Copy of review

Large Science Models:

Foundation Models for
Generalizable Insights Into Complex Systems

with Psycho-social Application

PI: Ishanu Chattopadhyay, PhD

Assistant Professor of Biomedical Informatics & Computer Science

University of Kentucky

DARPA-EA-25-02-05-MAGICS-PA-025

HR0011-26-3-E016

LSMs for complex systems arising in non-sequential multivariate measurements

Validation?

\upsilon(x,i) = \frac{\phi_i(x_i\mid x_{-i})} {\max_y \phi_i(y\mid x_{-i})}

\Upsilon(D) = \frac{1}{|D|\,n} \sum_{x\in D}\sum_{i=1}^{n}\upsilon(x,i)

\Upsilon^{*}(D) = \frac{1}{|D|\,n} \sum_{x\in D}\sum_{i=1}^{n} -\log_2\upsilon(x,i)

At each depth $d$, fit an LSM to the current population:

M_d=\mathrm{LSM}(R_d)

Score each observation by its conditional persistence:

P_d(x)=\frac1n\sum_{i=1}^n \log_2\phi_i^{(d)}(x_i\mid x_{-i})

Peel the low-persistence tail and refit:

R_{d+1} = \left\{ x\in R_d: \frac{P_d(x)-\mu_d}{\sigma_d} < -z_* \right\},\\ \qquad M_{d+1}=\mathrm{LSM}(R_{d+1})

R_0\supset R_1\supset R_2\supset\cdots

Repeatedly isolate progressively less-persistent observations and relearn the conditional structure at every depth.

Normalized conditional support:

\upsilon_{M_d}(x,i) = \frac{\phi_i^{(d)}(x_i\mid x_{-i})} {\max_a\phi_i^{(d)}(a\mid x_{-i})}

Conditional displacement:

\Delta C(d) = \mathbb E_x \left[ \frac1n\sum_i \left| \upsilon_{M_d}(x,i) - \upsilon_{M_{d+1}}(x,i) \right| \right]

Description length:

\ell_d(x) = -\frac1n\sum_i \log_2\phi_i^{(d)}(x_i\mid x_{-i})

Information displacement:

\Delta K^{\mathrm{pseudo}}(d) = \mathbb E_{R_{d+1}}[\ell_d] - \mathbb E_{R_d}[\ell_d]

multiscale spectrum is a data fingerprint

\mathcal S_X = \left\{ \left( \Delta C_X(d), \Delta K_X^{\mathrm{pseudo}}(d)\right) \right\}_{d}

The trajectory across depth fingerprints how organization is distributed from persistent to progressively less-persistent structure.

\displaystyle u_{X,i}^{(d)}=\frac{\phi_{X,i}^{(d)}(x_i\mid x_{-i})}{\max_{a\in\mathcal A_i}\phi_{X,i}^{(d)}(a\mid x_{-i})}\\ \displaystyle \Delta C_X(d)=\frac{1}{n}\sum_{i=1}^{n}\left|u_{X,i}^{(d+1)}-u_{X,i}^{(d)}\right|

DTAG: Global Digital Twin of Opinions

Recall Bail etal.

“Exposure to opposing views on social media can increase political polarization” by Christopher A. Bail et al., published in PNAS in September 2018 (Vol. 115, No. 37, pp. 9216–9221; DOI: 10.1073/pnas.1804840115)

We find more general possibilities: We can make world-views go more extreme or less extreme based on the line of questions and the persona

Perturbing with opposing views made conservatives more conservative (statistically significant), liberals more liberal (not statistically significant)

LSM

Fixed Question Sets Exist That Move Different Persona Towards Polarization/Depolarization

We can optimize question sequences to move the same persona in a chosen direction

Note: It is relatively easy to polarize than to depolarize

Stacked translucent layers illustrating a hierarchy of residual levels

SYSTEM-LEVEL DRIFT MEASUREMENT

The Characteristic Drift Spectrum

A task-free, multiscale fingerprint of systemic organization in complex systems

Systemic drift, not just more data from the same system.

Structural drift

Information drift

Kolmogorov drift

Ishanu Chattopadhyay

University of Kentucky

THE PROBLEM

More data is not drift

Characteristic Drift Spectra of Complex Systems

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Two samples from an unchanged system never match row for row. A drift measure must ignore this re-realization noise and respond when the system’s organization changes.

