing interactions between committed voters in
state $j$ and undecided voters in state $i$ in a phenomenological way. These
interactions could be direct (e.g., via conversations between a committed
voter in one state and an undecided voter in another state) or indirect (e.g.,
through news coverage). As another drawback, the model [VLPR20] has many more
parameters than the statistical approach [Abr88, Abr08].

###  5.2. Modeling Pedestrian Movement

Crowds of people exhibit rich collective behavior, including lane formation
and oscillating flows [HM95, HBJW05, HJ09, SSS17]. For example, as I show in
Figure 6(a), pedestrians may form lanes when two groups walk in opposing
directions in a narrow corridor. Like the application of election forecasting
in Section 5.1, studying the dynamics of crowds touches on many fields,
including engineering, sociology, psychology, physics, computer science, and
mathematics [BCG+16, SSS17, BR19, HJ09]. This interdisciplinarity stems from
the goals that can motivate models of pedestrian movement. Researchers may be
interested in designing functional buildings, testing how guidelines influence
disease transmission in a crowd, developing methods to improve evacuation in
emergency settings, or something else. Here I focus on the goal of
understanding under what conditions lanes emerge from pedestrian interactions,
and I assume accounting for the spatial organization of individuals in time is
important.

Figure 6. Example modeling approaches to pedestrian movement. Experiments in
settings such as (a) bidirectional movement in corridors [ZKSS12], (b)
movement through through an entrance without broader spatial barriers [SSS17],
and (c) movement through an entrance with spatial constraints [SSS17] produce
both quantitative and qualitative data. There are many approaches that we
could take to describe lane formation in (a). In (a) and (d)–(i), red denotes
pedestrians moving to the left and black denotes study participants moving to
the right. (d) Compartmental models, discussed in Section 5.1.3 could, for
example, track the fraction of study participants who are following others or
leading lines; this approach is non-spatial. (e) Macroscopic models track
pedestrian density and generally take the form of PDEs. (f) Microscopic on-
lattice models consider the positions of individuals in discrete space and
involve stochastic, computational rules. (g) Microscopic on-lattice models
track the positions of individuals in continuous space through coupled
differential equations. (h) Hybrid, multiscale approaches come in many forms;
for example, we could couple an agent-based model of pedestrian movement with
a compartmental model describing the feelings within each pedestrian, which,
in turn, influence their movement. (i) Fine-grid cellular automaton models use
multiple grid squares to represent each pedestrian, providing a more detailed
perspective on pedestrian position. Images (a)–(c) adapted (cropped) from
[SSS17] and licensed under CC-BY 4.0
(https://creativecommons.org/licenses/by/4.0/).

For this tutorial, I use crowd movement as a venue for discussing approaches
to modeling agent behavior in space, and highlighting some challenges
associated with qualitative data (Section 5.2.1). Pedestrian movement provides
an opportunity to illustrate a range of gray-box, spatial models, including
continuum models, cellular-automaton perspectives, and agent-based approaches
(Section 5.2.2). There are other data-driven approaches to crowd dynamics, and
I highlight [BR19] for a review of statistical models. From the perspective of
building simplified models (in particular, models that do not include concepts
from social psychology [SSS17]), similar challenges and approaches arise in
diverse examples of pattern formation and self-organization, including
migrating cells (e.g, [BEK20, Vol20b, GBKM20, HRM17, GG93]), animal
aggregations (e.g., [CKJ+02, PEK99, LLEK10]), swarming locusts (e.g., [AA15,
BCME+20, BT11]), and more general agents interacting in space (e.g., [CDM+07,
VCBJ+95, LRC01, DCBC06, MEK99, TBL06, CMW16]).

####  5.2.1. Pedestrian Data

Data on pedestrian movement comes in quantitative and qualitative forms,
including measurements of velocity [ZKSS12], questionnaires about pedestrian
experience [SSS17], and images of crowds [BHK+11]. This information may stem
from observations in the field or in controlled lab settings. For example,
Zhang _et al._ [ZKSS12] performed a series of experiments in which study
participants were instructed to move through corridors of different widths. As
I show in Figure 6(a), participants in red were asked to move to the left
through the corridor, and pedestrians wearing black were asked to move to the
right. Lanes—visible as red and black stripes in Figure 6(a)—emerged from the
interactions of the pedestrians in some settings [ZKSS12]. In addition to this
qualitative data, the experiments [ZKSS12] produced trajectories of each
participant’s position, along with measurements of velocity and density. As
another example, Sieben _et al._ [SSS17] performed a series of experiments to
better understand how pedestrians respond to different barriers as they seek
to pass through an entrance. The setups [SSS17] in Figure 6(b)–(c) are meant
to represent what might happen when people are entering a concert venue. After
extracting the positions of the white caps worn by pedestrians, Sieben _et
al_. [SSS17] collected trajectories of individuals. The authors [SSS17] also
asked the study participants questions about their experience of walking
through the entrance before and after watching a video of the experiment. This
survey [SSS17] produced data on how comfortable the heterogeneous participants
reported feeling and how just they felt the entrance process was, among other
things. When we view Figure 6(a), the presence of stripes is striking; while
it is not as visible in Figure 6(c), the trajectories of pedestrian movement
that Sieben _et al_. [SSS17] extracted from these experiments also show lanes
in some cases. This highlights one of the challenges associated with spatial
complex systems: many of the features in Figure 6(a)–(c) are qualitative. We
may see stripes, but how do we define these stripes objectively and
quantitatively in large sets of images? At different timepoints in the
experiment (see the videos in the supplementary material of [SSS17]), the
stripes are not as clear and do not extend across the full length of the
corridor. How do we define stripe width or the time when bands start or end
along the length of the corridor? The qualitative nature of data in spatial
complex systems presents new challenges when fitting and testing models.

