  A biological neural network is constituted by numerous subnetworks and
modules with different functionalities. For an artificial neural network, the
relationship between a network and its subnetworks is also important and useful
for both theoretical and algorithmic research, i.e. it can be exploited to
develop incremental network training algorithm or parallel network training
algorithm. In this paper we explore the relationship between an ELM neural
network and its subnetworks. To the best of our knowledge, we are the first to
prove a theorem that shows an ELM neural network can be scattered into
subnetworks and its optimal solution can be constructed recursively by the
optimal solutions of these subnetworks. Based on the theorem we also present
two algorithms to train a large ELM neural network efficiently: one is a
parallel network training algorithm and the other is an incremental network
training algorithm. The experimental results demonstrate the usefulness of the
theorem and the validity of the developed algorithms.
