Directional Bounded-Confidence Matrix Alignment in Vicsek Flocking

Abstract

This paper proposes a directional bounded-confidence variant of the discrete-time Vicsek multiagent model, in which pairwise heading alignment is modulated by a state-dependent, positive semidefinite matrix that independently gates orthogonal radial and tangential components in the instantaneous line-of-sight frame. We analyze the normalized, matrix-weighted system dynamics, proving a uniform non-vanishing bound on proposal vectors that guarantees global well-posedness across arbitrary network states. For local small-spread states under complete metric interaction, we prove linear geometric contraction of the heading diameter and establish an offline-verifiable spatial bootstrap condition that guarantees metric contact preservation and coincidence avoidance a priori. A counterexample demonstrates invariant polarized states on connected matrix graphs, delineating systemic structural limits. Numerical experiments against unfiltered Vicsek averaging, scalar tied confidence, trace-matched scalar baselines, and restricted-angle rules demonstrate component-selective filtering without rejected-channel leakage, along with improved fixed-bound alignment on multi-seed benchmarks across varying population densities.