I have a directed graph consisting of nodes and edges. A system of ordinary differential equations describes how the value at each node changes with respect to time, as a function of the value of the surrounding nodes that are connected by edges.
After numerical integration I can see how the value at each node changes as a function of time. Is there a way for me to see how each edge affects the change at a single node (using the Jacobian perhaps)?
In this example, if we pick node A, I want to know the relative contributions of each of those vertices (flow coming from nodes B, C and D) at each moment in time.
You can consider the partial derivative of $x_i(t)$ (the value at node $i$) with respect to a constant $a_{jk}$ of the system corresponding to edge $jk$. These will satisfy differential equations as well, since $$ \dfrac{d}{dt} \dfrac{\partial x_i}{\partial a_{jk}} = \dfrac{\partial}{\partial a_{jk}} \dfrac{d x_i}{dt} $$ You will thus end up with a larger system of differential equations involving the $x_i$ and $\partial x_i/\partial a_{jk}$, which can again be solved numerically.