Product of Tensors and Description of Networks

Any kind of network can be naturally represented by a Directed Acyclic Graph (DAG); additionally, activation functions can determine the reaction of each node of the network with respect to the signal(s) incoming. We study the characterization of the signal distribution in a network under the lens of tensor algebra. More specifically, we describe every activation function as tensor distributions with respect to the nodes, called $\mathit{\text{activation}}\mathit{\text{tensors}}$. The distribution of the signal is encoded in the $\mathit{\text{total}}\mathit{\text{tensor}}$ of the network. We formally prove that the total tensor can be obtained by computing the $\mathit{\text{Batthacharya-Mesner}}\mathit{\text{Product}}$ (BMP), an $n$-ary operation for tensors of order $n$, on the set of the activation tensors properly ordered and processed via two basic operations, that we call $\mathit{\text{blow}}$ and $\mathit{\text{forget}}$. Our theoretical framework can be validated through the related code developed in Python.