\clearpage
\begin{enumerate}
    \item \mai{Revised over the summer - I think all parties agree on these: }
    \item $p(G)$ - Denotes the true graph distribution of graphs
    \item $G$ - Denotes a graph random variable $G \sim p(G)$ 
    \item $m$ - The number of user specific graph properties we are interested in like average degree, number of triads, etc.
    \item  $\ell \in \{1,..,m\}$ - is a specific property of interest
    \item $h_\ell: \sG \to \sR$ - A deterministic function to calculate property $\ell$ for our graphs 
    \item  ${\bf{h}}(G)$ - ${\bf{h}}(G): \sG \rightarrow \sR^m$ - The vector function maps $G$ to its $m$ graph property values, i.e., ${\mathbf{h}}(G) = (h_1(G), h_2(G), \cdots, h_m(G))$
    \item $\mathbf{Z} = {\bf{h}}(G) = (Z_1,Z_2,\cdots, Z_m) = (h_1(G), h_2(G), \cdots, h_m(G))$, where the distribution of $Z$ is the pushforward of the graph distribution under $h$
    \item $F_\ell(Z_\ell)$ - The true CDF for each $Z_\ell$
    \item  $\mathbf{F}(\mathbf{Z})$ - Vectorized version of CDF of all m props $\mathbf{F}(\mathbf{Z})$ $ = (F_1(Z_1), F_2(Z_2), \cdots, F_m(Z_m))$
    \item $\mathbf{U} = \mathbf{F}(\mathbf{Z})$, therefore we have  $U_\ell = F_\ell(Z_\ell) = F_\ell(h_\ell(G))$ 
    \item $(G_i)_{i=1}^n$ - Our given dataset which has i.i.d. samples drawn from $p(G)$ and where each $G_i$ is a random variable.
    \item $k$ is the number of splits we chose 
    \item $S_{i,1}, S_{i,2}, \cdots, S_{i,m} \in \{1,2, \cdots,k\}$ -  which test split the $i$-th graph with the corresponding $\ell$-th graph property will fall into, each graph $G_i$ will have $m$ split variables corresponding to $m$ graph properties. 
    \item $p(S|G_i,  h_\ell)$ - the splitting distribution, $S_{i,\ell} \sim p(S|G_i,  h_\ell)$ such that $p(S_{i,\ell} = s|G_i, h_\ell) \neq p(S_{i,\ell}=s)$, ie. the distribution of $S_{i,\ell}$ is dependent on graph $G_i$ and its $\ell$-th graph property.
    \item $\Xtest^{(\ell,j)} = \{\{G_i | \forall i, S_{i,\ell} = j\}\}$ - held-out dataset from $(G_i)_{i=1}^n$ of the $j$-th split and $\ell$-th property
    \item  $\Xtrain^{(\ell,j)} =  \{\{G_i | \forall i, S_{i,\ell} \neq j\}\}$ - data that will be used to train the model
    \item  $\{\Gbar_i^{(\ell,j)}\}_{i=1}^{n_{l,j}}$ be i.i.d. samples from $q(\Gbar^{(\ell,j)}|\theta = \Omega(\Xtrain^{(\ell,j)}))$, where $q(\Gbar^{(\ell,j)}|\theta = \Omega(\Xtrain^{(\ell,j)}))$ is the graph distribution generated by the model trained on only $\Xtrain^{(\ell,j)}$, $\Omega$ is the algorithm of the training model, $\theta$ is the final parameter learned from the model.
    \item $\Xgen^{(\ell,j)} = \{\{\Gbar_i^{(\ell,j)} | \forall i\}\}$ be the samples generated by the $\Omega$ model which has been trained on $\Xtrain^{(\ell,j)}$. 
    \item  $Z_{i,\ell} = h_{\ell}(G_i)$ - the calculated $\ell$th property of interest for a graph $G_i$
    \item $U_{i,\ell} = F_{\ell}(Z_{i,\ell}) = F_{\ell}(h_{\ell}(G_i))$ by expanding the previous equation to involve i
    \item \mai{Very related to revisions over the summer with more expansions and additions:}
    \item $U_{i,\ell} \sim \textnormal{Unifrom}[0,1]$ in the above construction and is also the marginalization over our splits  $p(U_{i,\ell}) = \sum_{j} p(S_{i,\ell}=j) p(U_{i,\ell}| S_{i,\ell}=j)$
    \item $p(S_{i,\ell}|G_i,h_{\ell}) = p(S_{i,\ell}|U_{i,\ell})$ 
    \item $p(S_{i,\ell}) = 1/k$ - we want to ensure that this is the case to allow the splits to be relatively equal in size
    \item  $p(S_{i,\ell} | U_{i,\ell})= \frac{p(S_{i,\ell})p(U_{i,\ell} | S_{i,\ell})}{p(U_{i,\ell})}$, deconstructing by bayes rule, this will give us the freedom to choose a suitable $p(U_{i,\ell}|S_{i,\ell})$ under the constraint that $p(U_{i,\ell})$ is Unifrom
    \item $p_{\mathrm{Unif}[a,b]}(U) = \frac{1}{b-a}$ - A $\mathrm{Uniform}$ distribution between the interval $[a,b]$
    \item $p(U_{i,\ell} | S_{i,\ell} = j) = p_{\mathrm{Unif}[\frac{j-1}{k}, \frac{j}{k}]}(U_{i,\ell})$ - A choice for $p(U_{i,\ell}|S_{i,\ell})$ to achieve quantile-based splitting effect. 
