Federated learning emerges as a key enabler toward pervasive intelligence across IoT ecosystems with provable privacy guarantees. While recent efforts on client selection have been made for optimizing its communication efficiency in iterative model aggregation over resource-constrained networks, their scalability fundamentally breaks down in dense deployments. This limitation stems from the NP-hard complexity of congestion-aware scheduling, where co-channel interference creates exponentially growing solution spaces. In this context, we present a novel client selection framework with asymptotic scalability in massive IoT, which leverages the intrinsic graph topology with insights from statistical physics. First, this work formulate a universal client selection problem, capturing both positive network externalities derived from collaborative knowledge exchange and congestion effects induced by co-channel interference. This formulation is then transformed into a node classification task via Ising spin Hamiltonian mapping, establishing an explicit connection between federated learning, statistical physics, and graph optimization. Building on this foundation, we develop a lightweight graph neural solver that adaptively selects clients via recognizing node state with iterative neighbor aggregation of learnable embeddings. Comprehensive experiments validate that our approach maintains state-of-the-art scheduling performance, while scaling to network sizes orders of magnitude beyond what conventional methods can handle.