I was thinking of the following setting. Image we have a finite graph with nodes $V$, $|V| = n$. Furthermore we look at functions from each node into the real numbers, so $f:V\rightarrow \mathbb{R}$. To each function we associate edges $E_f$, so the edges can change for different $f$. Edges could be defined by weights $w_{ij}^f$ and collected into a matrix $A_f \in \mathbb{R}^{n \times n}$. I was trying to define a norm and/or inner product on the space of $(f,A_f)$. Is there some obvious or sensible way to do this ? I was thinking to define the squared distance between two elements $(f,A)$ and $(g,B)$ as $||A-B||^2_F+||A \cdot f-B \cdot g||^2_E$, where $||\cdot||_F$ is the Frobenius norm and $||\cdot||_2$ the Euclidean one. For example if we have two edge sets defined through $A$ resp $B$ with functions $f$ resp $g$ then if the Frobenius norm is zero we know they have the same edge set and if the euclidean one is zero we know the have the same image, For example take $A$ as some laplacian and $B$ as another one.
Forgive me for any ignorance on the subject. I am not even close to being knowledgable graph theory. Any literature and/or hints would be greatly appreciated.