<p>This study investigates scaled consensus problems in hybrid multi-agent systems (HMASs), focusing on edge dynamics regulated by pulse-modulated control. This work is different from traditional node-based consensus because it looks at synchronization at the edge level with time-varying scaling factors and uses line graph transformation to model interactions. A new pulse-modulated control framework is introduced. It combines sampled-data control and impulsive updates to make the system more stable, faster to converge, and better at communicating. We use stochastic matrix theory, graph Laplacians, and Lyapunov-based analysis to show that a spanning tree in the line graph guarantees synchronization and that there are necessary and sufficient conditions for achieving edge consensus. Theoretical results are confirmed by numerical simulations, which show that the protocol is strong. A unified control strategy that connects continuous and discrete-time dynamics, a generalized stability criterion, and rigorous connectivity analysis are some of the most important contributions. The results have an effect on robotic coordination, smart grids, and distributed sensor networks, and they also open the door to more research on switching topologies, time delays, and adversarial resilience.</p>