[[concept]]

Cut distance (arbitrary graphs)

Let $G_{n}$ and $G_{m}$ be two graphs with possibly different node counts $n$ and $m$ respectively. The cut distance is given by $δ_{□} (G_{n}, G_{m}) = δ_{□} (W_{n}, W_{m})$ Where $W_{n}, W_{m}$ are the induced graphons of the graphs.

Special Cases

cut distance (same node count, same labels)

Let $G$ and $G ’$ be graphs with the same $n$ and the same node sets $V = V ’$ (ie, same node labelling). The cut distance is given by $d_{□} (G, G ’) = ∣∣ A - A ’ ∣ ∣_{□}$ where $∣∣ \cdot ∣ ∣_{□}$ is the cut norm.

cut distance (same node count, potentially different labels)

Let $G$ and $G ’$ have the same number of nodes $n$ . The cut distance is given by $\hat{δ}_{□} (G, G ’) = min_{p \in P} ∣∣ A - P^{T} A ’ P ∣ ∣_{□}$ ie, we look at all possible permutations of the nodes of $G ’$ WRT the node labelings of $G$ .

Review

Arbitrary graphs {are not comparable as graphs on their own}, so we compare them as {induced graphons} instead

Mentions

Mentions

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