####  5.2.2. Example Spatial Modeling Approaches

Figure 6 shows some approaches to spatial modeling of complex systems,
including crowd movement, at different levels of detail. Here I focus on
introducing some broad gray-box, mathematical modeling approaches that we
could take to study lane formation, rather than discussing specific
references. (See the reviews [BCG+16, SCS+18, DDH13] and references therein
for more information about crowd dynamics.) These approaches—namely
macroscopic, microscopic on-lattice, microscopic off-lattice, and hybrid
(e.g., [KHB13]) models—are used to study a wealth of spatial dynamics. There
are also many perspectives that I do not discuss, including mesoscopic (e.g.,
[FTW18, BBK13]) and game-theoretic (e.g., [Dog10, LW11, BCD18]) approaches.
Macroscopic, continuum models of pedestrian movement often take the form of
partial differential equations (PDEs). As I show in Figure 6(e), this approach
stems from a zoomed-out perspective: instead of tracking the locations of
individuals in Figure 6(a), continuum models describe the evolution of density
in time. If we make the assumption that there are two populations in our
corridor example, a continuum model would track the density $r(\textbf{x},t)$
of “red-shirt-wearing” and $b(\textbf{x},t)$ “black-shirt-wearing” pedestrians
in space x and time $t$. One benefit of macroscopic models is that they are
often analytically tractable, and they provide a broad perspective on
overarching features that may be at work in a complex system. These models
often have few parameters, and researchers can perform bifurcation analysis to
understand how these parameters influence group dynamics. The drawback is that
PDE approaches may simplify the complex dynamics of heterogeneous pedestrians
significantly, and it can be challenging to relate the few parameters in these
models to specific agent behaviors. In contrast to macroscopic models,
microscopic approaches focus on the positions or features of individuals, and
two prominent frameworks are on-lattice and off-lattice models. These models
provide more detailed perspectives at the scale of individual agents, which
comes at the cost of more parameters. Spatial modeling is a place where
vocabulary differs some between fields, particularly in the case of
microscopic models. Depending on one’s perspective, the microscopic models in
Figure 6(f)–(e) may be described as individual- or agent-based models (IBMs or
ABMs), since these models track changes in the positions of agents. The term
“agent-based” also refers to more detailed models such as [BHK+11, BDM+09,
TFB+11]. Miller and Page [MP07] describe agent-based models as “bottom-up”
approaches, because the starting point is interactions of individuals. In
interdisciplinary—or even within-discipline—conversations, I suggest asking
questions to clarify what folks mean by ABMs and IBMs in their setting.
Microscopic on-lattice (cellular automaton) models consider space as a
lattice, and pedestrians can either occupy or not occupy positions on a grid
(e.g., [BKSZ01, VCM+07, BA01]); see Figure 6(f). Movement, as well as arrival
and exit, takes the form of stochastic, computational rules. Notationally, we
could denote whether the grid square in row $i$ and column $j$ at time $t_{k}$
is red (i.e., containing a pedestrian moving to the left in Figure 6(a)),
black (i.e., holding a right-moving pedestrian), or white (empty) by:

$\displaystyle x_{i,j}(t_{k})$ $\displaystyle=\begin{cases}-1&\text{~{}if grid
square is red}\\\ 0&\text{~{}if grid square is empty}\\\ 1&\text{~{}if grid
square is black}.\end{cases}$

For example, to model right-traveling pedestrians stepping to the side to
avoid collisions with left-moving study participants, we might select a grid
square $(i,j)$ uniformly at random from Figure 6(a) and implement the rule:

$\displaystyle\text{if}\underbrace{~{}\text{$x_{i,j}(t_{k})=-1$ and
$x_{i,j+1}(t_{k})=1$}}_{\text{conditions for a head-on
collision}}\text{~{}and~{}}\underbrace{\text{$x_{i+1,j}(t_{k})=0$}}_{\text{space
available}},$ (5.5)
$\displaystyle\text{then~{}}\underbrace{\text{$x_{i,j}(t_{k+1})=0$ and
$x_{i+1,j}(t_{k+1})=1$ with probability $p$}}_{\text{pedestrian may step to
the side}}.$

In one time step, we could iterate through a random perturbation of all of the
grid squares, implementing this and other model rules. There are many choices
and parameters in Rule (5.5), including the choice of probability $p$ and the
choice of neighborhoods considered (e.g., why should the pedestrian at space
$(i,j)$ only look one grid step ahead to space $(i,j+1)$? Maybe $(i,j+2)$ is
more appropriate?). Microscopic off-lattice models (e.g., [HM95, HBJW05]), in
comparison, assume that individuals move continuously in space; see Figure
6(g). In this case, movement is modeled through coupled ordinary or stochastic
differential equations, for example, of the form:

(5.6) $\displaystyle\frac{d\textbf{V}_{i}}{dt}$
$\displaystyle=\underbrace{\textbf{g}(\textbf{X}_{i},\textbf{V}_{i})}_{\text{pedestrian
$i$'s inherent
goals}}+\underbrace{\sum_{j=1}^{N}\textbf{f}(\textbf{X}_{i},\textbf{X}_{j},\textbf{V}_{i},\textbf{V}_{j})}_{\text{interactions
between pedestrians}}$ (5.7) $\displaystyle\frac{d\textbf{X}_{i}}{dt}$
$\displaystyle=\textbf{V}_{i},$

where $\textbf{X}_{i}(t)$ is the position of the $i$th pedestrian (e.g., a
point mass marking the $(x,y)$ coordinates of the pedestrian’s center of mass)
and $\textbf{V}_{i}(t)$ is that pedestrian’s velocity. So called “social-
force” models are a prominent off-lattice microscopic approach to pedestrian
dynamics [HBJW05, HM95]. In both on-lattice and off-lattice models, arrival
and exit of pedestrians from either side of the corridor in Figure 6(a) could
take the form of stochastic rules. Computationally, we might assume that a new
pedestrian enters the corridor at a randomly selected $(x,y)$ position near
the left or right edge of Figure 6(a) with probability $\alpha\Delta t$, where
$\Delta t$ is the time step of our simulations. While microscopic models offer
detailed perspectives on the behavior of individuals and can make
experimentally testable predictions, they have many more parameters than
macroscopic models do. In order to avoid overfitting and improve predictive
value, it is thus important to break our data into separate sets for model
development and testing. For example, we could fit the parameters in the
functions in Equations (5.6)–(5.7) based on measurements of
pedestrian–pedestrian distances and pedestrian velocities. We could specify
the rates at which pedestrians enter the corridor based on empirical data, and
we could use lane width to determine any unmeasurable parameters or guide the
form of model rules. To test our model, we could set aside certain experiments
(e.g., experiments with wider corridors) to simulate with our final model. We
could, for example, use our validated model to predict how the dynamics will
change when a pushier agent is introduced or when the structure of the
barriers and walls in Figure 6(a)–(c) is changed. Adding further difficulty,
microscopic models are often stochastic and not analytically tractable, and
they face some of the same challenges as qualitative data: how do we define
and quantitatively describe the stripes in Figure 6(f)–(g) in an automated,
objective way? To help address this challenge, topological techniques,
especially persistent homology [OPT+17, EH08, Car20], have recently been
combined with modeling to study complex systems, including aggregation [UZT19,
TZH15]. Additional examples of topological data analysis applied to biological
and social complex systems include [BMM+19, BCT20, MVS20, CJDM21, AQO+20,
NSF+21] and [FHP22, HJJ+22], respectively. Pair correlation functions [DBG18,
JC19, TSB+14] are another method for quantifying spatial data. Depending on
our goals and what our data suggest, building a hybrid, multiscale model that
accounts for dynamics within pedestrians may be appropriate; see Figure 6(h).
For example, in the off-lattice microscopic setting, we could introduce a
variable $P_{i}(t)$ that tracks how frustrated each individual is based on
their perceived justness of the crowd dynamics around them. We could define
$P_{i}(t)$ by comparing the distance that pedestrian $i$ has moved toward
their goal in some time interval to the estimated distance that the
individuals in a local neighborhood around $i$ are moving. There are many
other ways that we could define $P_{i}(t)$, and we could include feedback
between $P_{i}(t)$ and how pushy pedestrians choose to be, influencing our
ODEs for movement in an associated agent-based model. As a last example,
similar to cellular Potts models in biology [GG93, HRM17], fine-grid cellular
automaton represent each individual with a collection of grid squares (e.g.,
[SHT10]); see Figure 6(i). These detailed approaches are appropriate when
folks are interested in the spatial extent of agents. Representing each
pedestrian with $N>1$ grid squares, instead of just one as in Figure 6(f),
increases the number of parameters and the time that it will take to simulate
the model. This means fine-grid cellular automaton may make more sense when
the goal is to describe the behavior of a few pedestrians in a detailed way;
as we consider a larger crowd, agent-based or cellular automaton models become
more appropriate; and, as we zoom out further into very large, densely packed
crowds, macroscopic models are especially helpful. In crowd dynamics, as for
other complex social systems, there are many useful modeling approaches that
we could take, and it is a matter of choosing one that is parsimonious and
appropriate for our goals.

