A notion of homotopy for directed graphs and their flag complexes

Abstract

Directed graphs can be studied by their associated directed flag complex. The homology of this complex has been successful in applications as a topological invariant for digraphs. Through comparison with path homology theory, we derive a homotopy-like equivalence relation on digraph maps such that equivalent maps induce identical maps on the homology of the directed flag complex. Thus, we obtain an equivalence relation on digraphs such that equivalent digraphs have directed flag complexes with isomorphic homology. With the help of these relations, we can prove a generic stability theorem for the persistent homology of the directed flag complex of filtered digraphs. In particular, we show that the persistent homology of the directed flag complex of the shortest-path filtration of a weighted directed acyclic graph is stable to edge subdivision. In contrast, we also discuss some important instabilities that are not present in persistent path homology. We also derive similar equivalence relations for ordered simplicial complexes at large. Since such complexes can alternatively be viewed as simplicial sets, we verify that these two perspectives yield identical relations.

Figures (11)

• Figure 1: A simple example digraph $G$ for which $\mathop{\mathrm{\mathrm{dFl}}}\nolimits(G)\neq \mathcal{A}(G)$
• Figure 2: The categories defined in this section, and the functors between them. The black arrows denote inclusions of subcategories; the dashed arrows are wide inclusions whilst the solid arrows are full inclusions. The orange arrows denote $\mathop{\mathrm{\mathrm{dFl}}}\nolimits$ and the blue arrows denote $\mathcal{A}$.
• Figure 3: Visualisation of the lift $\mathfrak{L}(v_0 v_1 v_2 v_3 v_4)$. The lift is a sum of $5$ different $5$-paths, the green paths are added with coefficient $(+1)$ and the red paths are added with coefficient $(-1)$.
• Figure 4: Illustration of the difference between $G\boxdot H$ and $G\times H$. The box product $G\boxdot H$ consists of just the black edges. The cross product $G\times H$ additionally contains the red edges.
• Figure 5: The digraph $G\times {I}$, as relabelled in the proof of Proposition \ref{['prop:dfl_times_not_simeq_cyl']}. The tree, $T$, used to produce the basis for $\ker ^{nr} _1$ is shown in red.

Theorems & Definitions (160)

• Definition 2.1
• Example 2.2
• Definition 2.3
• Definition 2.4
• Remark 2.5
• Definition 2.6
• Definition 2.7
• Remark 2.8
• Lemma 2.9
• Lemma 2.10