Illustrative example: two survey items A and B, n = 1,000 per sample

SYSTEM P · SAMPLE 1

same organization

B yes

B no

A yes

396

104

A no

403

Yes-rates of A, B: 0.500, 0.493

P(B = yes | A = yes / no)

0.792 / 0.194

SYSTEM P · SAMPLE 2

same organization, new draw

B yes

B no

A yes

412

A no

104

391

Yes-rates of A, B: 0.505, 0.516

P(B = yes | A = yes / no)

0.816 / 0.210

SYSTEM Q · SAMPLE

items decouple; yes-rates unchanged

B yes

B no

A yes

247

254

A no

251

248

Yes-rates of A, B: 0.501, 0.498

P(B = yes | A = yes / no)

0.493 / 0.503

Marginal checks: the resample shifts more (up to 0.023) than the reorganized system (at most 0.005).

Conditional relation: 0.79 → 0.82 on resampling, but 0.79 → 0.49 when the items decouple.

Drift should mean the system changed, not that we drew new data.

WHAT DRIFT SHOULD MEAN

Re-realization or reorganization?

Characteristic Drift Spectra of Complex Systems

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We want a drift measure that is a property of the system, not of the particular sample: its estimate should settle as data accumulate, and move only when the organization moves.

SAME ORGANIZATION P

Re-realization

Independent samples differ only as realizations. With stable, consistent conditional learning and estimation:

d( S-hat_n(P), S-hat_m(P) ) -> 0 as n, m -> infinity

CHANGED ORGANIZATION P → Q

Reorganization

If the change alters the conditional organization, the residual hierarchy, or the information structure the probes interrogate, then S(P) and S(Q) need not coincide. The displacement persists rather than vanishing as a finite-sample effect.

Distance between estimated spectra

same organization: → 0

changed: persists

sample size n, m

schematic

Repeated realizations of the same organization approach the same fingerprint; organizational drift deforms that fingerprint.

CONDITIONAL ORGANIZATION

Learning the conditional organization

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For categorical state X = (X1, …, Xp) and row population R, a conditional learner returns target-wise predictors φi of Xi given X−i:

$M(R) = { phi_i( . | x_-i ) } for i in T_M$

For row x, the mean conditional log-persistence is

$P_M(x) = 1/n_M(x) sum over i in I_M(x) of log2 max{phi_i(x_i | x_-i), epsilon}$

PM(x) is the average log support assigned to the observed coordinates when each coordinate is predicted from the remainder of the state.

Low persistence identifies rows that are poorly reconstructed by the dominant learned conditional organization.

Learner-agnostic. The objective is the full family of conditionals, not one selected response. Any sufficiently accurate and stable conditional learner can instantiate the spectrum; the GSS results use Large Science Models (LSMs) [5, 6].

Masked-coordinate prediction

masked

⋯

MAP

observed

Conditional distribution over states a ∈ Ai

INTRINSIC RESIDUAL HIERARCHY

Peeling recovers a residual hierarchy

Characteristic Drift Spectra of Complex Systems

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⋮

a freshly fitted conditional model at every level

Fit and score

Robustly standardize lack of persistence

Peel and refit

At depth d, fit a conditional model Md = M(Rd) and compute the persistence of every row under Md.

$Z_d^lack(x) = (c_d - P_{M_d}(x)) / s_d$

$c_d = median P_{M_d}, s_d = kappa_MAD median |P_{M_d} - c_d|$

$kappa_MAD = {Phi^-1(3/4)}^-1 ≈ 1.4826$

$R_{d+1} = {x in R_d : Z_d^lack(x) > z*}, z* = 0.5$

Intuition. Peeling removes the statistically dominant core and refits on the surviving low-persistence population, so organization masked at the population level becomes visible. Depth is structural, not chronological.

COORDINATES OF THE SPECTRUM

Three nonredundant probes at every depth

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PROBE I

Structural drift

ASKS

Does MAP-normalized reconstruction change after refitting?

SENSITIVE TO

Reconstruction geometry

PROBE II

Information drift

ASKS

Does the residual population have a different conditional description length under the parent organization?

SENSITIVE TO

Conditional description length

PROBE III

Kolmogorov drift

ASKS

Are adjacent residual populations distinguishable through compression, without the fitted model as the measuring instrument?

SENSITIVE TO

Model-independent compressibility

Why three? The probes need not move together: information drift can be nonzero even when structural drift is zero. Collapsing them to one coordinate would discard distinguishable forms of reorganization.

PROBE I

Structural drift

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MAP-normalized masked-state reconstruction score

$u_M(x,i) = phi_i(x_i | x_-i) / max_{a in A_i} phi_i(a | x_-i)$

uM = 1 when the observed state attains the conditional MAP support; smaller values measure its support relative to that MAP state.

