Effect of Antagonistic Interactions on Agreement of Agents over a Hierarchical Ring Digraph

Hierarchical cyclic pursuit has garnered significant attention among researchers in recent years. Along similar lines, this work aims to study ring digraphs, with a hierarchical "necklace" structure, involving antagonistic interactions. In the first part, identical antagonistic interactions, represented by negative edge weights, within macro vertices are considered. A study on the effect of such negative interactions on the consensus of agents is presented and bounds on negative gain, for consensus, are derived. Subsequently, heterogeneous antagonistic interactions are considered and the effect of these interactions on the consensus of the agents is analyzed. The set of points where consensus is achievable, when the edge weights are varied within the derived permissible bounds, is also investigated. Finally, numerical examples and corresponding simulations are presented to validate the analytical results.