Leaderless Formation Motion Over Minimal Periodic Directed Rings: A Structural Imbalance Approach

This paper studies leaderless collective motion of a prescribed relative formation over periodically weighted directed rings. A period-2 weighting pattern (alternating positive edge weights) is shown to select the group's constant translational velocity without leaders, global references, or externally prescribed motion commands. For second-order agents, the classical modal stability condition is applied to the explicit period-2 spectrum to obtain a finite-spectrum gain threshold, whereas the collective velocity determined by the left null vector of the Laplacian remains invariant. This property is combined with relative shape regulation, yielding a distributed translation mechanism for the prescribed formation. An explicit velocity formula further reveals how the period-2 weight ratio blends the odd- and even-indexed average initial velocities, leading to a one-parameter inverse-design characterization of the open segment of achievable constant velocities between the two class averages. Beyond fixed-size networks, the paper studies pair insertion and pair deletion rules that preserve the alternating topology. In particular, assigning a local velocity offset during pair insertion produces a computable direction-consistent change of the collective velocity. The post-update dynamics then recover to the corresponding translating formation manifold. Comparative simulations with uniform-ring regulation, an additional thirty-agent case, and a gain-threshold sweep support the network-selected-motion interpretation and the finite-spectrum threshold.