Asymmetrically Weighted Dowker Persistence and Applications in Dynamical Systems

TL;DR

This work develops a directed Dowker persistence framework for time-series data by binning trajectories into weighted directed graphs and computing an asymmetric Dowker filtration using the shortest-path Lawvere metric. It provides a complete 1D persistence characterization for cycle graphs, derives how dominating sets dictate maximal complexes, and extends to wedge sums and cactus graphs, linking topology to underlying graph structure. The method is demonstrated on Lorenz '63 and Charney-DeVore systems, showing interpretable, long-lived 1D features that reflect dynamical regimes and are robust to missing data. The results offer a new topological toolkit for regime detection and structural analysis of high-dimensional dynamical systems, with potential extensions to hypergraphs and optimized binning strategies.

Abstract

By their nature it is difficult to differentiate chaotic dynamical systems through measurement. In recent years, work has begun on using methods of Topological Data Analysis (TDA) to qualitatively type dynamical data by approximating the topology of the underlying attracting set. This comes with the additional challenges of high dimensionality incurring computational complexity along with the lack of directional information encoded in the approximated topology. Due to the latter fact, standard methods of TDA for this high dimensional dynamical data do not differentiate between periodic cycles and non-periodic cycles in the attractor. We present a framework to address both of these challenges. We begin by binning the dynamical data, and capturing the sequential information in the form of a coarse-grained weighted and directed network. We then calculate the persistent Dowker homology of the asymmetric network, encoding spatial and temporal information. Analytically, we highlight the differences in periodic and non-periodic cycles by providing a full characterization of their one-dimensional Dowker persistences. We prove how the homologies of graph wedge sums can be described in terms of the wedge component homologies. Finally, we generalize our characterization to cactus graphs with arbitrary edge weights and orientations. Our analytical results give insight into how our method captures temporal information in its asymmetry, producing a persistence framework robust to noise and sensitive to dynamical structure.

Figures (24)

• Figure 1: A relation $R$, represented as a binary matrix, followed by the Dowker complex $\mathfrak{D}(R)$ and $\mathfrak{D}^{si}(R)$. Both of these complexes are homotopy equivalent to a wedge of two circles.
• Figure 2: In the source filtration, a source node of degree three contributes a $2$-simplex to the Dowker complex. In the sink filtration, three $1$-simplices are added instead.
• Figure 3: Above we have $3$ example cases. Each base graph has $6$ vertices. The choice of edge orientation determines the vertices which can form a minimal source dominating set in the path completion. The red vertices form minimal dominating sets and are in these cases unique. In A and B the blue shaded regions denote the simplices of the maximal Dowker complex without path completion. This allows us to compare and see that A has nontrivial first homology, allowed by its minimal dominating set of size $3$, whereas B has trivial homology with its minimal dominating set of size $1$. In C the red shaded region indicates the single fully connected maximal simplex of the maximal Dowker complex with path completion. The edges imparted by the path completion are in red as well.
• Figure 4:
• Figure 5: Example of a uniform weight $1$, consistently-oriented cycle with eight vertices (left) and its corresponding barcode (right). See the first homology class born at $\delta=1$ and die at $\delta=\lceil 8/2 \rceil =4$.

Theorems & Definitions (52)

• Definition 1
• Theorem 1: Dowker
• Definition 2
• Definition 3
• Definition 4: Dowker Filtration
• Proposition 2
• proof
• Definition 5
• Proposition 3
• proof