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Projection-based discrete-time consensus on the unit sphere

Johan Thunberg, Galina Sidorenko

TL;DR

This paper analyzes a projection-based discrete-time consensus algorithm for agents evolving on the unit sphere $\mathbb{S}^{d-1}$, updating via $x_i(k+1)=\frac{\sum_j a_{ij} x_j(k)}{\|\sum_j a_{ij} x_j(k)\|}$ under a weight matrix $A$ aligned with a strongly connected graph. By formulating a compact matrix form and examining the tangent-space differential, the authors establish conditions under which consensus points are the only stable fixed points: for $d\ge 3$, symmetric diagonally dominant $A$ on a symmetric connected graph yield unstable non-consensus fixed points, implying convergence to consensus for almost all initial conditions. They also show a measure-zero result for unit-circle cases (and complete graphs) indicating that fixed points not in consensus or antipodal configurations are rare across weight matrices, though symmetry can complicate fixed-point structure. Simulations support the theory and illustrate how symmetry and graph topology affect fixed points, underscoring robustness of consensus in broad settings while highlighting nuanced exceptions. Overall, the work advances understanding of manifold-valued consensus and its dependence on network symmetry, structure, and spectral properties.

Abstract

We address discrete-time consensus on the Euclidean unit sphere. For this purpose we consider a distributed algorithm comprising the iterative projection of a conical combination of neighboring states. Neighborhoods are represented by a strongly connected directed graph, and the conical combinations are represented by a (non-negative) weight matrix with a zero structure corresponding to the graph. A first result mirrors earlier results for gradient flows. Under the assumptions that each diagonal element of the weight matrix is more than $\sqrt{2}$ larger than the sum of the other elements in the corresponding row, the sphere dimension is greater or equal to 2, and the graph, as well as the weight matrix, is symmetric, we show that the algorithm comprises gradient ascent, stable fixed points are consensus points, and the set of initial points for which the algorithm converges to a non-consensus fixed point has measure zero. The second result is that for the unit circle and a strongly connected graph or for any unit sphere with dimension greater than or equal to $1$ and the complete graph, only for a measure zero set of weight matrices there are fixed points for the algorithm which do not have consensus or antipodal configurations.

Projection-based discrete-time consensus on the unit sphere

TL;DR

This paper analyzes a projection-based discrete-time consensus algorithm for agents evolving on the unit sphere , updating via under a weight matrix aligned with a strongly connected graph. By formulating a compact matrix form and examining the tangent-space differential, the authors establish conditions under which consensus points are the only stable fixed points: for , symmetric diagonally dominant on a symmetric connected graph yield unstable non-consensus fixed points, implying convergence to consensus for almost all initial conditions. They also show a measure-zero result for unit-circle cases (and complete graphs) indicating that fixed points not in consensus or antipodal configurations are rare across weight matrices, though symmetry can complicate fixed-point structure. Simulations support the theory and illustrate how symmetry and graph topology affect fixed points, underscoring robustness of consensus in broad settings while highlighting nuanced exceptions. Overall, the work advances understanding of manifold-valued consensus and its dependence on network symmetry, structure, and spectral properties.

Abstract

We address discrete-time consensus on the Euclidean unit sphere. For this purpose we consider a distributed algorithm comprising the iterative projection of a conical combination of neighboring states. Neighborhoods are represented by a strongly connected directed graph, and the conical combinations are represented by a (non-negative) weight matrix with a zero structure corresponding to the graph. A first result mirrors earlier results for gradient flows. Under the assumptions that each diagonal element of the weight matrix is more than larger than the sum of the other elements in the corresponding row, the sphere dimension is greater or equal to 2, and the graph, as well as the weight matrix, is symmetric, we show that the algorithm comprises gradient ascent, stable fixed points are consensus points, and the set of initial points for which the algorithm converges to a non-consensus fixed point has measure zero. The second result is that for the unit circle and a strongly connected graph or for any unit sphere with dimension greater than or equal to and the complete graph, only for a measure zero set of weight matrices there are fixed points for the algorithm which do not have consensus or antipodal configurations.
Paper Structure (19 sections, 6 theorems, 78 equations, 1 table)

This paper contains 19 sections, 6 theorems, 78 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Consensus algorithm
  4. Compact representation of consensus algorithm
  5. Fixed points and consensus points
  6. Tangent space, differential and unstable fixed points
  7. Properties of the differential
  8. Consensus for symmetric weight matrices
  9. Gradient ascent
  10. Main results
  11. General weight matrices
  12. Main result
  13. Simulations
  14. Conclusions
  15. Proofs
  16. ...and 4 more sections

Key Result

Proposition 1

If $d \geq 2$ and $A$ is a weight matrix for a strongly connected directed graph $\mathcal{G}(n) = (\mathcal{V}(n), \mathcal{E})$ which satisfies then $\text{det}(\mathcal{D}f(x)) \neq 0$ for all $x\in (\mathbb{S}^{d-1})^n$.

Theorems & Definitions (6)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Theorem 1
  • Proposition 4
  • Theorem 2