With a balanced probe cloud Qd and common evaluable targets Id(x), average the change between adjacent depths:

Same probe, adjacent depths

MAP state

observed state

$Delta_C^(d) = 1/|Q_d| sum_{x in Q_d} 1/|I_d(x)| sum_{i in I_d(x)} |u_{M_d}(x,i) - u_{M_{d+1}}(x,i)|$

Intuition. Structural drift measures how the system’s learned ability to reconstruct masked coordinates changes across structural depth.

PROBE II

Information drift

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A data-adaptive code from the learned conditional structure

$l_M(x) = -1/n_M(x) sum over i in I_M(x) of log2 max{phi_i(x_i | x_-i), epsilon}$

Mean code length (bits) of row x when each coordinate is encoded with the fitted conditional predictor.

Information drift between adjacent residual levels

$Delta_K^pseudo,(d) = E_{R_{d+1}} l_{M_d} - E_{R_d} l_{M_d}$

Both expectations use the parent model Md: rows of Rd+1 are coded under the organization learned on Rd.

Mean code length under Md

$E_{R_d} l_{M_d}$

$E_{R_{d+1}} l_{M_d}$

schematic

Intuition. A conditional-model-based description-length probe: information displacement is measured using structure learned from the system itself.

PROBE III

Kolmogorov drift

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At depth d, draw matched row blocks A1, A2 ⊂ Rd and B1, B2 ⊂ Rd+1. For serialized blocks a, b and compressor length C(·), the normalized compression distance is

CD(a,b) = (C(a||b) - min{C(a),C(b)}) / max{C(a),C(b)}

Within-depth distances establish the intrinsic compression scale; the across-depth distance measures separation between adjacent levels:

$R_{d+1}$

within depth: intrinsic scale

across depth: separation

$Delta_K^gzip,(d) = E[ CD(A1,B1) - 1/2 {CD(A1,A2) + CD(B1,B2)} ]$

Intuition. Compression-calibrated information displacement between adjacent residual levels, without using the fitted conditional model as the measuring instrument.

POSITIVE CONTROL

GSS: recovering a known data shift

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0.0738–0.0764

2012–2018 · four waves, one design

Re-realization: nearly the same value

0.1531

2021 · primarily web-based

Shift ≈ 30× the 2012–2018 spread

0.1420 → 0.1117

2022, 2024 · mixed web/face-to-face

Partial return toward the earlier regime

Conditionals estimated with our LSM implementation, one instantiation of the conditional-learning objective rather than a requirement of the spectrum.

Repeated waves under one design give nearly the same fingerprint; the documented design change displaces it, without training to discriminate years or modes.

SYNTHESIS

The characteristic drift spectrum

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At every depth d, the three numbers form

D_X(d) = (Delta_C^(d), Delta_K^pseudo,(d), Delta_K^gzip,(d))

and stacking them across depth gives a depth-by-coordinate matrix:

⋮

rows: structural depth d = 0, …, D−1 columns: the three probes

A task-free, multiscale fingerprint: repeated realizations of the same organization approach the same matrix, persistent displacement signals reorganization, and depth localizes where it occurs.

POSITIVE CONTROL

Two projections of the GSS spectrum

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Fig. 1a: polar rendering of the raw information drift versus structural drift plane for GSS waves 2012 to 2024

Fig. 1b: structural drift versus exp of Kolmogorov drift across depth for GSS waves 2012 to 2024

a Polar rendering of the raw (information drift, structural drift) plane. A tight 2012–2018 regime, a pronounced 2021 displacement, and partial return in 2022 and 2024.

b gzip/NCD information displacement against the Dirichlet-posterior mean of structural drift; the horizontal axis shows exp(Kolmogorov drift).

MULTISCALE COMPARISON

Spectra as fingerprints: 2018 vs 2024

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d = 0

.0971

.0749

.00061

d = 1

.1130

.0460

.00004

d = 2

.1164

.0395

−.00090

d = 0

.1165

.1117

.00015

d = 1

.1383

.0754

−.00008

d = 2

.1472

.0794

−.00035

First three transitions; cell shading scales with value within each of the first two columns.

Reading the fingerprints

Shallowest transition: information drift rises from .0749 (2018) to .1117 (2024).
2024 exceeds 2018 in both structural and information drift at all three depths.
Structural drift grows with depth in both years; Kolmogorov drift stays below 0.001 in magnitude.
Repeated observations can be compared through their full spectra rather than a single classifier or one global distance.

SECOND SURVEY PROGRAM

Eurobarometer: ten waves, 2018–2021

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Eurobarometer polar rendering of the raw information drift versus structural drift plane for ten waves, 2018 to 2021

Eurobarometer structural drift versus exp of Kolmogorov drift, with error bars, for ten waves, 2018 to 2021

DIRECTION

All ten arrows point within ≈30–43° in the (information drift, structural drift) plane.

