This is a talk on opinion dynamics (especially bounded-confidence models) on generalized networks. It is part of the MIX-NEXT III (Multiscale & Integrative compleX Networks: EXperiments & Theories) satellite at NetSci 2022. (Thursday 14 July 2022)

1. Opinion Dynamics on
Generalized Networks
Mason A. Porter (@masonporter)
Department of Mathematics, UCLA
Santa Fe Institute

2. Outline
•Introduction: Modeling of social dynamics and opinions on networks
•Bounded-confidence models (BCMs) of opinion dynamics
•A BCM on hypergraphs
•An adaptive BCM
•A BCM with node weights
•Opinion dynamics coupled to disease dynamics
•Conclusions

3. Some Review Articles
•Hossein Noorazar, Kevin R. Vixie, Arghavan Talebanpour, & Yunfeng Hu
[2020], “From classical to modern opinion dynamics”, International
Journal of Modern Physics C, Vol. 31, No. 07: 2050101
•Claudio Castellano, Santo Fortunato, & Vittorio Loreto [2009], “Statistical
physics of social dynamics”, Reviews of Modern Physics, Vol. 81, No. 2:
pp. 591–646
•Sune Lehmann & Yong-Yeol Ahn [2018], Complex Spreading Phenomena
in Social Systems: Influence and Contagion on Real-World Social
Networks, Springer International Publishing

4. What People Study in Models of Social Dynamics
• Note: Researchers focus on different things in different types of models
• Consensus vs Polarization vs Fragmentation
• How do you measure polarization and fragmentation?
• What is the convergence time to a steady state (if one reaches one)?
• Cascades and virality
• How far and how fast do things (e.g., a meme) spread? When do things go viral, and when do they not?
• Measuring virality in theory (e.g., percolation and giant components) versus in practice
• Incorporating behavior into models of the spread of diseases
• Just concluding that model social dynamics is impossible to do well and giving up on it isn’t an option
for studying certain problems
• More general: Investigate effects of network structure on dynamical processes (and vice versa)
• Making good choices of synthetic networks to consider is often helpful for obtaining insights

5. Dynamical Processes on Networks
•Incorporate which individuals (nodes) interact with which other
individuals via their ties (edges).
•This yields a dynamical system on a network.
•Basic question: How does network structure affect dynamics (and
vice versa)?
•MAP & J. P Gleeson [2016], “Dynamical Systems on Networks: A
Tutorial”, Frontiers in Applied Dynamical Systems: Reviews and
Tutorials, Vol. 4

6. Some Challenges in Modeling Social Dynamics
• How “correct” can these models ever be?
• But maybe they can be insightful or helpful?
• How does one connect the models and the behavior of those models with real life and real
data?
• Example: Can one measure somebody’s opinion as some scalar in the interval [–1,1] based
on their online “fingerprints” or survey answers?
• Comparing outputs like spreading trees of tweets from a model and reality, rather than
comparing node states themselves?
• Ethical considerations in measurements in attempts to evaluate models of social dynamics
with real data
• More general: complexity of models versus mathematical analysis of them?

7. Types of Social-Dynamics Models
• Compartmental models (hijacked from disease dynamics), threshold models
(percolation-like), voter models, majority-vote models, DeGroot models, bounded-
confidence models, games on networks, …
• Discrete states versus continuous-valued states
• Deterministic update rules versus stochastic update rules
• Dynamical systems versus stochastic processes
• Synchronous updating of node states versus asynchronous updating
• Note: Some of the different types of models can be related to each other
• Example: Certain threshold models have been written in game-theoretic terms

8. Generalizing Network Structures
• Weighted networks, multilayer networks, temporal networks, adaptive networks, hypergraphs (and,
more generally, polyadic interactions), etc.
• How do such more general structures affect dynamics?
• What new phenomena occur that cannot arise in simpler situations?
• There are multiple choices for how to do the generalizations, and they matter significantly
• When is consensus more likely, and when is it less likely?
• When is convergence to a steady state accelerated and when is it slowed down?
• When is virality more likely, and when is it less likely?
• If you do the “same type of generalization” on different types of models (e.g., a voter model vs a
bounded-confidence model), when does it have a similar/different effect on the qualitative dynamics?
• Example: Under what conditions do polyadic interactions promote consensus and when do they make it harder?
How does this answer differ — does it? — in different types of social-dynamics models?