##  6\. Conclusions

I conclude with the best piece of advice that I have been given as a modeler:
don’t be afraid to be wrong. In particular, developing a model that correctly
describes all of the unknown, intricate details of a complex social system
would come down to sheer luck, since the space of possible models is huge.
This can be discouraging. Instead, I have found it freeing to recognize that
all of my models have been and will continue to be “wrong” in some sense. What
matters is getting it “wrong” in a meaningful way. By building a parsimonious
model, balancing our assumptions with the amount of data available, and
designing a clear method for testing the model, we can make a meaningful
contribution and generate new insights despite being inevitably “wrong” (or
“right” in a simplified way). If the first model of a complex system does not
cross disciplinary boundaries, it can lay the groundwork for a bridge that
brings disciplines together in the future. Whether our starting point is a
rich data set or a nearly blank space, modeling complex systems is an
iterative, creative, and interdisciplinary process. It involves being aware of
the choices that we are making to simplify the problem, choosing model
complexity based on our data, carefully considering the bias in the data and
model, and identifying a plan for model building and validation. Through data
collection, model development, prediction, communication, and generating new
questions, we can push the field forward, help address societal challenges,
develop mathematical approaches, and bring disciplines together in new ways.

## References

  * [AA15] G Ariel and A Ayali, _Locust collective motion and its modeling_ , PLOS Comput Biol 11 (2015), no. 12, e1004522.
  * [AAA+21] T Alshaabi, J L Adams, M V Arnold, J R Minot, D R Dewhurst, A J Reagan, C M Danforth, and P S Dodds, _Storywrangler: A massive exploratorium for sociolinguistic, cultural, socioeconomic, and political timelines using Twitter_ , Sci Adv 7 (2021), no. 29, eabe6534.
  * [Abr88] A I Abramowitz, _An improved model for predicting presidential election outcomes_ , PS Political Sci Politics 21 (1988), no. 4, 843–847.
  * [Abr08] by same author, _Forecasting the 2008 Presidential Election with the Time-for-Change Model_ , PS Political Sci Politics 41 (2008), no. 4, 691–695.
  * [Aka74] H Akaike, _A new look at the statistical model identification_ , IEEE Trans Automat Contr 19 (1974), no. 6, 716–723.
  * [Aka98] by same author, _Information theory and an extension of the maximum likelihood principle_ , Selected Papers of Hirotugu Akaike (E Parzen, K Tanabe, and G Kitagawa, eds.), Springer, New York, 1998, pp. 199–213.
  * [AQO+20] E J Amézquita, M Y Quigley, T Ophelders, E Munch, and D H Chitwood, _The shape of things to come: Topological Data Analysis and biology, from molecules to organisms_ , Dev Dyn 249 (2020), no. 7, 816–833.
  * [BA01] V J Blue and J L Adler, _Cellular automata microsimulation for modeling bi-directional pedestrian walkways_ , Transp Res B: Methodol 35 (2001), no. 3, 293–312.
  * [BBG+22] R Best, A Bycoffe, C Groskopf, R King, E Koeze, D Mehta, J Mithani, M Radcliffe, A Wiederkehr, J Wolfe, A Jones-Rooy, N Rakich, D Shan, S Frostenson, J Mason, A Mangan, and C Yee, _FiveThirtyEight: Latest Polls_ , https://projects.fivethirtyeight.com/polls/, 2022, Last accessed: 06-30-2022.
  * [BBK13] N Bellomo, A Bellouquid, and D Knopoff, _From the microscale to collective crowd dynamics_ , Multiscale Model Simul 11 (2013), no. 3, 943–963.
  * [BCAKCC06] L M A Bettencourt, A Cintrón-Arias, D I Kaiser, and C Castillo-Chavéz, _The power of a good idea: Quantitative modeling of the spread of ideas from epidemiological models_ , Physica A 364 (2006), 513–536.
  * [BCC+08] M Ballerini, N Cabibbo, R Candelier, A Cavagna, E Cisbani, I Giardina, A Orlandi, G Parisi, A Procaccini, M Viale, and V Zdravkovic, _Empirical investigation of starling flocks: a benchmark study in collective animal behaviour_ , Anim Behav 76 (2008), no. 1, 201–215.
  * [BCC12] R Brauer and C Castillo-Chavez, _Mathematical Models in Population Biology and Epidemiology_ , 2nd ed., Springer-Verlag, Heidelberg, Germany, 2012\.
  * [BCD18] R Bailo, J A Carrillo, and P Degond, _Pedestrian models based on rational behaviour_ , Crowd Dynamics, Volume 1: Theory, Models, and Safety Problems (L Gibelli and N Bellomo, eds.), Springer, Cham, 2018, pp. 259–292.
  * [BCG+16] N Bellomo, D Clarke, L Gibelli, P Townsend, and B J Vreugdenhil, _Human behaviours in evacuation crowd dynamics: From modelling to “big data” toward crisis management_ , Phys Life Rev 18 (2016), 1–21.
  * [BCME+20] A J Bernoff, M Culshaw-Maurer, R A Everett, M E Hohn, W C Strickland, and J Weinburd, _Agent-based and continuous models of hopper bands for the Australian plague locust: How resource consumption mediates pulse formation and geometry_ , PLOS Comput Biol 16 (2020), no. 5, e1007820.
  * [BCT20] L L Bonilla, A Carpio, and C Trenado, _Tracking collective cell motion by topological data analysis_ , PLOS Comp Biol 16 (2020), no. 12, e1008407.
  * [BD11] N Bellomo and C Dogbe, _On the modeling of traffic and crowds: A survey of models, speculations, and perspectives_ , SIAM Rev 53 (2011), no. 3, 409–463.
  * [BdA17] D Braha and M A M de Aguiar, _Voting contagion: Modeling and analysis of a century of U.S. presidential elections_ , PLOS ONE 12 (2017), no. 5, e0177970.
  * [BDM+09] T Bosse, R Duell, Z A Memon, J Treur, C N van der Wal, J Otamendi, A Bargiela, J L Montes, and L M D Peera, _A Multi-Agent Model for Mutual Absorption of Emotions_ , Proceedings of the 23rd European Conference on Modelling and Simulation (ECMS’09), European Council on Modeling and Simulation, 2009, pp. 212–218.
