I am looking for the standard terminology for a mathematical structure consisting of a directed graph where each edge "applies a function passing through it".
Formal Definition
Let $G=(V, E)$ be a directed graph with $N$ vertices, indexed $1, \dots, N$. For each edge $(j, i) \in E$, let there be an associated function $a_{ij}: \mathbb{R} \to \mathbb{R}$. (If there is no edge from $j$ to $i$, we can define $a_{ij}(z) = 0$).
The main application I have in mind is a discrete-time nonlinear dynamical system defined on the graph. Let the state of the system at time $k$ be a vector $x(k) \in \mathbb{R}^N$, where $x_i(k)$ is a value associated with vertex $i$. The system evolves according to the equation: $$x_i(k+1) = \sum_{j \in N(i)} a_{ij}(x_j(k))$$ where $N(i)$ is the set of in-neighbors of vertex $i$. This can be written more compactly using a "functional adjacency matrix" $A = (a_{ij})$: $$x_i(k+1) = \sum_{j=1}^N a_{ij}(x_j(k)).$$
Note that if all functions $a_{ij}$ are linear, i.e., $a_{ij}(z) = c_{ij}z$, this system reduces to the standard linear dynamical system $x(k+1) = C x(k)$, where $C$ is the conventional weighted adjacency matrix. This structure can be seen as a type of nonlinear network flow or a recurrent neural network with specified activation functions on the edges.
Question:
My system is a discrete analogue of an integral operator of the form $(Tf)(x) = \int K(x, y) f(y) dy$. My formulation is a nonlinear version of this: $(Tx)_i = \sum_j K_{ij}(x_j)$. The usage in the paper I found (Reference with "Kernel" (p.2)) seems to align with this analysis context. Is it the common name?
Any reference would be appreciated.
