Social Dynamics on Networks

Social Dynamics on Networks
Mason A. Porter (@masonporter)
Department of Mathematics, UCLA

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

Some Things that People Study
in Models of Social Dynamics
• Notes:
• Researchers focus on different things in different types of models
• I’mbringing up what comes to mind. I amrelying on the audience to bring up other examples.
• 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

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?
• Juan Fernández-Gracia, Krzysztof Suchecki, José J. Ramasco, Maxi San Miguel, & Víctor M. Eguíluz
[2014], “Is the voter model a model for voters?”, Physical Review Letters, Vol. 112, No. 15: 158701
• 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?

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 opinion 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

Researchers Study Different Types of
Phenonema in the Different Types of Models
•Examine cascades, virality, and influence maximization in threshold
models
•Examine consensus vs polarization in voter models
•Examine consensus vs polarization vs fragmentation in bounded-confidence
models

Different Mathematical Approaches
in Different Types of Models
• What mean-field theories looks like can be rather different in different types of
models
• For example, bounded-confidence models (kinetic theories, like in studies of
collective behavior, but with different kernels) vs degree-based mean-field theories,
pair approximations, etc. in threshold models
• Branching-process calculations and percolation-based methods are often useful for
threshold models.
• Approximate master equations
• Dynamical-systems approaches vs probabilistic approaches

Generalizing Network Structures
• 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?
• 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 sped up 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 the same type of generalization have a similar 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?

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”

Other Things
•Using ideas like text analysis and sentiment analysis to infer opinions from
textual data
• Perhaps helpful for model evaluation but also to e.g. inspire inputs (such as
ideological values of “media nodes” that influence other nodes) into models of
social dynamics?
•Other connections with tools from machine learning, statistics, natural-
language processing (NLP), etc.
• Topic modeling, etc.

Social Networks
• Typically (but not always), nodes represent individuals
• Depending on the application, edges can represent one (or more) of various types
of social connections: offline interactions, phone calls, Facebook ‘friendships’,
Twitter followership, etc.
• Notions of actual social ties, but also notions of communication
• Different things propagate on different types of networks
• For example: information spreading versus disease spreading
• Complicated mixture of regular and ‘random’ structures
• Good random-graph models provide baselines for comparison

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.
•A fundamental 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

A General Note About Time Scales and Modeling
Dynamical Systems on Dynamical Networks
• Relative time scales of evolution of states versus evolution of network structure
• States change much faster than structure?
• Faster: Dynamical systems on static networks (“quenched”)
• MUCH faster (too rapidly): Can only trust statistical properties of states
• Structure changes much faster than states?
• Faster: Temporal networks
• MUCH faster (too rapidly): Can only trust statistical properties of network structure (“annealed”)
• Comparable time scales?
• “Adaptive” networks (aka “coevolving” networks)
• Dynamics of states of network nodes (or edges) coupled to dynamics of network structure

Spreading and Opinion Models
•There are many types of models. Some examples:
• Compartmental models (hijacked from disease dynamics)
• Convenient because of a long history of work on analyzing them
• Threshold models
• A type of model with discrete states (usually two of them) that models social
reinforcement in contagious spreading processes in a minimalist way
• Voter models
• Discrete-valued opinions, although not really a model for “voters”
• Bounded-confidence models
• Continuous-valued opinions

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

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.

Some of the Equations for the
Evolution of Pairs

Threshold Models
Example: Watts Threshold Model
• D. J.Watts, PNAS, 2002
• Each node j has a (frozen) threshold Rj drawn from some distribution and can be in one of two states (0 or 1)
• Choose a seed fraction ρ(0) of nodes (e.g. uniformly at random) to initially be in state 1 (“infected”,“active”,
etc.)
• Updating can be either:
• Synchronous: discrete time; update all nodes at once
• Asynchronous:“continuous” time; update some fraction of nodes in time step dt (e.g., using a Gillespie
algorithm)
• Update rule: Compare fraction of infected neighbors (m/kj) to Rj. Node j becomes infected if m/kj ≥ Rj.
Otherwise no change.
• Variant (Centola–Macy): Look at number of active neighbors (m) rather than fraction of active neighbors
• Monotonicity: Nodes in state 1 stay there forever.
J. P. Gleeson, PRX,Vol. 3, 021004 (2013): has a table of more than 20 binary-state models (WTM, percolation models, etc.)

