An approach is proposed in the present work in which the desirable properties of the Artificial Neural Networks (ANN) could be achieved through a network flow on an arbitrary graph with no necessity of multi-layer ordering of neurons, i.e. on an arbitrary network structure. The network-flow model of an ANN being proposed in the present work is more general than the multilayer variants of the latter and may lead to finding of more general properties of this network as a powerful mathematical apparatus of methods, algorithms, and software tools exists for the network flows. As the multi-layer ANN are a part of the network flow interpretation all results received in the framework of the multi-layer ANN are preserved.