relative perturbation edge changes are tied to node degree

[[concept]]

Let $\tilde{S}=S+DS+SD$ be a relative perturbation on $S$. Locally, we have

\tilde{S}_{ij} &= S_{ij} + (DS)_{ij} + (SD)_{ij} \\ &= S_{ij} + \sum_{k \in N(j)} D_{ik}S_{kj} + \sum_{k \in N(i)} S_{ik}D_{ki} \\ &= S_{ij} + (\propto \text{deg}(j)) + (\propto \text{deg}(i)) \end{aligned}$$ This tells us that the edge changes in the [[Concept Wiki/perturbations on a graph shift operator\|perturbed]] graph is tied to the degrees of the nodes. ### Mentions ```dataview TABLE FROM [[]] FLATTEN choice(contains(artist, this.file.link), 1, "") + choice(contains(author, this.file.link), 1, "") + choice(contains(director, this.file.link), 1, "") + choice(contains(source, this.file.link), 1, "") as direct_source WHERE !direct_source ```