<p>This paper investigates the problem of scaled edge consensus in directed hybrid multi-agent systems (HMASs) characterized by both continuous-time and discrete-time edge dynamics. Unlike classical consensus models, we introduce edge-specific scaling factors and propose novel distributed event-triggered and self-triggered control protocols that enable each edge to update its control input based solely on locally available information. By modeling edge interactions using a directed line graph and the corresponding edge Laplacian, we establish sufficient conditions for asymptotic consensus via Lyapunov-based analysis. The proposed strategies significantly reduce communication and computation by eliminating the need for continuous monitoring or centralized scheduling. Numerical simulations validate the theoretical results, confirming that all scaled edge states converge to a common value under both triggering mechanisms. The work generalizes traditional edge consensus frameworks by incorporating heterogeneous scaling and asynchronous control, and provides a scalable, resource-efficient solution applicable to a wide range of networked dynamical systems.</p>