Yong-Yeol (YY) Ahn
murmuration over Bulgaria—Katunchi, CC BY-SA 4.0
@spavel.bsky.social
Individual cognition and beliefs—group-level beliefs and norms—Social structure
youtube.com/watch?v=NyDDyT1lDhA
75% give at least one incorrect answer
I heard smoking causes cancer. But I enjoy smoking… The science may be wrong?
I spent so much money… Wait—this is actually good!
e.g., a bounded-confidence model.
Deffuant et al. (2000); Hegselmann & Krause (2002).
Agents want a few like-minded neighbors. Random start → stable segregation.
Schelling (1971).
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how does a single mind decide?
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how does the rule of contagion interact with social structure?
often blind to social structure
often blind to cognition
ALS Ice Bucket Challenge, 2014. Photo: Chris Rand, CC BY-SA 4.0.
Walk to Defeat ALS
Shouldn't they be the same?
It takes repeated exposure to take hold.
Worn steps—Tim Green, CC BY 2.0.
Su Renshan, Tiger (1849).
Infection is a probabilistic process that does not depend on previous exposures.
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Your friends have more friends than you.
Feld, American Journal of Sociology (1991)
Your friends are more cited, more prolific, more influential than you.
Eom & Jo, Scientific Reports (2014): generalized friendship paradox
A network with degree distribution pkp_kpk.
↓
Sample through edges instead, and you draw from:
∼k pk\sim k\,p_k ∼kpk
So you keep landing on the high-degree hubs.
Each new infection follows an edge, so it lands on a node in proportion to its degree.
→ The more connections you have, the more likely you catch it.
Early on, we're sampling from the edge-biased degree distribution.
→ Hubs get infected first.
Barthélemy et al., Phys. Rev. Lett. 92, 178701 (2004)
Hubs are the easiest to reach, and the ones that infect the most.
So a simple contagion is very hard to stop, and spreads from hubs to periphery.
Each agent has a threshold: a third of its neighbors must be the same type, say. Fall below it, and the agent moves.
Macro structure that no one chose or wanted.
Schelling, Dynamics of Models of Segregation (1971).
A radical needs none. A holdout needs almost everyone.
Granovetter, Threshold Models of Collective Behavior (1978).
Each infected neighbor transmits independently, with probability ppp.
Escape one exposure: 1−p1-p1−p. Escape all nnn: (1−p)n(1-p)^n(1−p)n.
Padopt≈1−(1−p)nP_{\text{adopt}} \approx 1 - (1-p)^n Padopt≈1−(1−p)n
The curve rises fast, then saturates.
Exposures aren't independent: what matters is crossing a threshold θ\thetaθ.
One exposure does almost nothing; you need social proof from several sources.
Padopt≈Θ(n)P_{\text{adopt}} \approx \Theta(n) Padopt≈Θ(n)
Θ\ThetaΘ is threshold-like: near-zero until the threshold θ\thetaθ, then it snaps to adoption (a step in the limit).
Simple saturates quickly; complex lags, then surges.
clustered well-mixed = better diffusion
High clustering packs a node's neighbors together, so reinforcing exposures pile onto the same person. Low clustering scatters them below the threshold.
clustered = better initiation well-mixed
Better inter-community spreading
Optimal?
Better intra-community spreading
Too clustered → trapped locally; too random → never ignites.
Nematzadeh, Ferrara, Flammini & Ahn, PRL (2014).
Centola, Science (2010).
Solid = clustered-lattice; open = random.
Weng, Menczer & Ahn, Scientific Reports (2013).
A hashtag is a meme we can watch spread, person to person, across the follower network.
We kept pushing buttons in our stats software until all our results had stars next to them. #OverlyHonestMethods
Data are available upon request, but we really hope no one will ask. #OverlyHonestMethods
We put "Bayesian" in the title so the other scientists think we're cool. #OverlyHonestMethods
Your bibliography is a giant selfie. #SixWordPeerReview
Here's a paper to cite: mine. #SixWordPeerReview
Too long; didn't read. Looks good. #SixWordPeerReview
If memes are complex contagions, the community structure of the follower network should leave a fingerprint on how they spread.
