Isoperimetric Certificates for Flocking: Weighted Cheeger Bounds on Cohesion, Tracking, and Fragmentation in State-Dependent Networks

Abstract

The continuous-time flocking law of Olfati-Saber is, in its alignment term, a flow generated by a state-dependent weighted graph Laplacian. We exploit this identification to bring the weighted Cheeger constants and Neumann/Dirichlet isoperimetric inequalities of Chung and Oden to bear on the analysis of second-order multi-agent dynamics. The control literature typically certifies flocking through direct spectral quantities—the algebraic connectivity (Fiedler value) of the interaction Laplacian. We instead derive cut-based certificates: (i) a Neumann isoperimetric lower bound on the rate of velocity (heading) consensus; (ii) a Dirichlet isoperimetric lower bound on the rate at which the flock locks onto a navigational target, recovering and refining grounded-Laplacian leader–follower estimates; and (iii) an isoperimetric fragmentation criterion that ties imminent splitting of the swarm to the collapse of the weighted Cheeger constant along a sparse cut. Because a cut is a local quantity, these certificates are monitorable without a global eigensolve; we make this operational with an analysis of computational complexity, distributed implementation, and robustness to measurement noise, and we characterize the regimes in which each bound is tight. In the mean-field limit we conjecture, and pose as an open problem, how the discrete weighted Cheeger constant converges to a continuum Cheeger constant, so that the discrete certificates would appear as consistent discretizations of the Riemannian Cheeger–Buser inequality. Finally we propose an isoperimetric connectivity-maintenance controller that regulates the weighted Cheeger constant directly, and contrast it with algebraic-connectivity maximization. An expanded numerical suite (varying agent number, density regime, and informed fraction) confirms every certificate and quantifies its slack.