Product of Tensors and Description of Networks

TL;DR

This work reframes network signal propagation as a tensor-algebra problem on directed acyclic graphs, introducing activation tensors for each node and defining a total network tensor that encodes the final state. The authors construct a rigorous pipeline in which activation tensors are inflated via blow and forget operations to form order-$q$ node tensors, and then combined with the Batthacharya-Mesner Product (BMP), a generalized $d$-ary tensor product, to produce the total tensor. The central result states that the final network tensor $N$ is exactly the BMP of these transformed node tensors, providing a precise algebraic bridge between network structure and high-order tensor operations. The work also discusses geometric perspectives (toric varieties) for observable networks, compares BMP properties to classical matrix algebra, and provides a practical Python implementation to compute the BMP for networks of realistic size, enabling analysis and visualization of complex network state distributions.

Abstract

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 \textit{activation tensors}. The distribution of the signal is encoded in the \textit{total tensor} of the network. We formally prove that the total tensor can be obtained by computing the \textit{Batthacharya-Mesner 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 \textit{blow} and \textit{forget}. Our theoretical framework can be validated through the related code developed in Python.

Figures (7)

• Figure 1: Notation. The labeling notation we consider for a tensor $T \in \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$.
• Figure 4: Markov chain with three nodes. Tensor network with three nodes $a, b, c$. Node $a$ transmits a binary signal to node $b$ which, in turn, sends a signal to node $a$.
• Figure 5: Total tensor of the Markov chain with three nodes. Representation of the total network of a binary Markov chain with three nodes $a, b, c$ and parameters $\alpha, \beta\in \mathbb{C}$. A possible reading of the values of the total tensor is the following: $N_{000}$ means that $a$ casts $0$ ($\alpha$), then $b$ obeys and casts $0$ ($\alpha$ again), and $c$ obeys and takes the $0$ state (another $\alpha$), so $N_{000}=\alpha^3$; $N_{011}$ means that $a$ casts $0$ ($\alpha$), then $b$ disobeys and casts $1$ ($\beta$), while $c$, which receives $1$, obeys and takes the $1$ state (another $\alpha$), so $N_{011}=\alpha^2\beta$; and so on.
• Figure 6: Acyclic triangle with three nodes. Tensor network with three nodes $a, b, c$ and their associated activation tensors $A, B, C$. Node $a$ transmits a binary signal to node $b$ and node $c$. In turn, node $b$ sends a signal to node $c$.
• Figure 7: Total tensor of the acyclic triangle. Representation of the total network of a binary acyclic triangle with nodes $a, b, c$ and parameters $\alpha, \beta\in \mathbb{C}$.

Theorems & Definitions (30)

• Definition 1
• Definition 2
• Definition 3
• Remark 4
• Example 5
• Definition 6
• Example 7
• Example 8
• Definition 9
• Remark 10