M3 and M4 add a community-trapping mechanism on top of the network.
Entropy of how a meme's tweets are distributed across communities, vs. total popularity.
Weng, Menczer & Ahn, Scientific Reports (2013); Weng et al. (2014).
assumes the dynamics
dynamics are imposed, not derived from micro-mechanisms
too simple a representation
the models don't allow enough complexity
This is too simple!
"the individual strives toward consistency within himself."
— Leon Festinger.
"The most fundamental values in a culture will be coherent with the metaphorical structure of the most fundamental concepts in the culture."
— George Lakoff.
"take his tax-hiking, government-expanding, latte-drinking, sushi-eating, Volvo-driving, New York Times-reading, body-piercing, Hollywood-loving, left-wing freak show back to Vermont, where it belongs."
Political advertisement, Des Moines TV (2004).
Lattes are not logically connected to tax policy—yet they cluster reliably.
Goldberg & Stein, Beyond Social Contagion: Associative Diffusion (2018).
Common ways to model a belief state:
…but neither captures how beliefs constrain each other.
Dalege et al.; Galesic et al.; Vlasceanu et al.; Friedkin et al.
Each belief is a node; edges encode dependency, support, or conflict—most importantly, correlation.
An Ising-like energy:
D=−∑ahaxa−12∑a≠cJacxaxcD = -\sum_a h_a x_a - \tfrac{1}{2}\sum_{a\ne c} J_{ac} x_a x_c D=−a∑haxa−21a=c∑Jacxaxc
Dalege; Galesic; Vlasceanu et al.
Correlated beliefs pull together (+); opposed ones push apart (−).
A mind is a graph of concepts; the beliefs are the signed (weighted) links
Heider (1946): "the enemy of my enemy is my friend."
An energy over signed triads:
D=−∑(u,v,z)K wuv wvz wuzD = -\sum_{(u,v,z)} K\, w_{uv}\, w_{vz}\, w_{uz} D=−(u,v,z)∑Kwuvwvzwuz
Heider (1946); Antal, Krapivsky, Redner (2005).
Greenwald, Banaji, Rudman, Farnham, Nosek, Mellott, Psych. Review (2002).
The original model couples internal balance with social input—beliefs are signed (±\pm±).
Rodriguez, Bollen, Ahn (2016).
The extension makes beliefs weighted as well as signed.
Aiyappa, Flammini, Ahn, Sci Adv (2024).
A belief steps toward a noisy target:
bxi(t+1)=bxi(t)+Δbxb_x^i(t{+}1) = b_x^i(t) + \Delta b_x bxi(t+1)=bxi(t)+Δbx
Δbx∼N (μ, σ2)\Delta b_x \sim \mathcal{N}\!\big(\mu,\ \sigma^2\big) Δbx∼N(μ, σ2)
μ=α bxj − β ∂Ei(t)∂bxi\mu = \alpha\, b_x^j \;-\; \beta\, \frac{\partial E^i(t)}{\partial b_x^i} μ=αbxj−β∂bxi∂Ei(t)
How are they related to simple and complex contagion?
Spreading peaks at intermediate modularity—too random, no reinforcement; too clustered, no bridges.
Nematzadeh, Ferrara, Flammini, Ahn, PRL (2014).
Aiyappa, Flammini, Ahn, Sci Adv (2024). Cf. Centola, Science (2010).
Seckin, Aiyappa, Vlasceanu, Menczer, Flammini, Ahn (2026, submitted).
Bob and Alice start neutral; coherence pressure pulls latte toward Group A.
Seckin et al. (2026).
Initially centered on zero; the population ends up linking Group A with latte.
Opinion polarization PO→2P_O \to 2PO→2 as average internal dissonance ⟨d(Bi)⟩\langle d(B_i)\rangle⟨d(Bi)⟩ drops.
Seckin et al. (2026). Cf. associative diffusion—Goldberg & Stein (2018).
Polarization can emerge spontaneously—even without any concrete ideological discourse.
Enough to induce both stereotyping of a neutral topic and polarization.