  * [BEK20] A Buttenschön and L Edelstein-Keshet, _Bridging from single to collective cell migration: A review of models and links to experiments_ , PLOS Comput Biol 16 (2020), no. 12, e1008411.
  * [BFG14] K M Bliss, K R Fowler, and B J Galluzzo, _Math Modeling: Getting Started and Getting Solutions_ , SIAM, Philadelphia, https://m3challenge.siam.org/resources/modeling-handbook, 2014.
  * [BGBD+18] L Bonnasse-Gahot, H Berestycki, M-A Depuiset, M B Gordon, S Roché, N Rodriguez, and J-P Nadal, _Epidemiological modelling of the 2005 French riots: a spreading wave and the role of contagion_ , Sci Rep 8 (2018), no. 107.
  * [BHK+11] T Bosse, M Hoogendoorn, M C A Klein, J Treur, and C N van der Wal, _Agent-based analysis of patterns in crowd behaviour involving contagion of mental states_ , Lecture Notes in Computer Science, vol. 6704, Springer, Berlin Heidelberg, 2011, pp. 566–577.
  * [BHN+95] M Bando, K Hasebe, A Nakayama, A Shibata, and Y Sugiyama, _Dynamical model of traffic congestion and numerical simulation_ , Phys Rev E 51 (1995), no. 2, 1035–1042.
  * [BIR] _The University of British Columbia: BIRS Workshop Lecture Videos_ , https://open.library.ubc.ca/cIRcle/collections/48630, Last accessed: 06-30-2022.
  * [BK19] S L Brunton and J N Kutz, _Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control_ , Cambridge University Press, Cambridge, United Kingdom, 2019.
  * [BKGL18] K M Bliss, K F Kavanagh, B J Galluzzo, and R Levy, _Math Modeling: Computing and Communicating_ , SIAM, Philadelphia, https://m3challenge.siam.org/resources/modeling-handbook, 2018.
  * [BKSZ01] C Burstedde, K Klauck, A Schadschneider, and J Zittartz, _Simulation of pedestrian dynamics using a two-dimensional cellular automaton_ , Physica A 295 (2001), no. 3, 507–525.
  * [BMM+19] D Bhaskar, A Manhart, J Milzman, J T Nardini, K M Storey, C M Topaz, and L Ziegelmeier, _Analyzing collective motion with machine learning and topology_ , Chaos 29 (2019), 123125.
  * [Boc10] N Boccara, _Modeling complex systems_ , 2nd ed. ed., Graduate Texts in Physics, Springer, New York, NY, 2010.
  * [BP20a] H Z Brooks and M A Porter, _A model for the influence of media on the ideology of content in online social networks_ , Phys Rev Research 2 (2020), no. 2, 023041.
  * [BP20b] by same author, _Topological Data Analysis and Data-Driven Modeling in Complex Systems: A Minisymposium in the 2020 SIAM Conference on Mathematics of Data Science_ , https://www.youtube.com/playlist?list=PLnzqyg_akFM1U4KVL0E5IlhyVjAsvJ2Od, 2020, Last accessed: 06-30-2022.
  * [BPK16] S L Brunton, J L Proctor, and J N Kutz, _Discovering governing equations from data by sparse identification of nonlinear dynamical systems_ , Proc Natl Acad Sci USA 113 (2016), no. 15, 3932–3937.
  * [BPR18] A E Biondo, A Pluchino, and A Rapisarda, _Modeling surveys effects in political competitions_ , Phys A: Stat Mech Appl 503 (2018), 714–726.
  * [BR06] O Bandiera and I Rasul, _Social networks and technology adoption in northern Mozambique_ , Econ J 116 (2006), no. 514, 869–902.
  * [BR19] N W F Bode and E Ronchi, _Statistical model fitting and model selection in pedestrian dynamics research_ , Collective Dynamics 4 (2019), 1–32.
  * [Bro21] H Z Brooks, _Networks in Social Systems_ , https://zerodivzero.com/short_course/aaac8c66007a4d23a7aa14857a3b778c/title/5dd029b5e02146d1926c17d5184d8b63, 2021, Last accessed: 09-09-2022.
  * [Bro22] D Brockmann, _Complexity Exporables_ , https://www.complexity-explorables.org, 2022, Last accessed 06-17-2022.
  * [BRSW15] A L Bertozzi, J Rosado, M B Short, and L Wang, _Contagion shocks in one dimension_ , J Stat Phys 158 (2015), no. 3, 647–664.
  * [Bru] S Brunton, _Data-Driven Dynamical Systems with Machine Learning_ , https://www.youtube.com/playlist?list=PLMrJAkhIeNNR6DzT17-MM1GHLkuYVjhyt, Last accessed: 06-30-2022.
  * [BT11] A J Bernoff and C M Topaz, _A primer of swarm equilibria_ , SIAM J Appl Dyn Sys 10 (2011), no. 1, 212–250.
  * [BV05] M Bodnar and J J L Velazquez, _Derivation of macroscopic equations for individual cell-based models: a formal approach_ , Math Methods Appl Sci 28 (2005), no. 15, 1757–1779.
  * [Car20] G Carlsson, _Topological methods for data modelling_ , Nat Rev Phys 2 (2020), 697–708.
  * [CDFMG+21] M Cinelli, G De Francisci Morales, A Galeazzi, W Quattrociocchi, and M Starnini, _The echo chamber effect on social media_ , Proc Natl Acad Sci USA 118 (2021), no. 9, e2023301118.
  * [CDM+07] Y Chuang, M R D’Orsogna, D Marthaler, A L Bertozzi, and L S Chayes, _State transitions and the continuum limit for a 2D interacting, self-propelled particle system_ , Physica D 232 (2007), 33–47.
  * [CFL09] C Castellano, S Fortunato, and V Loreto, _Statistical physics of social dynamics_ , Rev Mod Phys 81 (2009), no. 2, 591–646.
  * [CFPSS19] W Cota, S C Ferreira, R Pastor-Satorras, and M Starnini, _Quantifying echo chamber effects in information spreading over political communication networks_ , EPJ Data Science 8 (2019), no. 35, 1–13.
  * [CJDM21] M-V Ciocanel, R Juenemann, A T Dawes, and S A McKinley, _Topological data analysis approaches to uncovering the timing of ring structure onset in filamentous networks_ , Bull Math Biol 83 (2021), no. 21.
  * [CKJ+02] I D Couzin, J Krause, R James, G D Ruxton, and N R Franks, _Collective memory and spatial sorting in animal groups_ , J Theor Biol 218 (2002), no. 1, 1–11.
  * [CMW16] J A Carrillo, S Martin, and M-T Wolfram, _An improved version of the Hughes model for pedestrian flow_ , Math Models Methods Appl Sci 26 (2016), no. 04, 671–697.
