Asynchronous Nonlinear Sheaf Diffusion for Multi-Agent Coordination

TL;DR

The paper tackles asynchronous coordination for heterogeneous multi-agent systems by embedding goals into coordination sheaves and formulating a Dirichlet-energy minimization problem. It introduces an asynchronous nonlinear sheaf diffusion algorithm based on the nonlinear sheaf Laplacian $L_{\uparrow\mathcal{F}}^{\nabla U}$, and proves global linear convergence under bounded delays $B$ with a rate tied to spectral properties of the sheaf Laplacian and edge potentials. The results hold from arbitrary initial conditions and bridge to classical consensus when specialized to the constant (linear) case, with numerical simulations validating the theory across various sheaf configurations. Practically, this work enables robust, distributed coordination in networks with communication and compute heterogeneity, by leveraging the algebraic-topological structure of cellular sheaves and their spectra. The findings highlight the importance of the Hessian/spectral characteristics of the sheaf Laplacian in governing convergence speed under asynchrony and suggest design principles to optimize these spectral properties for faster coordination.

Abstract

Cellular sheaves and sheaf Laplacians provide a far-reaching generalization of graphs and graph Laplacians, resulting in a wide array of applications ranging from machine learning to multi-agent control. In the context of multi-agent systems, so called coordination sheaves provide a unifying formalism that models heterogeneous agents and coordination goals over undirected communication topologies, and applying sheaf diffusion drives agents to achieve their coordination goals. Existing literature on sheaf diffusion assumes that agents can communicate and compute updates synchronously, which is an unrealistic assumption in many scenarios where communication delays or heterogeneous agents with different compute capabilities cause disagreement among agents. To address these challenges, we introduce asynchronous nonlinear sheaf diffusion. Specifically, we show that under mild assumptions on the coordination sheaf and bounded delays in communication and computation, nonlinear sheaf diffusion converges to a minimizer of the Dirichlet energy of the coordination sheaf at a linear rate proportional to the delay bound. We further show that this linear convergence is attained from arbitrary initial conditions and the analysis depends on the spectrum of the sheaf Laplacian in a manner that generalizes the standard graph Laplacian case. We provide several numerical simulations to validate our theoretical results.

Figures (5)

• Figure 1: Mental model of a cellular sheaf: over a cycle graph, agent's stalks are tethered together with restriction maps.
• Figure 2: In the above coordination sheaf, $\mathcal{F}_{i \mathop{\mathrm{\trianglelefteqslant}}\nolimits ij} = \mathcal{F}_{j \mathop{\mathrm{\trianglelefteqslant}}\nolimits ij}$ between leaders and leaders are projection onto velocity while the restriction maps between leaders and followers are projection onto position. The restriction maps between followers are the zero map. The diagram layout is evocative of the intended formation.
• Figure 3: Asynchronous convergence for various values of $B$. The top row plots energy while the bottom row plots relative error. The experiments were conducted on different sheaves over the same random 4-regular graph $G$. I) the constant sheaf $\underline{\mathbb{R}}^4$ over $G$. II) a sheaf with random restriction maps $\mathcal{F}_{i\mathop{\mathrm{\trianglelefteqslant}}\nolimits ij}\sim\mathrm{rand}(1,4)$. III) a sheaf with random matrix-weighted edges. The positive semi-definite matrix weights had a 0.2 probability of being strictly positive definite.
• Figure 4: Running sheaf diffusion over the same sheaf $\mathcal{F}$ from 100 different initial conditions with $B=50$. The sheaf is over a random 4-regular graph with random restriction maps $\mathcal{F}_{i\mathop{\mathrm{\trianglelefteqslant}}\nolimits ij}\sim \mathrm{rand}(1,4)$. The initial conditions are sampled randomly from a Gaussian centered at the origin with variance 10.
• Figure 5: (a) Distance between solution obtained via asynchronous sheaf diffusion and orthogonal projection of the initial condition $\mathbf{x}(0)$ onto $\Gamma(G;\mathcal{F})$ for various values of $B$. (b) Smallest non-zero eigenvalue of the sheaf Laplacian ($\lambda_2$) versus number of iterations for asynchronous sheaf diffusion to reach convergence ($t^*$) for Erdos-Renyi random graphs with 20 nodes and 0.3 connection probability. In all cases, the communication bound was fixed to $B=50$.

Theorems & Definitions (23)

• Definition 1
• Example 1: Constant Sheaf
• Remark 1
• Definition 2
• Definition 3
• Definition 4
• Example 2: Moving UAV formations
• Definition 5
• Example 3
• Lemma 1