This is my general-audience talk at DiscCon III (2021 WorldCon). My talk overlapped with the Hugo Award ceremony, but the video will be posted later on the DisCon website for attendees who want to see it.

1. Mathematical Models of the
Spread of Diseases, Opinions,
Information, and Misinformation
Mason A. Porter (@masonporter)
Department of Mathematics, UCLA

2. Outline
• Social networks
• The spread of diseases on social networks
• The spread of other stuff (opinions, information,
misinformation, etc.) on networks
• Summary

3. Social Networks
How individuals are connected to each other

4. Example: Networks of Connections and
Interactions on Social-Media Platforms

5. Networks
A network consists of
nodes, which represent
entities.
The nodes are
connected (“adjacent”)
to each other by
edges, which encode
social, communication,
or ties between them.
Members of a karate club Facebook friendships

6. Penguin Relationship Network
(data set assembled by Heather Zinn Brooks and Michelle Feng)

7. Mathematical Representations of a Network
• Adjacency matrix A
• This example: unweighted
• Aij = 1 if there is a connection between
nodes i and j
• Aij = 0 if no connection
• How do we generalize these
representations to account for edge
directions, edge weights, multiple
relationships, and changes in time?

8. Penguin Relationship Network
(data set assembled by Heather Zinn Brooks and Michelle Feng)

9. “Centrality” (i.e., importance) of nodes and edges
Can be important for spread of disease, information, etc.
• Centrality: different notions of importance
of entities (i.e., nodes) and interactions
(i.e., edges) in a network
• Number of friends (“degree centrality”)
• On many short paths (“betweenneess
centrality”)
• Adjacent to other important nodes
(“eigenvector centrality”)
• Petter Holme, MAP, & Hiroki Sayama [2019],
“Who Is the Most Important Character in
Frozen? What Networks Can Tell Us About the
World”, Frontiers for Young Minds
• https://kids.frontiersin.org/articles/10.3389
/frym.2019.00099
• Article for teens and preteens about
“centrality” in networks

10. Centrality in the Marvel Cinematic Universe
https://felixluginbuhl.com/blog/posts/2018-01-26-network/

11. Spreading of Diseases on Networks
How does social-network structure affect the spread of infectious
diseases?

12. Key Point: Social Network Structure Strongly
Affects the Spread of Diseases
• Heather Z. Brooks, Unchitta Kanjanasaratool, Yacoub H. Kureh, & MAP
[2021] “Disease Detectives: Using Mathematics to Forecast the Spread
of Infectious Diseases”, Frontiers for Young Minds
• https://kids.frontiersin.org/articles/10.3389/frym.2020.577741
• Article for teens and preteens about mathematical modeling of the spread of
infectious diseases

13. “Compartmental Model” of the Spread of a Disease

14. “Basic Reproduction Number” (R0)

15. More Complicated Compartmental Models
(e.g., this one used to help inform COVID-19 policy in Spain; Alex Arenas et al., Physical Review X, 2020)

16. Equations of Motion
(from Arenas et al. 2020)

17. Spreading of Other Stuff on Networks
How does social-network structure affect the spread of opinions,
information, misinformation, and so on.

18. Spread of “Fake News” on Social Networks

19. Opinions, Information, Disinformation, and
Misinformation
• I was interviewed for a 2021 article in The Walrus
• “How Do We Exit the Post-Truth Era?” (by Viviane Fairbank):
https://thewalrus.ca/how-do-we-exit-the-post-truth-era/
• Subtitled: “Why fact-checking alone won’t save us from fake news”
• Mathematical models can help uncover things like the differences in spreading patterns of
different types of content.
• Opinions: discrete values versus continuous values
• Example of discrete values: “yes” or “no”)
• Example of continuous values: numbers in the interval [–1,1], where –1 is
the most liberal and +1 is the most conservative
• Disinformation versus Misinformation
• Disinformation = deliberatively deceptive
• Misinformation = false, inaccurate, or misleading

