Triangle-Aware Graph Filtration

• Triangle-aware graph filtration is a topological descriptor for spatial graphs that integrates triangle persistence to overcome the limitations of clique-based methods.
• It computes a modified simplicial complex using all-pairs shortest paths and delayed triangle inclusion to capture essential cycles and ensure robust persistent homology.
• The approach is parameter-free, equivariant under spatial transformations, and supports scalable hierarchical clustering as demonstrated in TMFG-DBHT frameworks.

A triangle-aware graph filtration is a topological descriptor for spatial graphs that faithfully encodes the presence and persistence of triangular cycles. It modifies the standard filtration approach by incorporating information about the appearance of triangles in the underlying graph, thereby resolving the limitations of clique-based filtrations in detecting short cycles. This technique provides a parameter-free, equivariant, and computationally tractable tool for persistent homology-based analysis and coarsening of spatial graphs, and is also closely connected to chordal and triangle-preserving filtrations used in hierarchical clustering applications (Calissano et al., 30 Dec 2025, Raphael et al., 2024).

1. Formal Definition of Triangle-Aware Graph Filtration

Let $G = (V, E, X)$ be a finite undirected spatial graph, where $V$ is the vertex set, $E$ the edge set, and $X = (x_v)_{v \in V} \subset \mathbb{R}^p$ specifies the spatial positions of nodes. Each edge $e = (u,v) \in E$ is assigned its Euclidean length $\ell_{u,v} = \| x_u - x_v \|$. The shortest-path metric $d_G(u,v)$ is defined for node pairs as the minimal sum of edge lengths over all connecting paths.

A simplicial complex is then constructed over $V$ by assigning each simplex $\sigma \subseteq V$ a filtration value $f(\sigma) \in \mathbb{R}_+$. The nested sequence of simplicial complexes is given by

$$G(\alpha) = \{ \sigma \subseteq V \mid f(\sigma) \leq \alpha \}, \quad \alpha \geq 0,$$

generating a filtration $\{ G(\alpha) \}_{\alpha \geq 0}$.

The filtration values $f(\sigma)$ are defined as:

1. Vertices: $f(\{v\}) = 0$.
2. Edges: $f(\{u,v\}) = d_G(u,v) = \ell_{u,v}$.
3. Triangles: $$f(\{u,v,w\}) = \min \left\{ d_G(u,v) + d_G(v,w),\ d_G(u,v) + d_G(u,w),\ d_G(u,w) + d_G(v,w) \right\}$$, where $(u,v), (v,w), (u,w) \in E$.
4. $\dim \geq 3$ simplices: $$f(\sigma) = \max_{\tau \subset \sigma,\,\dim \tau < \dim \sigma} f(\tau)$$.

This filtration is called triangle-aware because the inclusion time of each 2-simplex (triangle) is delayed until at least two edges in the triangle have entered, accurately reflecting when the triangular loop is "filled." For higher simplices, a lower-star monotonicity ensures all faces are present before any higher simplex.

The process produces a nested sequence $$G(\alpha_0) \subset G(\alpha_1) \subset \cdots \subset G(\alpha_m)$$, parameterized by the threshold $\alpha$, which governs the growth of the complex.

2. Construction Procedure

The explicit procedure for building the triangle-aware filtration is as follows:

• Compute all-pairs shortest-path distances $d_G(u,v)$ over $(V,E,\ell_{u,v})$.
• Assemble the candidate simplices:
  • Vertices: all $v \in V$
  • Edges: all $e \in E$
  • Triangles: all $\{u,v,w\}$ such that $(u,v),(v,w),(u,w) \in E$
• Compute filtration times $f(\sigma)$ as above for $\sigma$ up to dimension $2$.
• Optionally, for dimension $k>2$, set $f(\sigma)$ as the maximal filtration of its proper faces.
• Sort all simplices by $f(\sigma)$.
• For a discrete set of scales $0 = \alpha_0 < \alpha_1 < \dots < \alpha_m$, define $$G(\alpha_i) = \{ \sigma : f(\sigma) \leq \alpha_i \}$$.

Pseudocode Outline

Compute all-pairs shortest-paths d_G[u,v] over (V,E,ℓ_uv) Initialize S = [] for v in V: f({v}) = 0 S.append((0, {v})) for e={u,v} in E: f(e) = d_G[u,v] S.append((f(e), e)) T = { {u,v,w} : (u,v), (v,w), (u,w) in E } for τ = {u,v,w} in T: f(τ) = min(d_G[u,v] + d_G[v,w], d_G[u,v] + d_G[u,w], d_G[u,w] + d_G[v,w]) S.append((f(τ), τ)) for σ of dim > 2: f(σ) = max_{faces τ⊂σ} f(τ) S.append((f(σ), σ)) Sort S by f(σ) return S

Typically, the construction is truncated to simplices of dimension $\leq 2$ for practical computation of $H_0$ and $H_1$. The all-pairs shortest path computation dominates the complexity at $O(n^3)$ or less with optimized methods (Calissano et al., 30 Dec 2025).

