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Rigidity Theory of Graphs with Heterogeneous Vertices

Yongjie Liu, Gangshan Jing, Long Wang

TL;DR

This paper develops SA-RoD rigidity theory for graphs with heterogeneous vertex types, showing that global rigidity and unique shape determination depend on bipartition-driven connectivity between SA and RoD constraints. It proves a duality for infinitesimal SA-RoD rigidity under partition swaps and provides constructive methods (vertex additions and merging) to build globally SA-RoD rigid frameworks with fewer edges than traditional rigidity frameworks. The SA-RoD-based SNL formulation translates to an edge-based problem where connectivity of the SA and RoD triplet sets enables decoupled linear subproblems for bearings and distances, or a low-dimensional polynomial approach when both are disconnected. Numerical simulations corroborate the theory, demonstrating reliable localization under various SA/RoD connectivity conditions and illustrating practical benefits for heterogeneous-sensor networks.

Abstract

Graph rigidity theory answers the question of whether a set of local constraints can uniquely determine the shape of a graph embedded in a Euclidean space, and has been recognized as a useful tool of examining solvability of sensor network localization (SNL) problems. In recent years, constraints involving signed angle (SA) and ratio of distance (RoD) have been adopted in SNL due to their ease of measurements and independence of the global coordinate. However, most prior works consider homogeneous nodes, i.e., all the sensors have the same perceptual abilities. Although mixed constraints have been considered recently, little is known about how the bipartition of nodes based on perceptual abilities affects the rigidity property of the network. In this paper, we propose a novel SA-RoD rigidity theory for graphs with heterogeneous vertices, where each vertex corresponds to a sensor node capturing either SA or RoD measurements. Unlike existing rigidity theory, the SA-RoD rigidity is shown to be strongly dependent on bipartitions of nodes, and exhibits a duality. Moreover, the shape of an SA-RoD constrained network can be uniquely determined up to uniform rotations, translations, and scalings (global SA-RoD rigidity) even if it is neither globally RoD rigid nor globally SA rigid. A scalable approach to construction of globally SA-RoD rigid frameworks is proposed. Localizability analysis and localization algorithm synthesis are both conducted based on weaker network topology conditions, compared with SAor RoD-based SNL. Numerical simulations are worked out to validate the theoretical results.

Rigidity Theory of Graphs with Heterogeneous Vertices

TL;DR

This paper develops SA-RoD rigidity theory for graphs with heterogeneous vertex types, showing that global rigidity and unique shape determination depend on bipartition-driven connectivity between SA and RoD constraints. It proves a duality for infinitesimal SA-RoD rigidity under partition swaps and provides constructive methods (vertex additions and merging) to build globally SA-RoD rigid frameworks with fewer edges than traditional rigidity frameworks. The SA-RoD-based SNL formulation translates to an edge-based problem where connectivity of the SA and RoD triplet sets enables decoupled linear subproblems for bearings and distances, or a low-dimensional polynomial approach when both are disconnected. Numerical simulations corroborate the theory, demonstrating reliable localization under various SA/RoD connectivity conditions and illustrating practical benefits for heterogeneous-sensor networks.

Abstract

Graph rigidity theory answers the question of whether a set of local constraints can uniquely determine the shape of a graph embedded in a Euclidean space, and has been recognized as a useful tool of examining solvability of sensor network localization (SNL) problems. In recent years, constraints involving signed angle (SA) and ratio of distance (RoD) have been adopted in SNL due to their ease of measurements and independence of the global coordinate. However, most prior works consider homogeneous nodes, i.e., all the sensors have the same perceptual abilities. Although mixed constraints have been considered recently, little is known about how the bipartition of nodes based on perceptual abilities affects the rigidity property of the network. In this paper, we propose a novel SA-RoD rigidity theory for graphs with heterogeneous vertices, where each vertex corresponds to a sensor node capturing either SA or RoD measurements. Unlike existing rigidity theory, the SA-RoD rigidity is shown to be strongly dependent on bipartitions of nodes, and exhibits a duality. Moreover, the shape of an SA-RoD constrained network can be uniquely determined up to uniform rotations, translations, and scalings (global SA-RoD rigidity) even if it is neither globally RoD rigid nor globally SA rigid. A scalable approach to construction of globally SA-RoD rigid frameworks is proposed. Localizability analysis and localization algorithm synthesis are both conducted based on weaker network topology conditions, compared with SAor RoD-based SNL. Numerical simulations are worked out to validate the theoretical results.

