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Peer-to-peer (P2P) trading is seen as a viable solution to handle the growing number of distributed energy resources in distribution networks. However, when dealing with large-scale consumers, there are several challenges that must be addressed. One of these challenges is limited communication capabilities. Additionally, prosumers may have specific preferences when it comes to trading. Both can result in serious asynchrony in peer-to-peer trading, potentially impacting the effectiveness of negotiations and hindering convergence before the market closes. This paper introduces a connection-aware P2P trading algorithm designed for extensive prosumer trading. The algorithm facilitates asynchronous trading while respecting prosumer's autonomy in trading peer selection, an often overlooked aspect in traditional models. In addition, to optimize the use of limited connection opportunities, a smart trading peer connection selection strategy is developed to guide consumers to communicate strategically to accelerate convergence. A theoretical convergence guarantee is provided for the connection-aware P2P trading algorithm, which further details how smart selection strategies enhance convergence efficiency. Numerical studies are carried out to validate the effectiveness of the connection-aware algorithm and the performance of smart selection strategies in reducing the overall convergence time. [Display omitted] • Develop the connection-aware P2P trading algorithm to address the asynchrony challenge in P2P trading. • Propose smart selection strategies within the context of connection limits in P2P trading. • Provide theoretical guarantees on the convergence of the connection-aware P2P trading algorithm. [ABSTRACT FROM AUTHOR]