This letter addresses the challenge of network synchronization under limited communication, involving heterogeneous agents with different dynamics and various network topologies, to achieve a consensus. We investigate the distributed adaptive control for interconnected unknown linear subsystems with a leader and followers, with the presence of input-output disturbance. We enhance the communication within multi-agent systems to achieve consensus under the leadership's guidance. While the measured variable is similar among the followers, the incoming measurements are weighted and constructed based on their proximity to the leader. We also explore the convergence rates across various balanced topologies (Star-like, Cyclic-like, Path, Random), featuring different numbers of agents, using distributed first and high-order tuners. Moreover, we conduct several numerical simulations across various networks, agents and tuners to evaluate the effects of sparsity in the interaction between subsystems using the L-2-norm and L-infinity-norm. Some networks exhibit a trend where an increasing number of agents results in smaller errors, although this is not universally the case. Additionally, patterns observed at initial times may not reliably predict overall performance across different networks. Finally, we demonstrate that the proposed modified high-order tuners outperform its counterpart, and we provide related insights along with our conclusions.
