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Notation and Definitions
Let $G=(V, E)$ be a graph. For $X, Y\subseteq V$, we write $e_G(X, Y)$ to denote the number of edges in $G$ with one end-point in $X$ and the other end-point in $Y$; the edges with both end-points in $X\cap Y$ are counted twice. For a weighted graph $G$ on vertex set $V$, $e_G(X, Y)$ is the sum of the weights of all the edges having one end-point in $X$ and the other end-point in $Y$. Further, we define $$ d_G(X, Y) = \frac{e_G(X, Y)}{|X||Y|} $$ For a partition $\mc P=\set{V_1 , \ldots, V_k}$ we define a weighted graph $G_{\mc P}$ on $V$ by taking the complete graph on $V$ and assigning the edge $uv$ of $G$ a weight of $d_G(V_i, V_j)$ if $u\in V_i$ and $v\in V_j$.
A graphon is a symmetric measurable function $W:I^2\to I$, where $I$ is the unit interval $[0, 1]$. Here symmetry means that $W(x, y)=W(y, x)$. For a finite partition $\mc P=\set{S_1 , \ldots, S_q}$ of $I$ by measurable subsets of positive Lebesgue measure, we write $W_{\mc P}$ to denote the graphon which on $S_i\times S_j$ takes the value $$ \frac{1}{\lambda(S_i)\lambda(S_j)}\int_{S_i\times S_j}W(x, y)\ dxdy $$
The Cut Distance
For two weighted graphs $G$ and $H$ on the same vertex set $V$ of size $n$, we define $$ d_\Box(G, H) = \sup_{A, B\subseteq V} \frac{|e_G(A, B) - e_H(A, B)|}{n^2} $$ For two graphons $W$ and $U$ we define $$ d_\Box(W, U) = \sup_{A, B\subseteq I} \left| \int_{A\times B} W-\int_{A\times B} U \right| $$
Weak Regularity Lemmas
Weak Regularity Lemma for Graphons[*] For a graphon $W$, and for $k\geq 1$, there is a partition $\mc P$ of $I$ into at most $k$-sets with positive measure such that $$ d_\Box(W,W_{\mc P})_\Box \leq \frac{2}{\sqrt{\log k}} $$
Weak Regularity Lemma for Graphs[**] For a graph $G=(V, E)$, and for $k\geq 1$, there is a partition of $V$ into at most $k$ classes such that $$ d_\Box(G, G_{\mc P}) \leq \frac{2}{\sqrt{\log k}} $$
Question.
Can one derive the regularity lemma for graphs using the regularity lemma for graphons?
EDIT: Here is some progress.
Let $G=(V, E)$ be a finite graph on $n$ vertices. A fractional partition of $V$ is a collection $\rho=\set{\rho_1 , \ldots, \rho_q}$ of maps $\rho_i:V\to [0, 1]$ such that for each $v\in V$ we have $\sum_{i=1}^q \rho_i(v)=1$. When each $\rho_i$ takes values in $\set{0, 1}$ then we can think of $\rho$ as a true partition of $V$. We now define a weighted graph $G_\rho$ on $V$. For each $i, j\in [q]$, let $$ e_{ij}(G, \rho) = \sum_{u, v\in V} \rho_i(u)\rho_j(v)\chi_E(u, v) $$ and $$ d_{ij}(G, \rho) = \frac{e_{ij}(G, \rho)}{\lrp{\sum_{v\in V} \rho_i(u)} \lrp{\sum_{v\in V} \rho_j(v)}} $$ Finally, the weight $w_{uv}$ of the edge $uv$ is defined as $$ w_{uv}(G, \rho) = \sum_{i, j=1}^q d_{ij}(G, \rho)\rho_i(u) \rho_j(v) $$ The complete graph on $V$ coupled with this