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This manuscript presents a novel distributed greedy framework applicable to a network of thermostatically controlled loads (TCLs) to desynchronize the network's aggregated power consumption. Compared to the existing literature, our proposed framework offers two distinct novelties. First, our proposed algorithm relaxes the restrictive assumptions associated with the communication graph among TCLs. To elaborate, our algorithm only requires a connected graph to execute control, a condition less demanding than its counterpart algorithms that mandate a star architecture, K-regular graphs, or undirected connected graphs. Second, a significant novel feature is the relaxation of the obligation to share private information, such as each unit's local power consumption and appliance temperatures, either with a central coordinator or neighboring TCLs. The findings presented in this brief are validated through simulations conducted over a network comprising 1000 TCLs.