  * [CP21] J A Carrillo and Choi Y P, _Mean-field limits: From particle descriptions to macroscopic equations_ , Arch Ration Mech Anal 241 (2021), no. 3, 1529–1573.
  * [CR13] J D Clinton and S Rogers, _Robo-polls: Taking cues from traditional sources?_ , PS Political Sci Politics 46 (2013), no. 2, 333–337.
  * [Cre16] K Cressman, _Chapter 4.2 - desert locust_ , Biological and Environmental Hazards, Risks, and Disasters (J F Shroder and R Sivanpillai, eds.), Academic Press, Boston, 2016, pp. 87–105.
  * [CTS+20] M-V Ciocanel, C M Topaz, R Santorella, S Sen, C M Smith, and A Hufstetler, _JUSTFAIR: Judicial System Transparency through Federal Archive Inferred Records_ , PLOS ONE 15 (2020), no. 10, e0241381.
  * [CY16] P Cihon and T Yasseri, _A Biased Review of Biases in Twitter Studies on Political Collective Action_ , Front Phys 4 (2016).
  * [DAB+20] S Dodson, B Abrahms, S J Bograd, J Fiechter, and E L Hazen, _Disentangling the biotic and abiotic drivers of emergent migratory behavior using individual-based models_ , Ecol Model 432 (2020), 109225.
  * [DBC+19] M De Domenico, D Brockmann, C Camargo, C Gershenson, D Goldsmith, S Jeschonnek, L Kay, S Nichele, J R Nicolás, T Schmickl, M Stella, J Brandoff, A J Martínez Salinas, and H Sayama, _Complexity Explained_ , https://complexityexplained.github.io, 2019, Last accessed 10-15-2022.
  * [DBG18] S Dini, B J Binder, and J E F Green, _Understanding interactions between populations: Individual based modelling and quantification using pair correlation functions_ , J Theor Biol 439 (2018), 50–64.
  * [DCBC06] M R D’Orsogna, Y L Chuang, A L Bertozzi, and L S Chayes, _Self-propelled particles with soft-core interactions: Patterns, stability, and collapse_ , Phys Rev Lett 96 (2006), 104302.
  * [DDH13] D C Duives, W Daamen, and S P Hoogendoorn, _State-of-the-art crowd motion simulation models_ , Transp Res C: Emerg Technol 37 (2013), 193–209.
  * [DeF21] D R DeFord, _Python Tutorial on Networks_ , https://zerodivzero.com/short_course/aaac8c66007a4d23a7aa14857a3b778c/title/628602c8994746e491872a9380676b62, 2021, Last accessed: 09-09-2022.
  * [DH00] O Diekmann and J A P Heesterbeek, _Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation_ , John Wiley & Sons, Inc., New York City, NY, USA, 2000.
  * [DNAW00] G Deffuant, D Neau, F Amblard, and G Weisbuch, _Mixing beliefs among interacting agents_ , Adv Complex Syst 3 (2000), no. 1n04, 87–98.
  * [Dog10] C Dogbé, _Modeling crowd dynamics by the mean-field limit approach_ , Math Comput Model 52 (2010), no. 9, 1506–1520.
  * [EF18] T Evans and F Fu, _Opinion formation on dynamic networks: identifying conditions for the emergence of partisan echo chambers_ , Royal Soc Open Sci 5 (2018), 181122.
  * [eGH20] _The Economist_ , A Gelman, and M Heidemanns, _Forecasting the US elections: How The Economist presidential forecast works_ , https://projects.economist.com/us-2020-forecast/president/how-this-works, 2020, Last accessed: 09-07-2022.
  * [EH08] H Edelsbrunner and J Harer, _Persistent homology – a survey_ , Contemp Math 453 (2008), 257–282.
  * [epS11] epSos.de, _Driving cars in a traffic jam_ , Available at Wikimedia Commons: https://commons.wikimedia.org/wiki/File:Driving_Cars_in_a_Traffic_Jam.jpg, 2011, Accessed 10-12-2022.
  * [FGSR+14] J Fernández-Gracia, K Suchecki, J J Ramasco, M San Miguel, and V M Eguiluz, _Is the Voter Model a Model for Voters?_ , Phys Rev Lett 112 (2014), 158701.
  * [FHP22] M Feng, A Hickok, and M A Porter, _Topological data analysis of spatial systems_ , Higher-Order Systems: Understanding Complex Systems (F Battiston and G Petri, eds.), Springer, Cham, 2022, pp. 389–399.
  * [FPS+21] S L Franconeri, L M Padilla, P Shah, J M Zacks, and J Hullman, _The science of visual data communication: What works_ , Psychol Sci Public Interest 22 (2021), no. 3, 110–161.
  * [FTW18] A Festa, A Tosin, and M Wolfram, _Kinetic description of collision avoidance in pedestrian crowds by sidestepping_ , Kinet Relat Models 11 (2018), no. 3, 491.
  * [GAI19] _GAIMME: Guidelines for Assessment and Instruction in Mathematical Modeling Education, Second Edition_ , S Garfunkel and M Montgomery (eds.), COMAP and SIAM, Philadelphia, https://m3challenge.siam.org/resources/teaching-modeling, 2019.
  * [GBC18] D Guilbeault, J Becker, and D Centola, _Complex contagions: A decade in review_ , Complex Spreading Phenomena in Social Systems. Computational Social Sciences (S Lehmann and Y Y Ahn, eds.), Springer, Cham, 2018, pp. 3–25.
  * [GBKM20] R Gini$\bar{\text{u}}$nait$\dot{\text{e}}$, R E Baker, P M Kulesa, and P K Maini, _Modelling collective cell migration: neural crest as a model paradigm_ , J Math Biol 80 (2020), 481–504.
  * [GG93] J A Glazier and F Graner, _Simulation of the differential adhesion driven rearrangement of biological cells_ , Phys Rev E 47 (1993), 2128–2154.
  * [GHWM20] A Gelman, J Hullman, C Wlezien, and G E Morris, _Information, incentives, and goals in election forecasts_ , Judgm Decis Mak 15 (2020), no. 5, 863–880.
  * [GK93] A Gelman and G King, _Why are American Presidential Election Campaign Polls So Variable When Votes Are So Predictable?_ , Br J Political Sci 23 (1993), 409–451.
  * [HBJW05] D Helbing, L Buzna, A Johansson, and T Werner, _Self-organized pedestrian crowd dynamics: Experiments, simulations, and design solutions_ , Transp Sci 39 (2005), no. 1, 1–24.
  * [Het00] H W Hethcote, _The Mathematics of Infectious Diseases_ , SIAM Rev 42 (2000), no. 4, 599–653.
  * [HJ09] D Helbing and A Johansson, _Pedestrian, crowd and evacuation dynamics_ , Encyclopedia of Complexity and Systems Science (R A Meyers, ed.), Springer New York, 2009, pp. 6476–6495.
