graph sequence converges if and only if the induced graphon sequence converges

[[concept]]

Theorem

Sequences of graphs $\{G_n\}$ converging to graphon $W$ also converge in the cut norm: $G_n\to W⟺||W_n-W|{|}_{\square }\to 0\text{ as }n\to \infty$ Where $W_n$ is the induced graphon of $G_n$.

In fact, to show that left and right convergence with the homomorphism density are equivalent is via convergence wrt the cut metric/cut norm as above.

Proofs are in the book Large networks and convergent graph sequences by Lavàsz (online). For the equivalence in homomorphism density, see section counting and inverse counting lemmas.

Mentions

Mentions

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