Let $\{G_n{\}}_n$ be a graph sequence such that $\underset{n\to \mathrm{\infty }}{lim}t(F,G_n)=\tilde{t}(F,W)$ for all motifs $F$ where

• $t(\cdot )$ is the graph homomorphism density
• $\tilde{t}(\cdot )$ is the graphon homomorphism density and
• $W$ is a graphon

Then $\{G_n\}$ is a convergent graph sequence with limit $W$.

This is a {“local”} idea of convergence since it checks to see if {sampled subgraphs converge "in distribution" up to the limiting object}. This is called {left convergence since it deals with left homomorphism densities} $t(F,G_n)$ and $\tilde{t}(F,W)$.

• (we count the occurrences of the motif within the graph/graphon)
  ^note-1

This is not the only way to identify a (dense) convergent graph sequence. Another definition is based on the convergence of {min-cuts or right homomorphisms} $t(G_n,F)$ and $\tilde{t}(W,F)$.

• This is a {more “global”} notion of convergence that is used sometimes by graph theorists or in physics (micro-canonical ground state energy)
  ^note-2

For dense graphs, left and right convergence are equivalent (for the metric we like) without proof.
^note-3