  * [HJJ+22] A Hickok, B Jarman, M Johnson, J Luo, and M A Porter, _Persistent homology for resource coverage: A case study of access to polling sites_ , https://arxiv.org/abs/2206.04834, 2022.
  * [HK02] R Hegselmann and U Krause, _Opinion dynamics and bounded confidence: Models, analysis and simulation_ , J Artif Soc Soc Simul 5 (2002), no. 3.
  * [HLW16] J Humpherys, R Levy, and T Witelski, _Directions for Graduate and Undergraduate Modeling Courses_ , https://www.pathlms.com/siam/courses/3028/sections/4132, 2016, Last accessed 10-13-2022.
  * [HM95] D Helbing and P Molnár, _Social force model for pedestrian dynamics_ , Phys Rev E 51 (1995), 4282–4286.
  * [HR14] P Hummel and D Rothschild, _Fundamental models for forecasting elections at the state level_ , Elect Stud 35 (2014), 123–139.
  * [HRM17] T Hirashima, E G Rens, and R M H Merks, _Cellular Potts modeling of complex multicellular behaviors in tissue morphogenesis_ , Dev Growth Differ 59 (2017), no. 5, 329–339.
  * [Huf22a] _HuffPost Pollster_ , https://elections.huffingtonpost.com/pollster, (2022), Last accessed: 10-16-2022.
  * [Huf22b] _HuffPost Pollster API v2_ , https://elections.huffingtonpost.com/pollster/api/v2, 2022, Last accessed: 10-16-2022.
  * [IPBL19] I Iacopini, G Petri, A Barrat, and V Latora, _Simplicial models of social contagion_ , Nat Commun 10 (2019), no. 1, 2485.
  * [Jac05] S Jackman, _Pooling the polls over an election campaign_ , Aust J Political Sci 40 (2005), no. 4, 499–517.
  * [JC19] S T Johnston and E J Crampin, _Corrected pair correlation functions for environments with obstacles_ , Phys Rev E 99 (2019), 032124.
  * [JHZ+14] R Jiang, M-B Hu, H M Zhang, Z-Y Gao, B Jia, Q-S Wu, B Wang, and M Yang, _Traffic experiment reveals the nature of car-following_ , PLOS ONE 9 (2014), no. 4, e94351.
  * [KBBP16] J N Kutz, S L Brunton, B W Brunton, and J L Proctor, _Dynamic mode decomposition: data-driven modeling of complex systems_ , Other Titles in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 2016.
  * [KBF17] J Kursawe, R E Baker, and A G Fletcher, _Impact of implementation choices on quantitative predictions of cell-based computational models_ , J Comp Phys 345 (2017), 752–767.
  * [KBT+22] F P Kemeth, T Bertalan, T Thiem, F Dietrich, S J Moon, C R Laing, and I G Kevrekidis, _Learning emergent partial differential equations in a learned emergent space_ , Nat Commun 13 (2022), no. 1, 3318–3318.
  * [KHB13] A Kneidl, D Hartmann, and A Borrmann, _A hybrid multi-scale approach for simulation of pedestrian dynamics_ , Transp Res C: Emerg Technol 37 (2013), 223–237.
  * [KM27] W O Kermack and A G McKendrick, _A contribution to the mathematical theory of epidemics_ , Proc R Soc London 115 (1927), no. 772, 700–721.
  * [KM32] by same author, _Contributions to the mathematical theory of epidemics. II. –The problem of endemicity_ , Proc R Soc London 138 (1932), no. 834, 55–83.
  * [KM33] by same author, _Contributions to the mathematical theory of epidemics. III. –Further studies of the problem of endemicity_ , Proc R Soc London 141 (1933), no. 843, 94–122.
  * [KS20] M Kalt and D Scott, _Twitter Data Curation Primer: Data Curation Network GitHub Repository_ , https://github.com/DataCurationNetwork/data-primers/blob/master/Twitter%20Data%20Curation%20Primer/twitter-data-curation-primer.md, 2020, Last accessed: 10-16-2022.
  * [KTI+11] Y Katz, K Tunstrom, C C Ioannou, C Huepe, and I D Couzin, _Inferring the structure and dynamics of interactions in schooling fish_ , Proc Natl Acad Sci USA 108 (2011), no. 46, 18720–18725.
  * [Kut] N Kutz, _Nathan Kutz Videos_ , https://www.youtube.com/c/NathanKutzAMATH/videos, Last accessed: 06-30-2022.
  * [Kut13] J N Kutz, _Data-Driven Modeling & Scientific Computation: Methods for Complex Systems & Big Data_, Oxford University Press, Oxford, England, 2013\.
  * [LBBH98] Y Lecun, L Bottou, Y Bengio, and P Haffner, _Gradient-based learning applied to document recognition_ , Proc IEEE 86 (1998), no. 11, 2278–2324.
  * [Lin13] D A Linzer, _Dynamic Bayesian Forecasting of Presidential Elections in the States_ , J Am Stat Assoc 108 (2013), no. 501, 124–134.
  * [LLEK10] R Lukeman, Y-X Li, and L Edelstein-Keshet, _Inferring individual rules from collective behavior_ , Proc Natl Acad Sci USA 107 (2010), no. 28, 12576–12580.
  * [Loc] _Plague of Locusts Timelapse — Wild Africa — BBC Earth_ , https://www.youtube.com/watch?v=1YNy2R3hg2Q, Uploaded by BBC Earth on 08-21-2009. Last accessed 06-17-2022.
  * [LRC01] H Levine, W J Rappel, and I Cohen, _Self-organization in systems of self-propelled particles_ , Phys Rev E 63 (2001), 017101.
  * [LSZ15] K Law, A Stuart, and K Zygalakis, _Data Assimilation: A Mathematical Introduction_ , Texts in Applied Mathematics, vol. 63, Springer, Cham, 2015.
  * [LW11] A Lachapelle and M-T Wolfram, _On a mean field game approach modeling congestion and aversion in pedestrian crowds_ , Transp Res B: Methodol 45 (2011), no. 10, 1572–1589.
  * [LYY18] S Lehmann and A Yong-Yeol, _Complex Spreading Phenomena in Social Systems: Influence and Contagion in Real-World Social Networks_ , Springer, Cham, 2018.
  * [MBN+16] Mirrorme22, Brythones, Nilfanion, TUBS, and Sting, _United Kingdom EU referendum 2016 area_ , Available at Wikimedia Commons: https://commons.wikimedia.org/wiki/File:United_Kingdom_EU_referendum_2016_area_results.svg, 2016, Accessed 10-12-2022.
  * [MBPK16] N M Mangan, S L Brunton, J L Proctor, and J N Kutz, _Inferring biological networks by sparse identification of nonlinear dynamics_ , IEEE Trans Mol Biol Multi-Scale Commun 2 (2016), no. 1, 52–63.