3. Persistent Homology Framework

At each filtration level $\alpha$, the abstract simplicial complex $G(\alpha)$ supports a chain complex structure in degrees $k$:

$$C_k(G(\alpha)) = \bigg\{ \sum_i c_i \sigma_i \;\bigg|\; \sigma_i \in G(\alpha),\, \dim \sigma_i = k,\, c_i \in \Bbbk \bigg\},$$

for a field $\Bbbk$. The standard boundary operator $\partial_k: C_k \to C_{k-1}$ is used, and the $k$-th homology group is $$H_k(G(\alpha)) = \ker \partial_k / \operatorname{im} \partial_{k+1}$$. The filtration induces persistent homology: inclusion maps $G(\alpha) \hookrightarrow G(\beta)$ ($\alpha \leq \beta$) induce induced maps on homology, with the birth and death times of each class recorded in a persistence diagram.

Stability with respect to input perturbations is measured by the bottleneck distance between persistence diagrams:

$$d_B(D_1, D_2) = \inf_\pi \sup_{x \in D_1 \cup \Delta} \| x - \pi(x)\|_\infty$$

where points can be matched to the diagonal to account for noise.

Triangle-aware graph filtrations thus provide richer persistent homology by preserving the persistence of triangular cycles that are otherwise invisible in clique-based filtrations (Calissano et al., 30 Dec 2025).

4. Equivariance and Invariance Properties

Triangle-aware filtrations are equivariant under isometries and scalings of node positions. For a spatial transformation $\psi_{R,A,k}: x_v \mapsto k R x_v + A$ with $R \in O(p)$ (rotation/reflection), $A \in \mathbb{R}^p$ (translation), and $k > 0$ (scaling):

• Under $R,A$, edge lengths $\ell'_{u,v} = \ell_{u,v}$, so $d'_G(u,v) = d_G(u,v)$ and thus $f'(\sigma) = f(\sigma)$.
• Under scaling by $k$, all lengths are multiplied: $d'_G(u,v) = k\, d_G(u,v)$, so for all simplices $f'(\sigma) = k f(\sigma)$.

Consequently, the persistence diagram $D'$ of the transformed graph is $k \cdot D$ (all birth-death pairs scaled by $k$) (Calissano et al., 30 Dec 2025).

5. Comparison with Standard Filtrations

In standard Vietoris–Rips filtrations over the shortest-path metric $d_G$, a triangle $\{u,v,w\}$ enters the complex as soon as all three edges are present (at $r = \max(\ell_{u,v}, \ell_{v,w}, \ell_{u,w})$). The corresponding 1-cycle is killed immediately, yielding zero persistence and making short, tight cycles "invisible."

In the triangle-aware filtration, a triangle $\{u,v,w\}$ with edge lengths $\ell_{uv} = 1$, $\ell_{vw} = 2$, $\ell_{uw} = 2$ would be included at $f(\{u,v,w\}) = \min\{1+2, 1+2, 2+2\} = 3$, allowing the 1-cycle to persist from $r=2$ (when the last edge appears) until $r=3$, so its persistence is $1$.

Method	Triangle Entrance (r)	1-Cycle Persistence
Vietoris–Rips	$2$	$0$
Triangle-aware	$3$	$1$

This improved persistence for triangular 1-cycles enables the method to detect both "large-scale" and "small" cycles in spatial graphs, enhancing robustness against topological noise (Calissano et al., 30 Dec 2025).

6. Triangle-Awareness in Chordal Filtration and Clustering

Triangle-aware ideas are also central to the construction of chordal filtered graphs for hierarchical clustering, notably the Triangular Maximally Filtered Graph (TMFG) and Directed Bubble Hierarchy Tree (DBHT) framework. TMFG builds a chordal (triangulated) subgraph that maximizes aggregate edge weights, incrementally inserting nodes into triangular faces to preserve a high-quality planar backbone of strong triadic correlations. This triangle-preserving filtration supports robust shortest-path computations and stable, accurate hierarchical clustering (Raphael et al., 2024).

The TMFG-DBHT method operates as follows:

• Begins from an initial 4-clique, forming faces (triangles) as the basic units.
• Iteratively inserts unplaced vertices into existing triangular faces to maximize connection strength, maintaining planarity and chordality.
• Constructs the hierarchical tree from overlaps of 4-cliques ("bubbles") via DBHT; clustering decisions exploit the triangle-preserving filtration to reflect the strongest triadic correlations in the data.

Efficient parallel algorithms leverage triangle-awareness to accelerate construction and clustering, with further speedups obtained from lazy updates and approximate shortest-paths. The approach yields high clustering quality, as measured by Adjusted Rand Index, while maintaining computational scalability (Raphael et al., 2024).

7. Practical Significance and Applications

Triangle-aware graph filtrations provide a topology-sensitive, parameter-free approach for coarsening and simplifying spatial graphs while preserving essential topological features. In spatial graph reduction, the method allows substantial size reduction without significant loss of topological information, as demonstrated empirically on synthetic and real data sets (Calissano et al., 30 Dec 2025). In clustering, triangle-preserving filtrations yield chordal backbones that underpin robust hierarchical cluster assignments (Raphael et al., 2024).

A plausible implication is that triangle-awareness supplies a joint topological and combinatorial backbone for further geometric analysis, graph summarization, and hierarchical learning in settings where small cycles encode critical structural information. Limitations include the $O(n^2)$ storage required for dense similarity matrices and lack of global optimality guarantees in greedy triangle-augmentation procedures.

Together, these advances position triangle-aware graph filtrations as foundational tools for topological data analysis, structure-preserving coarsening, and scalable clustering of complex spatial graphs.