network-shaped.

Three graph product operations have been considered so far: Cartesian product, direct (tensor) product, and strong product [@sayama2016estimation]. We will also discuss their generalization later in this paper. Let $G=(V_G, E_G)$ and $H=(V_H, E_H)$ be two simple graphs as factor networks, where $V_G$ (or $V_H$) and $E_G$ (or $E_H$) are the sets of nodes and edges of $G$ (or $H$), respectively. Also let $A_G$ and $A_H$ be the adjacency matrices of $G$ and $H$, respectively. For all of the aforementioned three graph products, a new node set is given by a Cartesian product of $V_G$ and $V_H$, i.e., $\{(g,h) \; | \; g\in V_G , \, h\in V_H\}$. Then the three graph products are defined as follows:

Cartesian product $G \operatorname*{\Box}H$

:   A network in which two nodes $(g,
      h)$ and $(g', h')$ are connected if and only if $g = g'$ and $(h,
      h') \in E_H$, or $h = h'$ and $(g, g') \in E_G$. Its adjacency matrix is given by $$A_{G \operatorname*{\Box}H} = A_G \operatorname*{\oplus}A_H = A_G \operatorname*{\otimes}I_{|V_H|} + I_{|V_G|} \operatorname*{\otimes}A_H, \label{eq:cp}$$ where $\operatorname*{\oplus}$ and $\operatorname*{\otimes}$ are Kronecker sum and Kronecker product operators, respectively, and $I_n$ is an $n \times n$ identity matrix.

Direct (tensor) product $G 