  * [MCM+22] J R Minot, N Cheney, M Maier, D C Elbers, C M Danforth, and P S Dodds, _Interpretable bias mitigation for textual data: Reducing genderization in patient notes while maintaining classification performance_ , ACM Trans Comput Healthcare (2022), Just Accepted.
  * [MEK99] A Mogilner and L Edelstein-Keshet, _A non-local model for a swarm_ , J Math Biol 38 (1999), 534–570.
  * [Mit09] M Mitchell, _Complexity: A Guided Tour_ , Oxford University Press, 2009.
  * [MJ] M Madhav and E Johnson, _What do Your Data Say? A course to help you better understand your data_ , https://www.whatdoyourdatasay.com, Last accessed: 06-30-2022.
  * [MKBP17] N M Mangan, J N Kutz, S L Brunton, and J L Proctor, _Model selection for dynamical systems via sparse regression and information criteria_ , Proc R Soc A 473 (2017), 20170009.
  * [MP07] J H Miller and S E Page, _Complex Adaptive Systems: An Introduction to Computational Models of Social Life_ , Princeton Studies in Complexity, Princeton University Press, Princeton, 2007.
  * [MPLC14] F Morstatter, J Pfeffer, H Liu, and K M Carley, _Is the Sample Good Enough? Comparing Data from Twitter’s Streaming API with Twitter’s Firehose_ , Proceedings of the Seventh International AAAI Conference on Weblogs and Social Media, 2014, pp. 400–408.
  * [MVS20] M R McGuirl, A Volkening, and B Sandstede, _Topological data analysis of zebrafish patterns_ , Proc Natl Acad Sci USA 117 (2020), no. 10, 5113–5124.
  * [NBSF21] J T Nardini, R E Baker, M J Simpson, and K B Flores, _Learning differential equation models from stochastic agent-based model simulations_ , J R Soc Interface 18 (2021), 20200987.
  * [New10] M E J Newman, _Networks: An Introduction_ , Oxford University Press, Oxford, 2010.
  * [New11] by same author, _Complex Systems: A Survey_ , Am J Phys 79 (2011), no. 8, 800–810.
  * [NS92] K Nagel and M Schreckenberg, _A cellular automaton model for freeway traffic_ , J Phys, I 2 (1992), no. 12, 2221–2229.
  * [NSF+21] J T Nardini, B J Stolz, K B Flores, H A Harrington, and H M Byrne, _Topological data analysis distinguishes parameter regimes in the Anderson-Chaplain model of angiogenesis_ , PLOS Comput Biol 17 (2021), e1009094.
  * [OPT+17] N Otter, M A Porter, U Tillmann, P Grindrod, and H A Harrington, _A roadmap for the computation of persistent homology_ , EPJ Data Science 6 (2017), no. 17.
  * [OT12] E Oster and R Thornton, _Determinants of technology adoption: peer effects in menstrual cup take-up_ , J Eur Econ Assoc 10 (2012), no. 6, 1263–1293.
  * [PEK99] J K Parrish and L Edelstein-Keshet, _Complexity, pattern, and evolutionary trade-offs in animal aggregation_ , Science 284 (1999), no. 5411, 99–101.
  * [PG16] M A Porter and J P Gleeson, _Dynamical Systems on Networks: A Tutorial_ , Front Appl Dyn Syst Rev Tutor 4 (2016).
  * [PMS17] H Peng, F Menczer, and K Sasahara, _EchoDemo: How echo chambers emerge from social media_ , https://osome.iu.edu/demos/echo/, 2017, Last accessed 06-17-2022.
  * [Por22] M A Porter, _A non-expert’s introduction to data ethics for mathematicians_ , https://arxiv.org/abs/2201.07794, 2022.
  * [PT03] D M Parichy and J M Turner, _Temporal and cellular requirements for Fms signaling during zebrafish adult pigment pattern development_ , Development 130 (2003), no. 5, 817–833.
  * [QSI] _Institute for the Quantitative Study of Inclusion, Diversity, and Equity_ , https://qsideinstitute.org, Last accessed: 06-30-2022.
  * [Rea22] _RealClearPolitcs: Polls_ , https://www.realclearpolitics.com/epolls/latest_polls/elections/, (2022), Last accessed: 10-13-2022.
  * [RRH09] J M Restrepo, R C Rael, and J M Hyman, _Modeling the influence of polls on elections: a population dynamics approach_ , Public Choice 140 (2009), 395–420.
  * [SCDM+18] R E Stern, S Cui, M L Delle Monache, R Bhadani, M Bunting, M Churchill, N Hamilton, R Haulcy, H Pohlmann, F Wu, B Piccoli, B Seibold, J Sprinkle, and D B Work, _Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments_ , Transp Res Part C Emerg Technol 89 (2018), 205–221.
  * [Sch73] T C Schelling, _Hockey helmets, concealed weapons, and daylight saving: A study of binary choices with externalities_ , J Conflict Resolut 17 (1973), no. 3, 381–428.
  * [Sch78] G Schwarz, _Estimating the dimension of a model_ , Ann Stat 6 (1978), no. 2, 461–464.
  * [SCP+21] K Sasahara, W Chen, H Peng, G L Ciampaglia, A Flammini, and F Menczer, _Social influence and unfollowing accelerate the emergence of echo chambers_ , J Comput Soc Sci 4 (2021), 381–402.
  * [SCS+18] A Schadschneider, M Chraibi, A Seyfried, A Tordeux, and J Zhang, _Pedestrian dynamics: From empirical results to modeling_ , pp. 63–102, Springer, Cham, 2018.
  * [SEK13] L A Segel and L Edelstein-Keshet, _A Primer on Mathematical Models in Biology_ , Society for Industrial and Applied Mathematics, Philadelphia, PA, 2013\.
  * [SFK+08] Y Sugiyama, M Fukui, M Kikuchi, K Hasebe, A Nakayama, K Nishinari, S Tadaki, and S Yukawa, _Traffic jams without bottlenecks—experimental evidence for the physical mechanism of the formation of a jam_ , New J Phys 10 (2008), 033001.
  * [SFK16] J Schleuss, J Fox, and P Krishnakumar, _California 2016 election precinct maps_ , https://github.com/datadesk/california-2016-election-precinct-maps, (2016), Accessed: 03-14-2019.
  * [Shm10] G Shmueli, _To Explain or to Predict?_ , Statist Sci 25 (2010), no. 3, 289–310.
  * [SHP16] B J Stolz, H A Harrington, and M A Porter, _The Topological “Shape” of Brexit_ , https://arxiv.org/abs/1610.00752, 2016.
