Constructing Nested Self-Amplifying Multiperiod Hypergraphs through Mathematical Optimization

TL;DR

This paper introduces an optimization framework for analyzing multiperiod hypergraphs to identify self-amplifying structures that enable endogenous growth in complex systems, combining combinatorial and dynamic modeling.

Contribution

It develops a unified mixed integer optimization model with linear and nonlinear formulations to analyze nested activation and flow dynamics in complex networks.

Findings

The linear model effectively captures structural amplification.

The nonlinear model incorporates synergistic flow laws similar to chemical kinetics.

Case studies demonstrate the model's ability to identify growth sectors and bottlenecks.

Abstract

This paper proposes an optimization-based framework for the analysis of multiperiod directed multihypergraphs aimed at identifying self-amplifying structures that sustain endogenous growth in complex systems. The approach captures the progressive and nested activation of nodes and hyperarcs, providing a dynamic representation of evolving production and reaction networks. We formulate the problem as a mixed integer optimization model. First, we introduce a tractable linear formulation that captures structural amplification. We then extend this model to a mixed integer nonlinear setting that incorporates a synergistic flow law that generalizes mass-action kinetics in Chemical Reaction Networks and that accounts for interaction effects. This nonlinear formulation is handled through logarithmic transformations and piecewise-linear outer approximations. The framework unifies combinatorial structure selection and flow dynamics, bridging Mathematical Optimization with applications in Economics and Chemistry, including autocatalytic systems related to the Origin of Life. Computational experiments on synthetic instances demonstrate scalability, while an input--output case study illustrates the ability of the model to identify growth-enabling sectors, interdependencies, and structural bottlenecks across different periods, providing actionable insights for the analysis and management of complex systems.