Steady-State Levels of Adoption

A Threshold Model with Hipsters
• J. S. Juul & MAP [2019], “ Hipsters on Networks: How a Minority Group of Individuals Can Lead to an
Antiestablishment Majority”, Physical Review E, Vol. 99: 022313
• WTM rules to adopt some product (A or B)
• Conformist node: Adopts majority opinion from local neighborhood
• Hipster node: Adopts minority opinion (from full network, like a best-seller list) from ! times ago

5-Regular Configuration-Model Networks
How can a minority
opinion dominate?

Spread of “Fake News” on Social Networks

“The” Voter Model
• S. Redner [2019], “Reality Inspired Voter Models: A Mini-Review”, Comptes
Rendus Physique, Vol. 20:275–292
• In an update step, an individual updates their opinion based on the opinion of a
neighbor
• One choice: asynchronous versus synchronous updating
• Select a random node (e.g., uniformly at random) and then a random neighbor
• Another choice: node-based models versus edge-based models
• Select a random edge (or perhaps a random “discordant” edge)
• In Kureh & Porter (2020), we use asynchronous, edge-based updates.

A Nonlinear Coevolving Voter Model
• Y. Kureh & MAP [2020], “Fitting In
and Breaking Up: A Nonlinear Version
of Coevolving Voter Models”, Physical
Review E, Vol. 101, No. 6: 062303
• We consider versions of the model with
three types of changes in network
structure.
• Each step: probability !q of rewiring
step and complementary probability 1 –
!q of opinion update
• q = nonlinearity parameter

Example: Rewire-to-Random Model
on G(N,p) Erdős–Rényi Networks

RTR with Two-Community Structure
and Core–Periphery Structure

Figure from L. G. S. Jeub et al., Phys. Rev. E., 2015

Majority Illusion and Echo Chambers
• “Liberal Facebook” versus
“Conservative Facebook”:
http://graphics.wsj.com/blue-feed-
red-feed/
• “Majority illusion”: K. Lerman, X.
Yan, & X.-Z. Wu, PLoS ONE, Vol.
11, No. 2: e0147617 2016
• Such network structures form
naturally from homophily and are
exacerbated further by heated
arguments in contentious times.

“Majority Illusion” and “Minority
Illusion” in our Coevolving Voter Model

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

Bounded-Confidence Model 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

Influence of Media
• 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
• Distance 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.

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

Conclusions
• Many different types of models of social dynamics
• Examples include threshold models, voter models, bounded-confidence models, and others.
• Interactions between social dynamics and disease dynamics
• How does network structure affect dynamics?
• Is there a consensus? How many opinion groups? How long does it take to converge to a steady state? Etc.
• How can we tell when one of these models is “good”?
• Recent papers and some works in progress
• A. Hickok, Y. H. Kureh, H. Z. Brooks, M. Feng, & MAP [2022]: “A Bounded-Confidence Model on Hypergraphs”, SIAM Journal on
Applications of Dynamical Systems, Vol. 21, No. 1: 1–32
• U. Kan, M. Feng, & MAP [2021]: “An Adaptive Bounded-Confidence Model”, arXiv: 2112.05856
• 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
• G. Li & MAP, “Bounded-Confidence Models of Opinion Dynamics with Heterogeneous Node-Activity Levels”, in preparation
• K. Peng & MAP, “Bifurcations in a Multiplex Majority-Vote Model”, in preparation

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