  * [SHT10] S Sarmady, F Haron, and A Z Talib, _Simulating crowd movements using fine grid cellular automata_ , 2010 12th International Conference on Computer Modelling and Simulation, IEEE, 2010, pp. 428–433.
  * [Sil12] N Silver, _The signal and the noise: Why so many predictions fail — but some don’t_ , Penguin Press, New York City, NY, USA, 2012.
  * [Sil14] by same author, _FiveThirtyEight: Here’s Proof Some Pollsters Are Putting A Thumb On The Scale_ , https://fivethirtyeight.com/features/heres-proof-some-pollsters-are-putting-a-thumb-on-the-scale/, 2014, Last accessed 06-17-2022.
  * [SMA20] D Sabin-Miller and D M Abrams, _When pull turns to shove: A continuous-time model for opinion dynamics_ , Phys Rev Researchf 2 (2020), 043001.
  * [Smi14] R C Smith, _Uncertainty Quantification: Theory, Implementation, and Applications_ , Computational science & engineering series, Society for Industrial and Applied Mathematics, Philadelphia, 2014.
  * [SNV15] A P Singh and C Nüsslein-Volhard, _Zebrafish stripes as a model for vertebrate colour pattern formation_ , Curr Biol 25 (2015), R81–R92.
  * [SSS17] A Sieben, J Schumann, and A Seyfried, _Collective phenomena in crowds–Where pedestrian dynamics need social psychology_ , PLOS ONE 12 (2017), no. 6, e0177328.
  * [Sto] _storywrangler: From the University of Vermont Computational Story Lab_ , https://storywrangling.org, Last accessed: 06-30-2022.
  * [Str01] S H Strogatz, _Exploring complex networks_ , Nature 410 (2001), no. 6825, 268–276.
  * [Str15] by same author, _Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering_ , 2nd ed. ed., CRC Press, 2015.
  * [TBL06] C M Topaz, A L Bertozzi, and M A Lewis, _A nonlocal continuum model for biological aggregation_ , Bull Math Biol 68 (2006), 1601–1623.
  * [TECP20] J H Tien, M C Eisenberg, S T Cherng, and M A Porter, _Online reactions to the 2017 ‘Unite the right’ rally in Charlottesville: measuring polarization in Twitter networks using media followership_ , Appl Netw Sci 5 (2020), no. 10.
  * [TFB+11] J Tsai, N Fridman, E Bowring, M Brown, S Epstein, G Kaminka, S Marsella, A Ogden, I Rika, A Sheel, M Taylor, X Wang, A Zilka, and M Tambe, _ESCAPES - Evacuation simulation with children, authorities, parents, emotions, and social comparison_ , vol. 1, 2011, pp. 457–464.
  * [THK18] S Thurner, R Hanel, and P Klimek, _Introduction to the Theory of Complex Systems_ , Oxford University Press, 2018.
  * [Tib96] R Tibshirani, _Regression shrinkage and selection via the lasso_ , J R Stat Soc Ser B Methodol 58 (1996), no. 1, 267–288.
  * [TSB+14] K K Treloar, M J Simpson, B J Binder, D L S McElwain, and R E Baker, _Assessing the role of spatial correlations during collective cell spreading_ , Sci Rep 4 (2014), no. 1, 5713.
  * [Tuf14] Z Tufekci, _Big questions for social media big data: Representativeness, validity and other methodological pitfalls_ , Proceedings of the Eighth International AAAI Conference on Weblogs and Social Media, 2014, pp. 505–514.
  * [Twi] _Twitter Developer Platform: Tutorials_ , https://developer.twitter.com/en/docs/tutorials, Last accessed: 06-30-2022.
  * [TZH15] C M Topaz, L Ziegelmeier, and T Halverson, _Topological data analysis of biological aggregation models_ , PLOS ONE 10 (2015), no. 5, e0126383.
  * [UZT19] M Ulmer, L Ziegelmeier, and C M Topaz, _A topological approach to selecting models of biological experiments_ , PLOS ONE 14 (2019), no. 3, e0213679.
  * [VCBJ+95] T Vicsek, A Czirók, E Ben-Jacob, I Cohen, I, and O Shochet, _Novel type of phase transition in a system of self-driven particles_ , Phys Rev Lett 75 (1995), no. 6, 1226–1229.
  * [VCM+07] A Varas, M D Cornejo, D Mainemer, B Toledo, J Rogan, V Muñoz, and J A Valdivia, _Cellular automaton model for evacuation process with obstacles_ , Physica A 382 (2007), no. 2, 631–642.
  * [VLPR20] A Volkening, D F Linder, M A Porter, and G A Rempala, _Forecasting elections using compartmental models of infection_ , SIAM Rev 62 (2020), no. 4, 837–865.
  * [Vol17] A Volkening, _Modeling pattern formation on zebrafish_ , Ph.D. thesis, Brown University, 2017.
  * [Vol20a] A Volkening, _Intro to building models_ , https://northwestern.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=7d04a874-a292-4ff2-bd66-ac2500daeea1, 2020, Last accessed: 06-30-2022.
  * [Vol20b] A Volkening, _Linking genotype, cell behavior, and phenotype: multidisciplinary perspectives with a basis in zebrafish patterns_ , Curr Opin Genet Dev 63 (2020), 78–85.
  * [Vol21] by same author, _Data-Driven Modeling_ , https://zerodivzero.com/short_course/aaac8c66007a4d23a7aa14857a3b778c/title/d56faebff3a24f77a76085c1427038d8, 2021, Last accessed: 06-30-2022.
  * [VS15] A Volkening and B Sandstede, _Modelling stripe formation in zebrafish: an agent-based approach_ , J R Soc Interface 12 (2015), no. 112, 20150812\.
  * [VS18] by same author, _Iridophores as a source of robustness in zebrafish stripes and variability in _Danio_ patterns_, Nat Commun 9 (2018), no. 3231.
  * [WE02] C Wlezien and R S Erikson, _The Timeline of Presidential Election Campaigns_ , J Politics 64 (2002), no. 4, 969–993.
  * [WPCG+14] C H Weiss, J Poncela-Casasnovas, J I Glaser, A R Pah, S D Persell, D W Baker, R G Wunderink, and L A Nunes Amaral, _Adoption of a high-impact innovation in a homogeneous population_ , Phys Rev X 4 (2014), no. 4, 041008\.
  * [YAKM20] V C Yang, D M Abrams, G Kernell, and A E Motter, _Why Are U.S. Parties So Polarized? A “Satisficing” Dynamical Model_ , SIAM Rev 62 (2020), no. 3, 646–657.
  * [ZKSS12] J Zhang, W Klingsch, A Schadschneider, and A Seyfried, _Ordering in bidirectional pedestrian flows and its influence on the fundamental diagram_ , J Stat Mech 2012 (2012), no. 02, P02002.