A Generalized Gompertz Law for the Rise and Fall of Empires
Abstract
Human societies tend to evolve toward increasing scale and complexity, often followed by stagnation and decline. Historical empires provide a prominent example of this trajectory; however, existing quantitative models capture expansion and rarely account for decline within a unified framework. Here, we introduce a minimal dynamical model that describes both growth and collapse in a single formalism. The model yields a closed-form trajectory that generalizes the Gompertz law and admits equivalent maximum-entropy and optimal-control formulations. We show that this functional form captures the rise and fall of fourteen diverse empires across widely varying time scales. Despite substantial historical heterogeneity, all cases follow a common asymmetric rise-peak-decline pattern governed by a small number of interpretable parameters. We identify the mechanism driving this behavior as the chronophage: an exponentially accumulating burden of administrative and coordination costs that offsets the gains of expansion. These findings uncover a general quantitative principle, yielding new insights into how increasing complexity constrains large-scale social systems.
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BibTeX
@misc{roman2026generalized,
title = {{A Generalized Gompertz Law for the Rise and Fall of Empires}},
author = {Sabin Roman and Vaishnavi Jayakumar and Karoline Wiesner},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29202},
eprint = {2608.29202},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29202},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Generalized Gompertz Law for the Rise and Fall of Empires
AU - Sabin Roman
AU - Vaishnavi Jayakumar
AU - Karoline Wiesner
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2608.29202
UR - https://arxiv.org/abs/2608.29202
AB - Human societies tend to evolve toward increasing scale and complexity, often followed by stagnation and decline. Historical empires provide a prominent example of this trajectory; however, existing quantitative models capture expansion and rarely account for decline within a unified framework. Here, we introduce a minimal dynamical model that describes both growth and collapse in a single formalism. The model yields a closed-form trajectory that generalizes the Gompertz law and admits equivalent maximum-entropy and optimal-control formulations. We show that this functional form captures the rise and fall of fourteen diverse empires across widely varying time scales. Despite substantial historical heterogeneity, all cases follow a common asymmetric rise-peak-decline pattern governed by a small number of interpretable parameters. We identify the mechanism driving this behavior as the chronophage: an exponentially accumulating burden of administrative and coordination costs that offsets the gains of expansion. These findings uncover a general quantitative principle, yielding new insights into how increasing complexity constrains large-scale social systems.
ER -