MAGNITUDE

Arrow lengths range from ≈0.045 to ≈0.12; the 2021 wave, ZA7780, lies within the 2018–19 range.

KOLMOGOROV DRIFT

Within ≈ ±0.006 for every wave except one ZA7561 transition at ≈ +0.009.

RELATED FRAMEWORK

DTAG: digital twins of survey respondents

Characteristic Drift Spectra of Complex Systems

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A survey-grounded synthetic respondent and synthetic survey simulator built on the same LSMs. Public research release v0.1.0-rc1 [12].

Persona → respondent state

An LLM maps persona text to survey-variable assignments that seed the state.

Question → survey variables

Embedding and LLM matching against a survey-specific variable map.

Answer from the LSM

Responses come from the LSM’s conditional distributions given the current state; the LLM phrases them. State persists across questions.

Ideology tracking (GSS)

Position between liberal and conservative pole vectors, via the LSM’s q-distance:

I(s) = (d(s_L, s) - d(s_R, s)) / d(s_L, s_R)

RC1 coverage

GSS 2022: sex-specific LSMs; ideology tracking
GSS 2024: pooled LSM, persona-conditioned
WVS7: pooled LSM; country and year conditioning
Afrobarometer R1–R5: country conditioning (Nigeria, Ghana tested)
Eurobarometer: planned for a later release

Experiment harness: matched personas, forward, reverse and shuffled question orders, replicates, bootstrap CIs.

Proposed link to drift. A DTAG synthetic survey should be a re-realization of the real survey, not a reorganization of it: its drift spectrum should match the real one within finite-sample variation. The spectrum gives a task-free fidelity test.

OUTLOOK

Interpretation and next steps

Characteristic Drift Spectra of Complex Systems

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A system-level measurement object whose primary distinction is between re-realization and reorganization.

Its only learning requirement is a stable estimate of the target-wise conditionals. LSMs are one available realization; alternative learners can be substituted without changing the definition of the spectrum.

ROUTES FOR QUANTIFYING FINITE-SAMPLE VARIABILITY AND CALIBRATING EACH COORDINATE

Matched-size refit nulls

Separate finite-sample variability from real drift at matched sample sizes.

Independent peeling rules

Recover the hierarchy with alternative peeling rules to test robustness.

Planted-change simulations

Inject known changes to map the sensitivity of each coordinate.

Alternative conditional learners

Swap the conditional learner to test dependence on the algorithm.

Alternative compressors

Test how Kolmogorov drift depends on the choice of gzip.

Observation-design controls

Separate changes in how data are collected from changes in the system.

ACKNOWLEDGMENT & REFERENCES

References

Characteristic Drift Spectra of Complex Systems

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[1] J. G. Moreno-Torres et al., “A unifying view on dataset shift in classification,” Pattern Recognition 45(1):521–530, 2012.

[2] J. Gama et al., “A survey on concept drift adaptation,” ACM Computing Surveys 46(4), 2014.

[3] A. Gretton et al., “A kernel two-sample test,” JMLR 13:723–773, 2012.

[4] D. Lopez-Paz and M. Oquab, “Revisiting classifier two-sample tests,” ICLR, 2017.

[5] N. Sizemore et al., “A digital twin of the infant microbiome to predict neurodevelopmental deficits,” Science Advances 10(15):eadj0400, 2024.

[6] K. Wu, F. Li, and I. Chattopadhyay, “Emergenet: A digital twin of influenza A evolution for vaccine strain forecasting and emergence risk assessment,” Military Medicine, usag255, 2026.

[7] NORC at the University of Chicago, “General Social Survey 2021 codebook,” 2021.

[8] NORC, “General Social Survey 2022 codebook,” 2023.

[9] NORC, “General Social Survey 2024 codebook,” release 2, 2025.

[10] NORC, “Sequential data collection designs in the General Social Survey,” GSS Methodological Report 140, 2025.

[11] M. Li et al., “The similarity metric,” IEEE Trans. Information Theory 50(12):3250–3264, 2004.

[12] DTAG: Digital Twin Attitude Generator, v0.1.0-rc1, github.com/zeroknowledgediscovery/DTAG, 2026.

[13] Eurobarometer survey data, GESIS Data Archive, studies ZA7488–ZA7780.

Correspondence: ishanu_ch@uky.edu

Acknowledgment. This material is based upon work supported by the Defense Advanced Research Projects Agency (DARPA) under Agreement No. HR0011-26-3-E016. The views, opinions, and/or findings expressed are those of the authors and should not be interpreted as representing the official views or policies of the Department of Defense or the U.S. Government.