Paper Structure

This paper contains 41 sections, 41 theorems, 103 equations, 12 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. SA-RoD rigidity theory
  3. Bipartition of the vertex set
  4. Inherent properties under connectivity constraints
  5. Duality
  6. Shape determination
  7. Characterizations of shape uniqueness
  8. Shape uniqueness of quadrilaterals
  9. Generic property
  10. Combinatorial construction of globally SA-RoD rigid frameworks
  11. Vertex addition
  12. Construction of globally SA-RoD rigid frameworks
  13. Merging globally SA-RoD rigid frameworks
  14. SA-RoD-Based SNL
  15. Problem Formulation
  16. ...and 26 more sections

Key Result

Proposition 1

The SA-RoD rigidity matrix $R_{\mathcal{G}(A,D)}(p)$ can be expressed as where $\bar{H}=H\otimes I_{2}$, $\bar{R}_{A}(p)\in\mathbb{R}^{|\mathcal{T}_{A}|\times 2|\mathcal{E}|}$ and $\bar{R}_{D}(p)\in\mathbb{R}^{|\mathcal{T}_{D}|\times 2|\mathcal{E}|}$ can be written as

Figures (12)

  • Figure 1: Examples of quadrilateral frameworks. The dots in red represent nodes in $\mathcal{V}_{A}$, while hollow dots represent nodes in $\mathcal{V}_{D}$. (a) A pure RoD constrained framework that is not infinitesimally RoD rigid. (b) A pure SA constrained framework that is not infinitesimally SA rigid. (c) A framework with $\mathcal{V}_{A}=\{1\}$ whose shape can not be uniquely determined under the given bipartition. (d) A framework with $\mathcal{V}_{A}=\{1,2\}$ whose shape can not be uniquely determined under the given bipartition. The coordinates of nodes $1,2,3,4,2^{\prime},3^{\prime}$ are $(0,0),(4,0),(3,1),(2,1),(2,0),(1,1)$, respectively. (e) A framework with with $\mathcal{V}_{A}=\{1,3\}$ whose shape can not be uniquely determined under the given bipartition. The coordinates of nodes $1,2,3,4,3^{\prime},4^{\prime}$ are $(1,\sqrt{3}),(0,0),(4,0),(2,\sqrt{3}),(2,-2\sqrt{3}),(3,\sqrt{3})$, respectively. (f) A framework with $\mathcal{V}_{A}=\{1,2,3\}$ whose shape can be uniquely determined under the given bipartition.
  • Figure 2: (a) A globally and infinitesimally SA-RoD rigid framework corresponding to a degenerate triangle with $\mathcal{V}_{A}=\{2\}$. (b) A globally and infinitesimally SA-RoD rigid framework corresponding to a quadrilateral with $\mathcal{V}_{A}=\{1,3,4\}$. (c) A globally SA-RoD rigid framework that is not infinitesimally SA-RoD rigid corresponding to a degenerate quadrilateral with $\mathcal{V}_{A}=\{1\}$. (d) An infinitesimally SA-RoD rigid framework that is not globally SA-RoD rigid on a Laman graph with $\mathcal{V}_{A}=\{2\}$. The dots in red represent nodes in $\mathcal{V}_{A}$.
  • Figure 3: An example of a graph associated with a given bipartition, the SA index graph, and the RoD index graph. There exist five connected components in the SA index graph, while the RoD index graph is connected. The dots in red represent nodes in $\mathcal{V}_{A}$. The vertices in SA/RoD index graphs satisfy $2=a_{12}, 4=a_{14}, 9=a_{23}, 11=a_{25}, 12=a_{26}, 16=a_{34}, 23=a_{45}, 30=a_{56}$.
  • Figure 4: Various types of 1-vertex additions. (a)(b) Type $A_{1}$ 1-vertex addition. (c)(d) Type $D_{1}$ 1-vertex addition. (e)Type $A_{2}$ 1-vertex addition. (f) Type $D_{2}$ 1-vertex addition. The dots in red represent nodes in $\mathcal{V}_{A}$.
  • Figure 5: 2-vertex additions. (a) The newly added two nodes belong to $\mathcal{V}_{A}$. (b) The newly added two nodes belong to $\mathcal{V}_{D}$. The dots in red represent nodes in $\mathcal{V}_{A}$.
  • ...and 7 more figures

Theorems & Definitions (80)

  • Definition 2.1
  • Definition 2.2
  • Proposition 1
  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Theorem 2
  • proof
  • Definition 2.3: Triple index graph
  • ...and 70 more