Quantifying the structural stability of simplicial homology

TL;DR

The paper addresses the problem of quantifying the structural stability of a simplicial complex by identifying the smallest edge perturbation that increases the dimension of the $1$-st homology group. It casts this as a spectral matrix nearness problem using weighted, normalized higher-order Laplacians and tackles it with a bilevel optimization: a constrained-gradient inner loop minimizes a spectral functional $F(\varepsilon,E)$, while an outer level adjusts the perturbation size $\varepsilon$ to drive $F$ to zero. A key contribution is the functional $F(\varepsilon,E) = \tfrac{1}{2} \lambda_+(\varepsilon,E)^2 + \tfrac{\alpha}{2}\max\{0,1-\mu_2(\varepsilon,E)/\mu\}^2$, which penalizes near-disconnectedness while driving the up-Laplacian eigenvalue to zero, thereby inducing a new hole with minimal perturbation. The framework is validated on synthetic quasi-triangulations and real transportation networks, illustrating the method’s ability to quantify topological instability and guiding edge-pruning decisions in higher-order relational data.

Abstract

The homology groups of a simplicial complex reveal fundamental properties of the topology of the data or the system and the notion of topological stability naturally poses an important yet not fully investigated question. In the current work, we study the stability in terms of the smallest perturbation sufficient to change the dimensionality of the corresponding homology group. Such definition requires an appropriate weighting and normalizing procedure for the boundary operators acting on the Hodge algebra's homology groups. Using the resulting boundary operators, we then formulate the question of structural stability as a spectral matrix nearness problem for the corresponding higher-order graph Laplacian. We develop a bilevel optimization procedure suitable for the formulated matrix nearness problem and illustrate the method's performance on a variety of synthetic quasi-triangulation datasets and transportation networks.

Figures (9)

• Figure 1: Left-hand side panel: example of simplicial complex $\mathcal{K}$ on $7$ nodes, and of the action of $\partial_2$ on the 2-simplex $[1,2,3]$; $2$-simplices included in the complex are shown in red, arrows correspond to the orientation. Panels on the right: matrix forms $B_1$ and $B_2$ of boundary operators $\partial_1$ and $\partial_2$ respectively. \newlabelfig:bound_mat0
• Figure 1: Example of the homological pollution, \ref{['ex:connect']}, for the simplicial complex $\mathcal{K}$ on $7$ vertices; the existing hole is $[2,3,4,5]$ (left and center pane), all $3$ cliques are included in the simplicial complex and shown in blue. The left pane demonstrates the initial setup with $1$ hole; the center pane retains the hole exhibiting spectral pollution; the continuous transition to the eliminated edges with $\beta_1 = 0$ (no holes) is shown on the right pane.
• Figure 1: The scheme of alternating constrained (blue, $\| E(t) \| \equiv 1$) and free gradient (red) flows. Each stage inherits the final iteration of the previous stage as initial $E_0(\varepsilon_i)$ or $\widetilde{E}_0(\varepsilon_i)$ respectively; constrained gradient is integrated till the stationary point ($\| \nabla F(E) \| = 0$), free gradient is integrated until $\| \delta W_1 \| = \varepsilon_i + \Delta \varepsilon$. The scheme alternates until the target functional vanishes ($F(\varepsilon, E) =0$).
• Figure 1: Simplicial complex $\mathcal{K}$ on $8$ vertices for the illustrative run (on the left): all 2-simplices from $\mathcal{V}_2$ are shown in blue, the weight of each edge $w_1(e_i)$ is given on the figure. On the right: perturbed simplicial complex $\mathcal{K}$ through the elimination of the edge $[5,6]$ creating additional hole $[5, 6, 7, 8]$. \newlabelfig:illustrative_start0
• Figure 2: Continuous and analogous discrete manifolds with one $1$-dimensional hole ($\dim \overline{\mathcal{H}}_1=1$). Left pane: the continuous manifold; center pane: the discretization with mesh vertices; right pane: a simplicial complex built upon the mesh. Triangles in the simplicial complex $\mathcal{K}$ are colored gray (right). \newlabelfig:example_holes0

Theorems & Definitions (23)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Proposition 2.6
• Proof 1
• Theorem 2.7: Spectral inheritance of higher-order Laplacians
• Proof 2
• Example 3.2