I have a weighted graph and want to apply a diffusion step to it. I read this paper, where they formulate such a diffusion step for unweighted graphs:
$Z_i(t+1)=Z_i(t)+\alpha\sum_j \left[Z_j(t)-Z_i(t)\right],$
leading to
$Z(t+1) = (1+\alpha H)Z(t),$
where H is the negative of the Graph Laplacian. Now, please correct me if I'm wrong: $\sum_j \left[Z_j(t)-Z_i(t)\right]$ is the spatial derivative for unweighted graphs. The (approximation in) continuous form would be
$\frac{f(x)-f(x+\Delta x)}{\Delta x}$,
with $\Delta x=1$. So I want to adapt this to weighted graphs. They say in the paper that diffusion kernels are not restricted to unweighted graphs and H can just be constructed from the weight matrix instead of the adjaciency matrix. However, I'm not quite sure if this can be applied to the diffusion step I'm interested in, because clearly, for $\Delta x \not= 1$ (as would be the case for weighted graphs), I would need a division to get a proper spatial derivative.
Do you have any tips on how to do this? Maybe by creating a "weight matrix" based on similarities instead of distances? Ideally, I would like an elegant (but valid) operator like $(1+\alpha H)$.
Please let me know if things are unclear. Thanks for your help!
I realised that the weights in a weighted graph, in this case, are assumed to represent similarities, not distances. It therefore makes sense that this diffusion step can be directly extended to weighted graphs.