  The paradigm of federated learning (FL) to address data privacy concerns by
locally training parameters on resource-constrained clients in a distributed
manner has garnered significant attention. Nonetheless, FL is not applicable
when not all clients within the coverage of the FL server are registered with
the FL network. To bridge this gap, this paper proposes joint learner referral
aided federated client selection (LRef-FedCS), along with communications and
computing resource scheduling, and local model accuracy optimization (LMAO)
methods. These methods are designed to minimize the cost incurred by the
worst-case participant and ensure the long-term fairness of FL in hierarchical
Internet of Things (HieIoT) networks. Utilizing the Lyapunov optimization
technique, we reformulate the original problem into a stepwise joint
optimization problem (JOP). Subsequently, to tackle the mixed-integer
non-convex JOP, we separatively and iteratively address LRef-FedCS and LMAO
through the centralized method and self-adaptive global best harmony search
(SGHS) algorithm, respectively. To enhance scalability, we further propose a
distributed LRef-FedCS approach based on a matching game to replace the
centralized method described above. Numerical simulations and experimental
results on the MNIST/CIFAR-10 datasets demonstrate that our proposed LRef-FedCS
approach could achieve a good balance between pursuing high global accuracy and
reducing cost.
