Topological Properties on Neural Networks Using Graph Properties

A computer-based model called a neural network comprises neurons inspired by the human brain’s structure and operations. In graph theory, neurons are represented as nodes, and connections from one neuron to another neuron are represented as edges. Neural networks have been widely applied in artificial intelligence, machine learning, deep learning, biology, etc. Here, the approach of graph properties has been used for certain neural networks like convolutional neural networks (CVNNs), modular neural networks (MNNs), generalised regression neural networks (GRNNs), and Hopfield neural networks (HNNs). This article investigates nondeterministic polynomial-time (NP)-complete and NP-hard problems on several neural networks by considering their graphs. To explore their topological properties, we study their structures of maximum clique number, chromatic number, minimum vertex cover, independence number, perfect matching, and minimum domination number. We compute these graph properties on the considered neural networks and analyse the structural behaviour of the neural networks. For instance, the clique and chromatic numbers of CVNNs, MNNs, and GRNNs are 2, respectively, whereas for the Hopfield neural network, it is m. These results are useful in identifying densely connected group of neurons within the network architecture, which helps in model compression and interpretation of network functionality. The minimum domination numbers of MNNs, GRNNs and HNNs are 3, 4, and 1, respectively. These results are useful in model simplification, identifying critical vertices for diagnosis, and minimising network connections for improved efficiency. For every graph property, we have discussed an algorithm to verify the effectiveness of the theoretical results. Comparative analysis and future direction have been discussed in the article.