Merge Trees of Periodic Filtrations

TL;DR

The paper develops a rigorous framework for persistent topology on periodic data by introducing shadow monomials and the periodic merge tree, which augments standard 0-dimensional persistence with lattice-aware growth and shadows. An efficient algorithm with near-linear performance constructs periodic merge trees and their associated periodic barcodes, proving invariance under lattice changes and stability under perturbations. It then defines and analyzes the periodic 0-th barcode using an alternating Wasserstein distance, establishing invariance and stability results and providing a concrete 3D example to illustrate the concepts. The approach enables robust, scalable topological summaries of crystalline materials and other Λ-periodic structures across arbitrary dimensions. The core contributions include shadow monomial formalism, a lattice-aware construction algorithm, equivalence notions via splintering, and stable periodic barcodes with explicit distance bounds.

Abstract

Motivated by applications to crystalline materials, we generalize the merge tree and the related barcode of a filtered complex to the periodic setting in Euclidean space. They are invariant under isometries, changing bases, and indeed changing lattices. In addition, we prove stability under perturbations and provide an algorithm that under mild geometric conditions typically satisfied by crystalline materials takes $\mathcal{O}({(n+m) \log n})$ time, in which $n$ and $m$ are the numbers of vertices and edges in the quotient complex, respectively.

Figures (9)

• Figure 1: In a unit cell with periodic boundary conditions (the torus), we see a single snake that bites itself, both in the left and the right panel. There is however a significant difference in the periodically tiled plane, since the snakes on the right connect in infinite diagonal lines, while the snakes on the left remain isolated, a distinction we will quantify with the novel concept of a shadow monomial. The material properties of the two examples would indeed be rather different, with higher resistance to tearing on the right.
• Figure 2: Left panel: a periodic graph with two vertices and three edges inside a unit cell in the shape of the unit square in the upper left portion, and the corresponding graph in the $2$-dimensional torus in the upper right portion of the panel. The filter maps the vertices to their (real) labels and the edges to the values shown, which defines the merge tree at the bottom in the panel. Right panel: the same periodic graph as in the left panel, but now represented by a sublattice with a rectangular unit cell of twice the area. Correspondingly, the graph in the $2$-dimensional torus has twice the number of vertices and edges, and the merge tree is richer than in the left panel.
• Figure 3: Each shadow of the loop in the quotient complex is an infinite polygonal line with periodicity lattice spanned by the vector $(1,1)$. The length of its unit cell is $\sqrt{2}$, which implies that its shadow monomial is $2 \sqrt{2} R$.
• Figure 4: The same graphs and periodic merge trees as in Figure \ref{['fig:TwoTreesOne']} but with additional information. Edges with non-zero shift vectors (to be defined in Section \ref{['sec:3.2']}) are drawn as (directed) arcs and labeled with their shift vectors, while edges with zero shift vectors remain undirected and without vector. In addition to the appearances at $t = 1.0, 3.0$ and the merger at $t = 5.0$, there are two catenations at $t = 7.0, 9.0$ that define the shadow monomials decorating the beams of the periodic merge trees in the left panel. Note that the tree in the right panel has twice as many subtrees rooted at the point labeled $7.0$, and that the shadow monomials compensate for the increased number of beams. Indeed, we have two events at each of the first four values defining the periodic merge tree, with a merger followed by a catenation at $t = 7.0$.
• Figure 5: The edge connecting $x$ to $y$ cannot be in the Delaunay triangulation if $y$ lies outside the hyper-rectangle whose facets are centered at the points $x \pm u_i$. This hyper-rectangle has the volume of $2^d$ unit cells and overlaps $3^d$ of them, which for the displayed $2$-dimensional case are drawn with dotted lines.

Theorems & Definitions (22)

• Definition 1: Merge Tree
• Definition 2: Shadow Monomial
• Lemma 3: Counting Inside Sphere
• Definition 4: Periodic Merge Tree
• Lemma 5: Monotonicity
• Definition 6: Paths and Loops
• Lemma 7: Time for Reduction
• Lemma 8: Magnitude of Basis
• Theorem 9: Time for Construction
• Definition 10: Interleaving Distance