"AI will surpass human intelligence soon." "Machines will become smarter than people." "AGI is just around the corner." "We'll have human-level AI within a decade."
Two latent axes capture much of political belief:
Distance encodes (dis)agreement.
Belief tesseract?
LLMs already place propositions in a high-dimensional latent space where paraphrases cluster and meaningful directions emerge.
Beliefs as points; agreement as geometry.
A latent belief space where distance reflects compatibility.
Goldberg & Stein (2018); Lee et al., Nat Hum Behav (2025).
Fine-tune RoBERTa with anchor / positive / negative triples drawn from user voting patterns. Loss: max(∥sa−sp∥−∥sa−sn∥+ϵ, 0)\max(\lVert s_a - s_p\rVert - \lVert s_a - s_n\rVert + \epsilon,\, 0)max(∥sa−sp∥−∥sa−sn∥+ϵ,0).
Lee et al., Nat Hum Behav (2025).
Social and cultural proximity counts.
A career is a sentence; an institution is a word.
Murray, Yoon, Kojaku, Costas, Jung, Milojević & Ahn, "Unsupervised embedding of trajectories captures the latent structure of scientific migration," PNAS (2023).
Portuguese-speaking Brazil sits beside Portugal; Spanish-speaking Latin America clusters with Spain.
One axis lines up with institutional prestige: Spearman ρ=0.78\rho = 0.78ρ=0.78 against an independent ranking.
Tij=C mi mj f(rij)T_{ij} = C\,m_i\,m_j\,f(r_{ij}) Tij=Cmimjf(rij)
Flux between iii and jjj grows with their masses (population) and decays with distance rijr_{ij}rij.
"You are less likely to go somewhere far away than somewhere close."
Embedding proximity (R2=0.48R^2=0.48R2=0.48) predicts flux far better than geographic distance (R2=0.22R^2=0.22R2=0.22).
word2vec cosine distance outperforms MDS, spectral, PPR, and gravity-law-fit distances.
Tij⏟flux=1Z P0(i) P0(j)⏟mass×mass exp (vj⊤ui)⏟closeness\underbrace{T_{ij}}_{\text{flux}}=\frac{1}{Z}\,\underbrace{P_0(i)\,P_0(j)}_{\text{mass}\times\text{mass}}\;\underbrace{\exp\!\big(\mathbf v_j^{\top}\mathbf u_i\big)}_{\text{closeness}} fluxTij=Z1mass×massP0(i)P0(j)closenessexp(vj⊤ui)
A word2vec embedding is a gravity model: the degree bias in negative sampling becomes the mass term, and embedding closeness becomes the distance.
The belief embedding works the same way—a learned embedding space with meaningful distance, made meaningful through the theory.
Weng, Kim, Ahn & Moro, "Beyond Distance: Mobility Neural Embeddings Reveal Visible and Invisible Barriers in Urban Space" (2025).
Comparing effective distance vs. physical distance reveals barriers.
logitP (i↔j)∼dPOI+dPhy+dDemo+dCounty\operatorname{logit} P\!\left(i \leftrightarrow j\right)\sim d_{\text{POI}}+d_{\text{Phy}}+d_{\text{Demo}}+d_{\text{County}} logitP(i↔j)∼dPOI+dPhy+dDemo+dCounty
Kim, Kojaku & Ahn, "Uncovering simultaneous breakthroughs with a robust measure of disruptiveness," Sci Adv (2026).
A theory ties the embedding back to the data and allows interpretation.
Tycho Brahe's Mars observations, 1582–1600 · Johannes Kepler · Astronomia Nova (1609)
Hubble (2012) and Webb (2022)—the same patch of sky, SMACS 0723
NASA / ESA / CSA—Webb's "Cosmic Cliffs," Carina Nebula (NGC 3324), 2022
Competition among memes · Scientific Reports 2013 · ICWSM 2014
Optimal network modularity · Phys. Rev. Lett. 2014
Belief networks & coherence · PLOS ONE 2016
Embedding scientific migration · PNAS 2023
Belief embedding · Nature Human Behaviour 2025
Mobility & invisible borders · preprint 2025
Stereotypes & affective polarization · preprint 2026
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