Analyzing the topological structure of composite dynamical systems

TL;DR

The chapter addresses the challenge of modeling large, composite dynamical systems by fusing dynamical structural equation models (DSEMs) with topological sheaves. It introduces netlist encodings to translate DSEMs into sheaves and defines consistency-radius minimization as a robust, unified inference tool capable of testing consistency, imputing missing data, and forecasting within a coherent framework. A key theoretical result proves a bijection between DSEM solutions and global sections of the netlist sheaf, while autoregressive timeseries and missing data are naturally supported through extended stalks; the Bering Sea ecosystem serves as a concrete case study. The framework also develops a dual subsystem cosheaf view, showing how invariant sets and subsystems form a topological dual pair, and demonstrates practical implications for model reduction, uncertainty quantification, and multi-scale analysis with potential links to statistical hypothesis testing. Overall, the work provides a principled, graph-guided approach to decompose, analyze, and infer across interacting dynamical subsystems in ecology and beyond, with broad applicability to complex networked systems.

Abstract

This chapter explores dynamical structural equation models (DSEMs) and their nonlinear generalizations into sheaves of dynamical systems. It demonstrates these two disciplines on part of the food web in the Bering Sea. The translation from DSEMs to sheaves passes through a formal construction borrowed from electronics called a netlist that specifies how data route through a system. A sheaf can be considered a formal hypothesis about how variables interact, that then specifies how observations can be tested for consistency, how missing data can be inferred, and how uncertainty about the observations can be quantified. Sheaf modeling provides a coherent mathematical framework for studying the interaction of various dynamical subsystems that together determine a larger system.

Figures (14)

• Figure 1: (a) The DSEM model for part of a food web in the Bering Sea thorson2024dynamic, (b) its wiring hypergraph, (c) its netlist graph, and (d) its sheaf diagram. The arrows in each subfigure have different meanings: in (a) they denote causal, linear relationships (Sec. \ref{['sec:dsem_background']}); in (c), they point from netlist parts to nets (Sec. \ref{['sec:netlists']}); and in (d), they denote restriction functions (Sec. \ref{['sec:sheaf_background']}). While the DSEM also estimates a first-order autoregressive term for each variable (not shown in (a) to simplify presentation), there is no autoregressive structure assumed in the sheaf model. This remedied in Section \ref{['sec:sheaf_autoregression']}.
• Figure 2: A netlist for an electric circuit, described in Example \ref{['eg:eg_netlist']}.
• Figure 3: A linear regression problem as (a) a SEM, (b) a netlist with hardcoded coefficients, (c) a netlist with coefficients exposed as inputs, and (d) a sheaf. To solve the linear regression problem, the partial assignment supported on the darkest shaded region is supplied by the observations, and then the assignment is extended to the remaining stalks. Finally, the copies of $m$, $b$, and $x$ that should be constrained so that they are identical are shown by the three lighter shadings.
• Figure 4: Geometric meanings of the terms contributing to consistency radius in Equation \ref{['eq:linear_regression_cr']}.
• Figure 5: Modification to the sheaf in Figure \ref{['fig:linear_regression_netlist']}(d) to allow for missing data.

Theorems & Definitions (67)

• Definition 1
• Example 1
• Definition 2
• Definition 3
• Proposition 1
• proof
• Definition 4
• Corollary 2
• Proposition 3
• Definition 5