Unveiling hidden features of social evolution by inferring Langevin dynamics from data

TL;DR

This paper argues that stochastic differential equation models provide a principled language for capturing historical dynamics beyond deterministic snapshots, separating deterministic drift from intrinsic noise. It develops a latent-state SDE framework and diagnostic tools for time irreversibility, exogenous perturbations, and probabilistic imputation, enabling uncertainty-aware analysis of long-run trajectories. The authors implement two inference methods, Langevin Bayesian Networks and nonparametric Gaussian process SDEs, and apply them to two domains: the modern political economy and ancient civilizations via the Seshat Polaris data. The results reveal regime-structure and volatility patterns, identify crisis-associated irreversibility, and flag anomalies for targeted historical inquiry, illustrating the framework’s potential to unify structure, contingency, and agency in historical analysis.

Abstract

Are there hidden dynamical common patterns in the evolution of social and cultural history? While the growing availability of digitized social data invites us to answer this question, prevailing quantitative methods often rely on deterministic snapshots or average effects. Such approaches overlook the continuous and inherently uncertain nature of historical trajectories. In this paper, we propose a framework for modeling historical dynamics as stochastic processes described by stochastic differential equations (SDEs). By viewing historical change through the lens of continuous-time dynamics, this framework provides a natural language to describe how structural trends and inherent random fluctuations interact to shape societal evolution. This approach allows us to handle the uncertainty in fragmentary historical records, moving beyond the dichotomy of structural determinism versus pure chance. We demonstrate that adopting this stochastic perspective unlocks a rich suite of analytical capabilities unavailable to static models. Specifically, we introduce methods to: (1) quantify the irreversibility; (2) detect exogenous perturbations; (3) perform multiple imputation for missing historical records. This framework offers a unified methodology for dissecting the stability, contingency, and dynamics of historical change.

Figures (17)

• Figure 1: Conceptual overview of the framework. (Left) We start with historical raw observations (e.g., GDP, population) from multiple units. (Middle) These observations are represented as low-dimensional latent state trajectories (e.g., trajectories of Units 1, 2, and 3) evolving stochastically over time. (Right) The underlying dynamics of these latent states are modeled by an SDE, characterized by a drift field $\bm{F}(\bm{x})$ (blue arrows) and a diffusion matrix $\mathbf{D}(\bm{x})$ (orange ellipses), which we aim to infer.
• Figure 2: Trajectories of nations in the latent space of political economic development. The axes $\bm{x}_D$, $\bm{x}_I$, and $\bm{x}_G$ represent the principal components corresponding to Democracy (V-Dem), Inequality (WID), and the logarithm of real GDP/capita (MPD) using 2011 purchasing power parities benchmark, respectively. Colored paths trace the historical evolution of representative countries, illustrating diverse developmental patterns. The circle denotes the starting year, and the diamond indicates the most recent year. Grey lines depict the trajectories of other countries.
• Figure 3: (a) Comparison of the Autocorrelation Function (ACF) between the observed historical data $x_t$ (left) and the trajectories $\tilde{x}_t$ generated from simulations of the inferred dynamics (right). (b) ACF of the standardized residuals $(2\mathbf{D}(\bm{x}_t))^{-1/2}(\Delta \bm{x}_t - \bm{F}(\bm{x}_t)\Delta t)$. The immediate drop to zero for all variables, Democracy (blue), Inequality (orange), and $\log_{10} \text{GDP/Capita}$ (green), indicates that the residuals are temporally uncorrelated (white noise), supporting the Markov assumption. Shaded regions indicate 95% confidence intervals in (a) and (b).
• Figure 4: Inferred drift field $\bm{F}(\bm{x})$ on the Democracy-Inequality plane. Columns correspond to different ranges of $\log_{10}(\text{GDP/capita})$ (left to right: $2.9, 3.9, 5.0$). Black streamlines depict the mean drift field, the background color map indicates the magnitude $|\bm{F}(\bm{x})|$, and white dots represent the observed data points. The axes display values mapped from $\bm{x}$ back to the original variable space via linear fitting for visualization.
• Figure 5: Inferred diffusion field $\mathbf{D}(\bm{x})$ on the Democracy-Inequality plane. Columns correspond to different ranges of $\log_{10}(\text{GDP/capita})$ (left to right: $2.9, 3.9, 5.0$). Orange ellipses depict the local diffusion anisotropy, where the orientation and size correspond to the eigenvectors and eigenvalues of $\mathbf{D}(\bm{x})$, respectively. The background color map indicates the magnitude ${\rm Tr}[\mathbf{D}(\bm{x})]$, and white dots represent the observed data points. The axes display values mapped from $\bm{x}$ back to the original variable space via linear fitting for visualization.