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A graphon is a measurable symmetric function $W: [0,1]\to [0,1].$ By Lovasz's book "Large networks and graph limits" we know for any graph sequence $G_1, G_2, \dots G_i,\dots$ there exists a graphon $W$ such that $\lim ||G_n-W||_\square=0$, where $||\cdot||_\square$ is cut norm.

Note that every graph can be viewed as a graphon takes value only from $\{0,1\}$.

Thus graph sequence could be approximated by graphon. So, could graphon be approximated by a sequence of uniformally-weighted graph(graphon that takes value only from $\{0,d\}$ for some $0<d<1$)?

Question: For any graphon $W$ with $\int W \leq 0.5$ and any $d\geq 2\int W$, does there exist a sequence of graphon $W_1,W_2,\dots$ such that every $W_i$ takes value only from $\{0,d\}$ and $\lim ||W_n-W||_\square=0$?

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    $\begingroup$ Every graphon is the limit of a sequence of graphs. This follows from the completeness of the graphon space with the cut metric. $\endgroup$
    – TheBestMagician
    Commented Sep 20 at 15:50

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