Large Science Models: Mathematical Framework

\begin{aligned} \text{Observables:} \quad & \color{yellow}X = \{x^1, \ldots, x^N\}, \overbrace{x^i \in \Sigma^i}^{\text{finite alphabet}} \\ {\color{gray}\text{Notation:} }\quad & \color{gray} x^{-i} = \{x^j : j \ne i\}\\ \text{Crosstalk:} \quad & \forall i \ P(x^i) = \color{red} f_i(x^{-i}) \\ \text{System state:} \quad & \color{Cyan} \psi = \bigotimes_{i=1}^N \psi^i, \quad \psi^i \in \mathscr{D}(\Sigma^i) \cup \varnothing \\ %\textbf{Degenerate case:} \quad & \psi^i \text{ is a delta distribution (fully observed)} \\ {\color{gray}\text{Notation:} }\quad & \color{gray}\psi^{-i} = \bigotimes_{j \ne i} \psi^j \end{aligned}

	reliten	gunlaw	abany	---	grass
Person 1
Person 2
---
Person m

observables

samples

Distributions over alphabet $\Sigma^i$

\phi = \bigotimes_{i=1}^N \phi^i, \quad \phi^i(\psi^{-i}) \in \mathscr{D}(\Sigma^i) \\

Individual Predictor (CIT)

cross-talk

\phi(\psi) \vert \vert \psi

Tension between predicted and observed distribution drives change

Example

GSS topic: There should be more gun-control

$\psi^i$

strongly agree

agree

neutral

disagree

strongly disagree

\Sigma^i

Digital Twin

\phi^i(\psi^{-i}) \sim \widetilde{\psi}^i

$\phi$ estimates $\psi$

Examples: GSS, ANES, WVS, ESS, Eurobarometer, Afrobarometer, Asian Barometer etc

group

individual

estimate is always a non-empty non-degenerate distribution

missing observation

Large Science Models: Properties

LSM-Distance Metric*

\theta(\psi,\psi') \triangleq \frac{1}{N}\sum_{i=1}^{N} \sqrt{D_{JS}\Bigl(\phi^i(\psi^{-i}) \vert \vert \phi^i(\psi'^{-i})\Bigr)}

where $D_{JS}(P\vert \vert Q)$ is the Jensen-Shannon divergence.

g_{ij}(\psi) \;=\; \frac{1}{2}\,\frac{\partial^2}{\partial \psi^i\,\partial \psi^j}\,\theta^2(\psi,\psi')\Biggr|_{\psi'=\psi}

\left \lvert \ln \frac{\Pr(\psi\to \psi')}{\Pr(\psi' \rightarrow \psi')}\right \rvert \le \beta\,\theta(\psi,\psi')

Large Deviation Bound*

Induced Riemannian metric tensor

This bound connects ``closeness'' of samples to the odds of perturbing from one to the other, bridging geometry to dynamics

Ergodic Projection

\psi_\star \triangleq \bigotimes_{i=1}^N\phi^i\left (\prod_{1}^{N-1}\varnothing\right )

(Sanov's Theorem, Pinkser's Inequality)

$\psi$

$\psi'$

$\theta$

"spatial average": average of all plausible worldviews or states

* Sizemore, Nicholas, Kaitlyn Oliphant, Ruolin Zheng, Camilia R. Martin, Erika C. Claud, and Ishanu Chattopadhyay. "A digital twin of the infant microbiome to predict neurodevelopmental deficits." Science Advances 10, no. 15 (2024): eadj0400. https://www.science.org/doi/full/10.1126/sciadv.adj0400

persistence probability

Ergodic dispersion

\Psi_\star = \theta(\psi,\psi_\star)

Central to Model Drift Quantification

Start with opinion vector with all entries missing

This is a standard Physics construct, quantifying curvature of the underlying latent geometry

Pr(\psi \rightarrow \psi')

Easily computable in LSM framework!

Apply $\phi^i$

Random variable quantifying dispersion around the spatial average of worlviews

const. scaling as $N^2$

Datasets

DatasetAccess modelLicense / use constraints (typical for research use)

General Social Survey (GSS)	Open public download	Free for research use; citation required; no redistribution of modified datasets
American National Election Studies (ANES)	Public-use + restricted-use tiers	Public-use data freely available; restricted-use data requires application and secure handling
Cooperative Election Study (CES)	Public download (common content)	Free for academic research; team modules may have additional citation or use constraints
Eurobarometer	Registration-based access (GESIS)	Free for non-commercial research; user registration required; citation and compliance with GESIS terms
World Values Survey (WVS)	Registration-based download	Free for non-commercial research; attribution required; redistribution restricted
European Social Survey (ESS)	Registration-based download	Free for non-commercial research; strict citation and documentation compliance
Latinobarómetro	Controlled public access	Use subject to project terms; citation required; redistribution limitations apply
Afrobarometer	Public download	Free for research and policy use; attribution required; redistribution limited
Arab Barometer	Form-based access	Free for non-commercial research; short request form; citation required
Asian Barometer	Application-based access	Explicit permission required; usage and redistribution restrictions apply

