  Deep learning has been successfully adopted in mobile edge computing (MEC) to
optimize task offloading and resource allocation. However, the dynamics of edge
networks raise two challenges in neural network (NN)-based optimization
methods: low scalability and high training costs. Although conventional
node-output graph neural networks (GNN) can extract features of edge nodes when
the network scales, they fail to handle a new scalability issue whereas the
dimension of the decision space may change as the network scales. To address
the issue, in this paper, a novel link-output GNN (LOGNN)-based resource
management approach is proposed to flexibly optimize the resource allocation in
MEC for an arbitrary number of edge nodes with extremely low algorithm
inference delay. Moreover, a label-free unsupervised method is applied to train
the LOGNN efficiently, where the gradient of edge tasks processing delay with
respect to the LOGNN parameters is derived explicitly. In addition, a
theoretical analysis of the scalability of the node-output GNN and link-output
GNN is performed. Simulation results show that the proposed LOGNN can
efficiently optimize the MEC resource allocation problem in a scalable way,
with an arbitrary number of servers and users. In addition, the proposed
unsupervised training method has better convergence performance and speed than
supervised learning and reinforcement learning-based training methods. The code
is available at \url{https://github.com/UNIC-Lab/LOGNN}.