20. “Bounded-Confidence Models” of Opinion Dynamics
• Continuous-valued opinions on some space, such as [–1,1]
• When two agents interact:
• If their opinions are sufficiently close, they compromise by some amount
• Otherwise, their opinions don’t change
• Most traditionally studied without network structure
(i.e., all-to-all coupling of agents) and with a view
towards studying consensus
• By contrast, early motivation — but barely explored in practice — of
bounded-confidence models was to examine how extremist ideas, even when
seeded in a small proportion of a population, can take root in a population

21. Bounded-Confidence Models on Social Networks
• X. Flora Meng, Robert A. Van Gorder, & MAP [2018], “Opinion Formation
and Distribution in a Bounded-Confidence Model on Various Networks”,
Physical Review E, Vol. 97, No. 2: 022312
• Network structure has a major effect on the dynamics, including how many
distinct opinion groups form and how long they take to form
• At each discrete time, randomly select a pair of agents who are
adjacent in a network
• If their opinions are close enough, they compromise their opinion by an amount
proportional to the difference
• If their opinions are too far apart, they don’t change
• Complicated dynamics
• Does consensus occur? How many opinion groups are there at steady state? How
long does it take to converge to steady state? How does this depend on
parameters and network structure?
• Example: Convergence time seems to undergo a critical transition with respect to
opinion confidence bound (indicating compromise range) on some types of networks

22. Influence of Media in Bounded-Confidence Models
of Opinion Dynamics
• Heather Z. Brooks & MAP [2020], “A Model for the Influence of Media on
the Ideology of Content in Online Social Networks”, Physical Review
Research, Vol. 2, No. 2: 023041)
• Discrete events (sharing stories), but the probability to share them
(and thereby influence opinions of neighboring nodes) is based on a
bounded-confidence mechanism
• Based both on location in ideology space and on the level of quality of the
content that is being spread
• Include “media nodes” that have only out-edges
• How easily can media nodes with extreme ideological positions influence
opinions in a network?
• Future considerations: can also incorporate bots, sockpuppet accounts,
cyborg accounts, etc.

23. Example using Hand-Curated Media
Locations in (Ideology, Quality) Space

24. Coupling the Spread of Opinions/Behavior with the
Spread of a Disease
• Kaiyan Peng, Zheng Lu, Vanessa Lin, Michael R. Lindstrom, Christian
Parkinson, Chuntian Wang, Andrea L. Bertozzi, & Mason A. Porter
[2021], “A Multilayer Network Model of the Coevolution of the Spread
of a Disease and Competing Opinions”, Mathematical Models and Methods
in Applied Sciences
• Opinions (no opinion, pro-physical-distancing, and anti-physical-
distancing) spread on one layer of a multilayer network.
• An infectious disease spreads on the other layer. People who are anti-
physical-distancing are more likely to become infected.
• It is crucial to develop models in which human behavior is coupled to
disease spread. Models of disease spread need to incorporate behavior.

25. Infodemics
• From Wikipedia: “Infodemic is a portmanteau of "information"
and "epidemic" that typically refers to a rapid and far-
reaching spread of both accurate and inaccurate information
about something, such as a disease.”
• World Health Organization page on infodemic:
https://www.who.int/health-topics/infodemic#tab=tab_1
• In the context of a mathematical model of opinion dynamics and
information spread, one can calculate things (like basic
reproduction numbers) that are analogous to what one calculates
in models of disease spread.

26. Summary
The spread of diseases, opinions, information, and so on are
affected by the structure of social networks.

27. Summary
• The spread of diseases, opinions, information, and so on are
affected by the structure of social networks.
• Mathematical models of spreading processes on networks can help
elucidate these effects.
• These mathematical models can inform policy and interventions.
• Example: Physics distancing affects social-network structure, which in
turn affects how a disease spreads on that network.
• Many other generalizations: polyadic interactions (i.e.,
interactions with 3+ people), adaptive network models, time-
dependent networks, etc.