Access Difficulty

least easy

less easy

easy

Datasets: Global Coverage

World Value Survey (WVS) is global

$\checkmark$ Overlapping survey datasets

Digital Twin & Fidelity of Simulation

\mathcal{N}_\epsilon(\psi) \triangleq \big\{ \psi': {\color{red}\forall i \ \psi'_i \sim \phi^i\left ( \psi^{-i}\right )} \wedge {\color{yellow} \theta(\psi,\psi') \leqq \epsilon }\big \}

Sample predicted distributions

perturbed state within $\epsilon$ of $\psi$

Variable	Masked	Reconstructed
spkcom	allowed	allowed
colcom	not fired	not fired
spkmil	allowed	allowed
colmil	allowed	not allowed
libmil	not remove	not remove
libhomo	not remove	not remove
reliten	strong	no religion
pray	once a day	once a day
bible	inspired word	word of god
abhlth	yes	yes
abpoor	no	no
pillok	agree	agree
intmil	very interested	very interested
abpoorw	always wrong	not wrong at all
godchnge	believe now, always have	believe now, always have
prayfreq	several times a week	several times a week
religcon	strong disagree	disagree
religint	disagree	disagree

Variable	Masked	Reconstructed
spkcom	allowed	allowed
colcom	not fired	not fired
libmil	not remove	not remove
libhomo	not remove	not remove
gunlaw	favor	favor
reliten	no religion	no religion
prayer	approve	approve
bible	book of fables	inspired word
abnomore	yes	yes
abhlth	yes	yes
abpoor	yes	yes
abany	yes	yes
owngun	no	no
intmil	moderately interested	moderately interested
abpoorw	not wrong at all	not wrong at all
godchnge	believe now, didn't used to	believe now, always have
prayfreq	several times a week	several times a week

2018 GSS individual samples

Digital Twin

-Neighborhood of state $\psi$

\epsilon

Definition

Sample neighborhood to impute missing data

\psi

\epsilon

2018 GSS out-of-sample reconstruction

post-reconstruction error ratio (%)

LSM sampling: sampling the $\epsilon$-neighborhood of a state or worldview allows reconstruction of censored opinions

examples

Predictive ability of LSM quantified as ability to reconstruct censored out-of-sample opinions**

{\color{Tomato}\psi_\star }\rightarrow \psi \rightarrow \cdots \rightarrow \psi'

Null state (all missing observations)

Valid perturbations/ simulations

LSM sampling allows simulating opinion perturbations

Both Individuals and groups maybe modeled as digital twins$\dag$

Data Sufficiency via Conservation of Complexity

%K(x) = K(S) + K(x \vert S_\star) + O(1) = K(S') + K(x \vert S'_\star) +O(1) K(x \vert S_\star) = O(1) = K(S \vert x_\star)

The No-cheating Thorem: Generative models cannot cheat on complexity

Kolmogorov Complexity

Optimal Generative Model

compressed data representation

compressed model representation

Theorem

K(\textrm{data}) = K(\textrm{LSM}) +O(1)

Conservation Law arising from the continuous symmetry of typicality*

\mu_0(X) \triangleq \frac{\delta(\vert \langle S(X) \rangle \vert)}{\delta(\vert \langle X \rangle \vert)} \leq 1

Saturation relation:

Data Sufficiency Statistic $\mu_0$

We need LSM-sampling to calculate this

*Noether's Theorem

For every continuous symmetry of a physical system, there exists a corresponding conserved quantity

$\checkmark$ 6. Explain how you will measure the accuracy of predicted structure emergence and data sufficiency.

\vert \langle X' \rangle \vert \approx \max\{1,\mu_0(X)\} \vert \langle X \rangle \vert

How much more data do we need?

Data saturation

Data deficient

Needed

Current

Empirical Validation

Large Science Models & Ergodicity

No Convergence

(~50% belief mismatch between pairs)

2018 GSS survey belief vectors simulated via LSM sampling

No ergodicity assumption: LSMs are built for non-ergodic systems
Sampling and simuation "remembers" the start point (No convergence), demonstrating non-ergodic learned structure
Local potential fields vary across the space
Potential wells may arise, driven by the dynamics at hand, not via assumptions
"change" is driven by non-equilibrium (dissonance)

Model Drift Quantification

Ergodic dispersion

\Delta_\star = \theta(\Psi,\psi_\star)

z(\Delta_\star) = \frac{\Delta_\star^{[t]} - \langle \Delta_\star^{[t]} \rangle}{\sigma(\Delta_\star^{[t]} )}

Z-value of dispersion

Do new samples (survey respondents) still conform to the model?

