The unnormalised weighted Laplacian of a weighted simple graph $G(V,E)$ is:
$$ L_{ij} = \begin{cases} \begin{split} -w_{ij} & \quad \text{if} \quad i \sim j \\ w_{i} & \quad \text{if} \quad i = j \\ 0 \quad & \quad \text{otherwise} \end{split} \end{cases} $$
Where $w_i$ is the sum of the weights neighbouring the vertex $i$. $L$ acts on a function $\phi(V)$ of the vertices.
$$L \phi (i) = \sum_{j/ i \sim j} w_{ij} (\phi_j - \phi_i)$$
This is interpreted as the discrete Laplace operator acting on the graph. But what if $w_{ij}$ depends on $\phi_i$ and $\phi_j$? In particular it is a function of their mean $w_{ij}(\psi_{ij}) = w_{ij} ((\phi_i + \phi_j)/2)$.
$$L \phi (i) = \sum_{j/ i \sim j} w_{ij} (\phi_i, \phi_j) (\phi_j - \phi_i)$$
The motivation for this was was for $\phi_i$ to not only depend on the differences between neighbouring vertexes $ (\phi_j - \phi_i)$ but also on the neighbouring edges $\psi_{ij}$. Is there any theory about these types of systems?