  This paper addresses the challenge of network synchronization under limited
communication, involving heterogeneous agents with different dynamics and
various network topologies, to achieve consensus. We investigate the
distributed adaptive control for interconnected unknown linear subsystems with
a leader and followers, in 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 three distributed algorithms, ranging from
first- to high-order tuners to effectively address time-varying regressors. The
mathematical foundation is rigorously presented from the network designs of the
unknown agents following a leader, to the distributed methods. 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 $L_\infty-$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 tuner outperforms
its counterparts, and we provide related insights along with our conclusions.
