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Weighted Combinatorial Laplacian and its Application to Coverage Repair in Sensor Networks

Shunsaku Yadokoro, Subhrajit Bhattacharya

TL;DR

This work defines weighted combinatorial Laplacian operators on simplicial complexes and analyzes their spectral properties, showing that near-zero eigenvalues detect almost $n$-dimensional holes via $\tilde{\mathcal{L}}_n$, $\tilde{\mathcal{B}}_n$, and homology relations. By endowing simplices with real-valued weights and enforcing a filtration condition, the authors establish a robust, piecewise-smooth relationship between weights and spectral behavior, enabling gradient-descent based control for sensor-network coverage repair. The main theoretical contributions include the weighted boundary maps $\tilde{{\mathcal{B}}}_n$, the weighted Laplacian $\tilde{\mathcal{L}}_n$, a norm bound $\|\tilde{\mathcal{L}}_n\| \le (n+2)|X_n|$, and the result that small eigenvalues correspond to almost-holes, with extensions to multiple almost-holes. Practically, the framework supports dynamic coverage repair, hole creation (caging), and obstacle-aware planning via relative homology, using gradient-based updates driven by the spectral information to steer mobile sensors. The approach provides a continuum between topologies, scales to large networks, and integrates obstacle constraints through relative chain complexes, enabling robust, topology-aware sensor-network management.

Abstract

We define the weighted combinatorial Laplacian operators on a simplicial complex and investigate their spectral properties. Eigenvalues close to zero and the corresponding eigenvectors of them are especially of our interest, and we show that they can detect almost $n$-dimensional holes in the given complex. Real-valued weights on simplices allow gradient descent based optimization, which in turn gives an efficient dynamic coverage repair algorithm for the sensor network of a mobile robot team. Using the theory of relative homology, we also extend the problem of dynamic coverage repair to environments with obstacles.

Weighted Combinatorial Laplacian and its Application to Coverage Repair in Sensor Networks

TL;DR

This work defines weighted combinatorial Laplacian operators on simplicial complexes and analyzes their spectral properties, showing that near-zero eigenvalues detect almost -dimensional holes via , , and homology relations. By endowing simplices with real-valued weights and enforcing a filtration condition, the authors establish a robust, piecewise-smooth relationship between weights and spectral behavior, enabling gradient-descent based control for sensor-network coverage repair. The main theoretical contributions include the weighted boundary maps , the weighted Laplacian , a norm bound , and the result that small eigenvalues correspond to almost-holes, with extensions to multiple almost-holes. Practically, the framework supports dynamic coverage repair, hole creation (caging), and obstacle-aware planning via relative homology, using gradient-based updates driven by the spectral information to steer mobile sensors. The approach provides a continuum between topologies, scales to large networks, and integrates obstacle constraints through relative chain complexes, enabling robust, topology-aware sensor-network management.

Abstract

We define the weighted combinatorial Laplacian operators on a simplicial complex and investigate their spectral properties. Eigenvalues close to zero and the corresponding eigenvectors of them are especially of our interest, and we show that they can detect almost -dimensional holes in the given complex. Real-valued weights on simplices allow gradient descent based optimization, which in turn gives an efficient dynamic coverage repair algorithm for the sensor network of a mobile robot team. Using the theory of relative homology, we also extend the problem of dynamic coverage repair to environments with obstacles.
Paper Structure (17 sections, 8 theorems, 45 equations, 11 figures, 2 algorithms)

This paper contains 17 sections, 8 theorems, 45 equations, 11 figures, 2 algorithms.

Table of Contents

  1. Introduction and Related Work
  2. Motivation and Contribution
  3. Overview
  4. Background
  5. Notations and Conventions
  6. Weighted Graph, Incidence Matrix and Weighted Graph Laplacian
  7. Simplicial Complexes and Homology
  8. Combinatorial Laplacian
  9. Weighted Laplacian
  10. The Definition and the Basic Properties
  11. Norm
  12. Small Eigenvalues
  13. Results and Applications
  14. Dynamic Coverage Repair of Sensor Network
  15. Creation of Holes in Sensor Networks for Caging
  16. ...and 2 more sections

Key Result

Proposition 2.1

Let $X$ be a simplicial complex over a vertex set $V$, and let $C_n\,X$ be the vector spaces over $\mathbb R$ with basis the $n$-simplices of $X$ and let ${\mathcal{B}}_{n}: C_n \to C_{n-1}$ to be the linear map obtained by linearly extending the following formula: Then $C\, X:=\{C_n\}_{n \geq 0}$ is a chain complex.

Figures (11)

  • Figure 1: The Fiedler vector of a graph
  • Figure 2: Illustration of holes in a simplicial complex. There are two 1-dimensional holes (shown as orange circles) and one 2-dimensional hole (shown as a purple sphere)
  • Figure 3: Preview/Overview of the main technical contribution (actual result using our proposed Weighted Combinatorial Laplacian): A demonstration of the proposed 1st weighted Laplacian operator, $\tilde{\mathcal{L}}_1$. Although the simplicial complex does not literally have a hole in it (and hence the null-space of $\tilde{\mathcal{L}}_1$ is empty), the fact that the sensors/vertices are placed in the shape of three touching circular rings, creates low-weight simplices in the interior of the rings. This manifests as the three small eigenvalues of $\tilde{\mathcal{L}}_1$ shown in (b).
  • Figure 4: Just as weighted graphs allow us to reason about and control clustering and weak connection between clusters through analysis of the spectrum and eigenvectors of $\tilde{\mathcal{L}}_0$ using gradient-based algorithms, a weighted simplicial complex would allow us to reason about and control large-scale holes bounding low-weight $2$-simplices through the analysis of the spectrum and eigenvectors ($2$-nd-order modes) of $\tilde{\mathcal{L}}_1$. The main technical contribution of the paper is the development of the theory of weighted higher-order/combinatorial Laplacian, and apply it to the control of sensor networks to fill and create holes in sensor coverage.
  • Figure 5: Example of a simplicial complex of dimension $2$ with arbitrary orientations assigned to the $1$ and $2$ simplices. The corresponding $1$-st and $2$-nd order (combinatorial, unweighted) boundary matrices are shown. Note how the higher order Laplacian matrix, $L_1$, has a null-space of dimension $2$ corresponding to the two holes (cycles that do not form the boundary of a set of $2$-simplices) marked in the complex.
  • ...and 6 more figures

Theorems & Definitions (24)

  • Definition 1: Simplicial Complex
  • Example 2.1
  • Example 2.2
  • Definition 2: Chain Complex and Boundary Maps over Reals
  • Proposition 2.1: Simplicial Chain Complex hatcher
  • Definition 3: Homology with Real Coefficients
  • Proposition 2.2
  • Theorem 2.3: Discrete Hodge Theorem arnold2010finitederenick
  • Definition 4: Weights on Simplicial Complex
  • Definition 5: Weighted Chain Complex and Weighted Laplacian
  • ...and 14 more