weight function is the weighted graph $G_\rho$. Now let $G=(V, E)$ be a graph where $V=[n]$, and $W:=W_G$ be the corresponding graphon. Let $\mc P=\set{S_1 , \ldots, S_q}$ be a measurable partition of $I$. For each $v\in V=[n]$, let $I_v$ denote the interval $[(v-1)/n, v/n)$. The partition $\mc P$ defines a fractional partition $\rho=(\rho_1 , \ldots, \rho_q)$ of $V$ as $$ \rho_i(v) = \mu(I_v\cap S_i)/\mu(I_v) $$ for all $v\in V$. Let $\mc I$ be the partition $\set{I_v:\ v\in V}$ of $I$. Note that $$ W_{\mc P}(\alpha, \beta) = d_{ij}(G, \rho) \text{ when } \alpha\in S_i, \beta\in S_j $$ Also, let $u, v\in [n]$ and choose $\alpha\in I_u$ and $\beta\in I_v$. Write $W_{\mc P, \mc I}$ to denote $(W_{\mc P})_{\mc I}$. Then
$$ \begin{array}{rl} W_{\mc P, \mc I}(\alpha, \beta) &= \frac{\int_{I_u\times I_v} W_{\mc P}}{\mu(I_u)\mu(I_v)}\\ \\ &= n^2\int_{I_u\times I_v} W_{\mc P}\\ \\ &= n^2\sum_{i, j=1}^q \int_{(I_u\cap S_i)\times (I_v\cap S_j)} W_{\mc P}\\ \\ &= n^2\sum_{i, j=1}^q \int_{(I_u\cap S_i)\times (I_v\cap S_j)} d_{ij}(G, \rho)\\ \\ &= \sum_{i, j=1}^q n \mu(I_u\cap S_i)\cdot n\mu(I_v\cap S_j) d_{ij}(G, \rho)\\ \\ &= \sum_{i, j=1}^q \rho_i(u)\rho_j(v) d_{ij}(G, \rho) = w_{u, v}(G, \rho) \end{array} $$ Now if $A, B\subseteq V$, and $I_A=\bigcup_{u\in A}I_u$ and $B=\bigcup_{v\in B}I_v$, then we have
$$ \begin{array}{rl} \frac{e_G(A, B) - e_{G_\rho}(A, B)}{n^2} &= \frac{1}{n^2} e_G(A, B) - \frac{1}{n^2}\sum_{u\in A, v\in B} w_{uv}(G, \rho)\\ &= \int_{I_A\times I_B} W - \sum_{u\in A, v\in B} \int_{I_u\times I_v} W_{\mc P, \mc I}\\ &= \int_{I_A\times I_B} W - \sum_{u\in A, v\in B} \int_{I_u\times I_v} W_{\mc P}\\ &= \int_{I_A\times I_B} W - \int_{I_A\times I_B} W_{\mc P}\\ &= \int_{I_A\times I_B}(W-W_{\mc P}) \end{array} $$ Therefore, for all $A, B\subseteq V$, we have
$$ \lrmod{\frac{e_G(A, B)-e_{G_\rho}(A, B)}{n^2}} = \lrmod{\int_{I_A\times I_B} (W-W_{\mc P})} \leq d_\Box(W, W_{\mc P}) $$ and thus we have proved that
Let $G=([n], E)$ be a finite graph and let $W$ be the corresponding graphon. Let $\mc P$ be a measurable partition of $I$ and $\rho$ be the corresponding fractional partition of $V$. Then $$ d_\Box(G, G_\rho) \leq d_\Box(W ,W_{\mc P}) $$
For any given $k$, the Weak Regularity Lemma for Kernels allows us to find a partition $\mc P$ of $I$ of size at most $k$ such that $d_\Box(W, W_{\mc P})\leq 2/\sqrt{\log k}$, and thus from the above lemma we get a fractional partition $\rho$ of $G$ such that $d_\Box(G, G_\rho)\leq 2/\sqrt{\log k}$. So if we could somehow get a true partition $R$ of $V$ out of $\rho$ such that $d_\Box(G, G_R)\leq c/\sqrt{\log k}$, where $c$ is an absolute constant, then we can derive the finite graph version of weak regularity lemma.
[*]: Large Networks and Graph Limits, Lovasz, Corollary 9.13.
[**]: Large Networks and Graph Limits, Lovasz, Lemma 9.3.