9. Some Application-Related Questions
• Spread and mitigation of misinformation, disinformation, and “fake news”
• Formation of echo chambers
• Spread of extremist opinions
• Measuring and forecasting viral posts?
• Distinguishing internal effects from external ones (e.g., something gets popular
enough from retweets that it then shows up on mainstream media sources)
• Inverse problems
• Example: determining “patient 0” in the spread of content
• “Majority illusion” and “minority illusion”

10. Bounded-Confidence Models

11. Bounded-Confidence Models
• 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
• Two best-known variants
• Deffuant–Weisbuch (DW) model: asynchronous updating of node states
• Hegselmann–Krause (HK) model: synchronous updating of node states
• 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 has not been explored much 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

12. BCMs on 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 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

14. Example: G(N,p) ER Networks

15. A BCM on Hypergraphs
•A. Hickok, Y. Kureh, H. Z. Brooks, M. Feng, & MAP [2022], “A
Bounded-Confidence Model of Opinion Dynamics on Hypergraphs”, SIAM
Journal on Applied Dynamical Systems, Vol. 21, No. 1: 1–32.
•In addition to dyadic (i.e., pairwise) interactions between nodes, also
consider polyadic interactions
•Asynchronous updates (generalizes DW model)
•Quantify the disagreement among the nodes in a hyperedge with a
discordance function:

16. A BCM on Hypergraphs
•Select one hyperedge at each time and update node opinions to the mean
opinion of the hyperedge if the discordance function is less than a
confidence bound:

17. Phase Transition in Convergence Time

18. Two General Questions for
Networks with Polyadic Interactions
•1. When generalizing opinion dynamics to incorporate polyadic
interactions, which generalizations make consensus more likely and why
generalizations make consensus less likely?
•2. When generalizing opinion dynamics to incorporate polyadic
interactions, which generalizations speed up convergence to a steady state
and which generalizations slow it down?

19. An Adaptive BCM
• U. Kan, M. Feng, & MAP [2021], “An Adaptive Bounded-Confidence Model of
Opinion Dynamics on Networks”, arXiv:2112.05856
• Adaptive model: the opinions of the nodes are coupled to changes in the network
structure
• Generalize DW model with 1D opinions (on a finite interval)
• Both a confidence bound C and a second bound ! ≥ C that determines an opinion-
tolerance threshold for when we consider an edge to be “discordant”
• Discordant edges may rewire so that nodes connect to nodes with more similar
opinions (“homophilic rewiring”)

20. An Adaptive BCM
•Our adaptive BCM requires a larger confidence bound than the standard
DW model to achieve consensus.
•Our BCM includes ‘pseudo-consensus’ steady states, in which there exist
two subclusters within an opinion-consensus group that differ from each
other by a small amount.

21. An Adaptive BCM

22. Steady States and Convergence Time

23. A BCM with Node Weights
•Grace J. Li & MAP [2022], “A Bounded-Confidence Model of Opinion
Dynamics with Heterogeneous Node-Activity Levels”, arXiv:2206.09490
•Some individuals share their opinions more frequently than others
• Heterogeneous sociabilities, activity levels, prevalences to share opinions, etc.
•How do heterogeneous node-activity levels affect BCM dynamics?
• In our model, it results in (1) slower convergence to steady states and (2) more
opinion fragmentation than in a baseline DW model.
• More general question: How do node weights affect dynamical processes?

24. “Weighted networks”
should not automatically
refer to edge weights!
Public Service Announcement:The network-science community
should spend much more time on studying node weights and their
influence on dynamical processes on networks.

25. Incorporating Node Weights into
a DW Model

26. Node-Weight Distributions

27. Summary of Trends

28. Coupling the Spread of Opinions/Behavior
with the Spread of a Disease
• Jamie Bedson et al. [2021], “A review and agenda for integrated disease models
including social and behavioural factors”, Nature Human Behaviour, Vol. 5, No. 7:
834–846
• In a compartmental model, nodes have different states (i.e., “compartments”) and there
are rules for how to transition between states
• For example, in a stochastic SIR (susceptible–infected–recovered) model, nodes in S change to I
with some probability if they have a contact with a node in I. Nodes in I recover and change to
R with some probability.
• A rich history of work on mean-field theories (both homogeneous and heterogeneous
ones), pair approximations, and other approximations.
• István Z. Kiss, Joel C. Miller, & Péter L. Simon [2017], Mathematics of Epidemics on
Networks: From Exact to Approximate Models, Springer International Publishing

29. 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, Vol. 31, No. 12: 2455–2494
• 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.
• For simplicity (e.g., the same type of mathematical form in the right-hand sides for both layers), we
used compartmental models for each layer (SIR/SIR and SIR/SIRS). It is important to develop more
realistic models.

31. What Some of the Equations
of Motion Look Like

32. Conclusions
• Lots of cool stuff to study in models of social dynamics and opinions on networks
• How do generalized network structures affect opinion dynamics on networks?
• Consensus versus polarization versus fragmentation?
• Convergence times to steady states?
• When do generalizations affect things in one qualitative way (e.g., more consensus), and when do they affect things
in another way (e.g., less consensus)?
• Some other work and works in progress
• W. Chu & MAP [2022], “A Density Description of a Bounded-Confidence Model of Opinion Dynamics on Hypergraphs”,
arXiv:2203.12189
• W. Chu & MAP, “Non-Markovian Models of Opinion Dynamics with Random Jumps on Networks”, in preparation
• H. Z. Brooks & MAP, “Spreading Cascades in Bounded-Confidence Dynamics on Networks”, in preparation
• P. Chodrow, H. Z. Brooks, & MAP, “Bifurcations in Bounded-Confidence Models with Smooth Transition Functions”, in
preparation