I am reading this paper on treating learning on graph neural networks as solving a diffusion using PDEs. The definitions of the terms involved are as follows:
$G = (V, E)$ is defined as an undirected graph, with $|V| = n$ nodes and $|E| = e$ edges, where $V$ and $E$ are the set of vertices and edges respectively. Furthermore, $\vec{\textbf{x}}$ and $F$ denote the features defined on the vertices and edges respectively.
The gradient of $\vec{\textbf{x}}$ is defined as $\nabla{\vec{\textbf{x}}} = x_{j} - x_{i}$. The divergence of $\operatorname{div}(F)_{i}$ is given by $\sum_{j=1}^{n}w_{ij}F_{ij}$, where $w_{ij} = w_{ji} = 1$ iff $(i,j) \in E$.
The paper also defines the two operators to be adjoint. $\langle\nabla{\vec{\textbf{x}}}, F\rangle$ = $\langle\vec{\textbf{x}}, \operatorname{div}(F)\rangle$.
All the above are the definitions.
The diffusion equation is modeled as: $$\frac{\partial{\vec{\textbf{x}(t)}}}{\partial{t}} = \operatorname{div}[\textbf{G}(\vec{\textbf{x}(t)})\nabla{\vec{\textbf{x}(t)}}]$$ with an initial condition $\vec{\textbf{x}(0)} = 0$, and $\textbf{G} = \operatorname{diag}(a(x_{i}(t), x_j(t))$, which is a diagonal matrix with $e \times e$ elements.
Using the definition of the gradient and divergence, the above equation becomes $$\frac{\partial{\vec{\textbf{x(t)}}}}{\partial{t}} = \textbf{A}(\vec{\textbf{x}(t)})-\textbf{I})\vec{\textbf{x}(t)},$$ where $\textbf{A}$ is an $n \times n$ matrix.
I can follow how the first term in the second equation is arrived at. What I can't understand is the second term. If we look at the diffusion equation, $\textbf{G}(\vec{\textbf{x}(t)})\operatorname{div}(\nabla{\vec{\textbf{x}(t)}})$ is the product of the diffusion function $\textbf{G}$ and the Laplacian of the vector. I don't quite follow the result the authors have arrived at. I am pretty sure it is something trivial that I am missing out on. Would appreciate if someone could point me in the right direction.