GSS Model drift

ergodic projection (all missing values)

A random belief state (with possibly missing entries)

random variable

normal variate

\zeta(M) = \vert z(\Delta_\star^0) - z(\Delta_\star^{[t]}) \vert

Model drift stochastic process ($\zeta$)

\mathbf{E}(\zeta(M) )

assess if $\zeta$ is stationary: if not then new samples are not conforming to model

Example for GSS LSM inferred for year 2000

Global Emergent Structure via Clusters & Poles

2018 GSS

\theta_t(\psi_+,\psi_-)

Polar separation over time

2016 Presidential Election Vote Prediction

2004

abany	no	yes
abdefctw	always wrong	not wrong at all
abdefect	no	yes
abhlth	no	yes
abnomore	no	yes
abpoor	no	yes
abpoorw	always wrong	not wrong at all
abrape	no	yes
absingle	no	yes
bible	inspired word	book of fables
colcom	fired	not fired
colmil	not fired	not allowed
comfort	strongly agree	strongly disagree
conlabor	hardly any	a great deal
godchnge	believe now, always have	don't believe now, never have
grass	not legal	legal
gunlaw	oppose	favor
intmil	very interested	not at all interested
libcom	remove	not remove
libmil	not remove	remove
maboygrl	true	false
owngun	yes	no
pillok	agree	strongly agree
pilloky	strongly disagree	strongly agree
polabuse	no	yes
pray	several times a day	never
prayer	disapprove	approve
prayfreq	several times a day	never
religcon	strongly disagree	strongly agree
religint	strongly disagree	strongly agree
reliten	strong	no religion
rowngun	yes	no
shotgun	yes	no
spkcom	not allowed	allowed
spkmil	allowed	not allowed
taxrich	about right	much too low

conservative pole

\psi_+

liberal pole

\psi_-

Clustering LSM distance $\theta(x,y)$ between out-of-sample individuals

conservative

liberal

poles:

partial states aligning with extreme opposing worldviews

Compare across time and different GSS surveys
Derived features for individuals (ideology index)

I(x) = \frac{\theta(x,\psi_+) - \theta(x,\psi_-)}{\theta(\psi_+,\psi_-)}

Predict 2016 votes using ideology index

Emergent global structure

Reflexivity and State Collapse on Observation

Emergent Equations of Motion

L \triangleq \frac{1}{2} \sum_i g_{kl} P^k_p \dot{\psi}^p_i P^l_n \dot{\psi}^n_i - \theta(\psi, \phi)

Define Lagrangian*

\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\psi}^m_i} \right) - \frac{\partial L}{\partial \psi^m_i} = 0

Via the Euler-Lagrange Equations$^\dag$:

\ddot{\psi}^m_i = -g^{km} P^k_m \frac{1}{2N} \sum_j \frac{1}{\sqrt{D_{JS}(\psi^m_j \| \phi^m_j)}} \left[ \ln\left( \frac{2e\psi^m_j}{\psi^m_j + \phi^m_j} \right) - \frac{1}{2(\psi^m_j + \phi^m_j)} \right]

Over-damped Gradient flow Equation*

where $-g^{km}$ is the inverse metric tensor

kinetic energy

state collapse

strongly agree

agree

neutral

disagree

strongly disagree

strongly agree

agree

neutral

disagree

strongly disagree

Query/

Observation

$X_i$

Non-local Influence propagation on measurement/observation (QM-like)

\phi^i(\psi^{-i})

potential energy

* Einstein notation used

Goldstein, Herbert, et al. Classical Mechanics. 3rd ed., Pearson, 2002.

$^\dag$

Principle of stationary action

Dynamics

Local potential field eqn

Embedded Social Theories in LSM

How LSMs reflect and operationalize foundational social-scientific ideas

Belief about topic $i$ is expected to align with beliefs about other topics $\displaystyle\psi^{-i}$.
Deviations are exponentially improbable $\Rightarrow $ people/groups seek internal coherence.
Theory Link:
- Cognitive consistency theory – Abelson et al. (1968)
- Constraint satisfaction in beliefs – Read & Marcus-Newhall (1993)

Beliefs evolve to minimize tension between actual state and “expected” state.
Reflexive gradient flow — system reduces internal contradiction.
Theory Link:
- Cognitive Dissonance Theory – Festinger (1957)
- Homeostatic belief adjustment – Gawronski & Strack (2004)

Observing a belief changes it and affects all conditionals.
Direct encoding of feedback loops central to human systems.
Theory Link:
- Reflexivity in social systems – Giddens (1984), Soros (1994)
- Theory of mind / mutual modeling – Premack & Woodruff (1978)

Validation of Social Theory Questions:

Perception changes reality, which changes perception
The Constitution of Society
The Alchemy of Finance
Does a chimpanzee have a theory of mind?

Our system “wants” to reach a low-energy (low-dissonance) state — a direct computational analog of Festinger’s theory.

People strive to align beliefs and attitudes across related domains. Inconsistencies create cognitive discomfort, prompting adjustments across belief clusters to restore harmony.

Exploratory: Belief systems react measurably to exogenous events and shocks

Exploratory: Cross-dependencies between beliefs have observable effects on societal resilience.

Is Polarization an Inevitable Attractor?

Social Identity Theory vs. Belief Proximity

Large Science Models: Broader Applications

A General Framework for modeling Complex Systems

Genomic database: Missing heritability problem

Personalized Clinical Digital Twin, Virtual Patients

Any structured interview, PTSD fabrication

Assess sysmptom data and co-pathologies

Predict future mutations; which animal strain is closest to jumping to humans

Mental health diagnosis

Microbiome Analysis**

Algorithmic lie detector

Viral emergence

Teomims

Opinion Dynamics

Darkome

Generative model of complex microbial ecosystems, and their impact on health and disease

Data requirements

Tabular data
Potentially large number of features/covariates ($10^2 - 10^8 $)
Sufficient number of samples ($10^3 - 10^6$)
Small number of longitudinal samples (currently, $ < 100$)

Limitation	Mitigation / Response
Conventional time series is currently out-of-scope	Focus on cross-sectional interdependencies and belief geometry; time handled via drift
LSMs model statistical interdependence, not causal mechanisms	Use perturbation-based simulations to infer plausible influence pathways
Limited by observed belief variables	Integrate multiple surveys; use latent proxies and test sensitivity of digital twins
Social theory connections and interpretability may be challenging	Anchor dynamics with theory-driven constructs (e.g., ToM, cognitive dissonance)

LSMs for complex systems

**preliminary study published (https://www.science.org/doi/10.1126/sciadv.adj0400)

Large Science Models:

Foundation Models for
Generalizable Insights Into Complex Systems

with Psycho-social Application

DTAG: Global Digital Twin of Opinions

Recall Bail etal.

Large Science Models: Mathematical Framework

Digital Twin

Large Science Models: Properties

LSM-Distance Metric*

Large Deviation Bound*

Induced Riemannian metric tensor

Ergodic Projection

Ergodic dispersion

Datasets

Access Difficulty

Datasets: Global Coverage

Digital Twin & Fidelity of Simulation

Digital Twin

-Neighborhood of state \(\psi\)

Data Sufficiency via Conservation of Complexity

Large Science Models & Ergodicity

Model Drift Quantification

Ergodic dispersion

Z-value of dispersion

Model drift stochastic process (\(\zeta\))

Global Emergent Structure via Clusters & Poles

Reflexivity and State Collapse on Observation

Emergent Equations of Motion

Query/

Observation

Non-local Influence propagation on measurement/observation (QM-like)

Dynamics

Embedded Social Theories in LSM

Large Science Models: Broader Applications

Large Science Models:

Foundation Models for Generalizable Insights Into Complex Systems

with Psycho-social Application

DTAG: Global Digital Twin of Opinions

Recall Bail etal.

Large Science Models: Mathematical Framework

Digital Twin

Large Science Models: Properties

LSM-Distance Metric*

Large Deviation Bound*

Induced Riemannian metric tensor

Ergodic Projection

Ergodic dispersion

Datasets

Access Difficulty

Datasets: Global Coverage

Digital Twin & Fidelity of Simulation

Digital Twin

-Neighborhood of state \(\psi\)

Data Sufficiency via Conservation of Complexity

Large Science Models & Ergodicity

Model Drift Quantification

Ergodic dispersion

Z-value of dispersion

Model drift stochastic process (\(\zeta\))

Global Emergent Structure via Clusters & Poles

Reflexivity and State Collapse on Observation

Emergent Equations of Motion

Query/

Observation

Non-local Influence propagation on measurement/observation (QM-like)

Dynamics

Embedded Social Theories in LSM

Large Science Models: Broader Applications

Foundation Models for
Generalizable Insights Into Complex Systems