    \item \mai{Post summer notation satrts next:}
    \item \mai{First related to Beta Splits:}
    \item $\sum_j p_{\mathrm{Beta}[\alpha_j,\beta_j]}(U) = p_{\mathrm{Unif}[0,1]}(U) $ - This is just saying mixture of Betas is going to be Uniform ($U$ is not specific to our case)
    \item $p(U_{i,\ell}|S_{i,\ell}) = p_{\mathrm{Beta}[\alpha_j,\beta_j]}(U_{i,\ell})$, Another choice for $p(U_{i,\ell}|S_{i,\ell})$ which will lead to $\sum_{j} p(S_{i,\ell} = j) p_{\mathrm{Beta}[\alpha_j,\beta_j]}(U_{i,\ell}|S_{i,\ell}=j) = p_{\mathrm{Unif}[0,1]}(U_{i,\ell}) = p(U_{i,\ell})$ satisfying the criteria we want and achieving smoother splits
    \item $\sharpness$ - A scalar that indicates the total number of Beta distributions to combine for 1 split (this will help to make the boundaries of the splits sharper and less overlapping, see main paper for more)
    \item $p(U_{i,\ell}|S_{i,\ell} = j ) = \frac{1}{\sharpness}\sum_{i'=1}^{\sharpness} p_{\mathrm{Beta}[\alpha_j.\sharpness + i', \beta_j.\sharpness + i']}(U_{i,\ell})$ - Another choice for $p(U_{i,\ell}|S_{i,\ell} = j )$
    \item $\epsilon \in [0,1]$ - A mixing parameter (see the point below)
    \item $p(U_{i,\ell}|S_{i,\ell}) = (1-\epsilon) p(U_{i,\ell}|S_{i,\ell}) + \epsilon . p_{\mathrm{Unif}[0,1]}(U_{i,\ell})$ - Another choice for $p(U_{i,\ell}|S_{i,\ell} = j )$ to achieve min representation (more on justification in main paper)
    \item \mai{Second related to reweighting and metrics:}
    \item $p(Z_{i,\ell}|S_{i,\ell} \neq j)$ - true distribution of property $\ell$ in $\Xtrain^{(\ell,j)}$ 
    \item $p(Z_{i,\ell}|S_{i,\ell} = j)$ - true distribution of property $\ell$ in $\Xtest^{(\ell,j)}$
    \item $q(\bar{Z}_{i,\ell}|\theta = \Omega(\Xtrain^{(\ell,j)}))$ - true distribution of property $\ell$ in $\Xgen^{(\ell,j)}$
    \item $W^{(\ell,j)}(\Gbar_i) := W^{(\ell,j)}(\bar{Z}_i= h(\bar{G_i})) :=  \frac{p(\bar{Z}_{i,\ell}|S_{i,\ell}=j)}{q(\bar{Z}_{i,\ell}|\theta = \Omega(\Xtrain^{(\ell,j)}))}$ - importance weights to unbias the generated samples $\Gbar_i$
    \item $\mathcal{G}_W = \{\{ (G_i, W(G_i)): G_i \in \mathcal{G} \}\}$ - weighted version of a graph set, so for example $\Xgen$ we will have a corresponding $\mathcal{G}_{\sgen,W}$ \mai{Less cluttered notation ideas?}
    \item $\phi$ - A metric that can compare two distributions and handle weighted samples
    \item $\phi_{\textnormal{\name}}(\Xtest^{(\ell, j)}, \mathcal{G}_{\sgen,W}^{(\ell, j)}; \phi) = \phi(\Xtest^{(\ell, j)},\mathcal{G}_{\sgen,W}^{(\ell,j)})$ - what we want to measure
    \item $N_{\text{eff}}(\mathcal{G}_{\sgen,W}^{(\ell, j)})= \frac{(\sum_{(\Gbar_i, W(\Gbar_i)\in \mathcal{G}_{\sgen,W}^{(\ell, j)}} W^{(\ell,j)}(\Gbar_i))^2}{\sum_{(\Gbar_i, W(\Gbar_i)\in \mathcal{G}_{\sgen,W}^{(\ell, j)}} W^{(\ell,j)}(\Gbar_i)^2}.$ - The number of effective samples to use when sampling from our generative model.
    \item $\phi_{\text{KS}}$ - A choice for $\phi$ capable of handling weighted samples and comparing them
    \item $\cG_1$, $\cG_2$, $\cG_{1,W1}$, $\cG_{1,W2}$ - generic graph datasets and their weighted versions, will be used just to introduce the next 2 points
    \item $\hat{F}_{W1}(z) = \frac{1}{\sum_i W_i} \sum_i W_i {\bf 1}_{(Z_{i,\ell} \leq z)}$ is the weighted empirical CDF associated with $\cG_{1,W_1}$
    \item $\phi_{\text{KS}}(\cG_{1,W_1}, \cG_{2,W_2}; h_{\ell}) = \sup_{G_i} |\hat{F}_{W1}(h_{\ell}(G_i)) - \hat{F}_{W2}(h_{\ell}(G_i)) | \\
    = \sup_{Z_{i,\ell}} |\hat{F}_{W1}(Z_{i,\ell}) - \hat{F}_{W2}
    (Z_{i,\ell}) |$ - the weighted KS statistic between samples in $\cG_{1,W1}$ and $\cG_{1,W2}$
    \item $\phi_{\textnormal{\name}}(\Xtest^{(\ell, j)}, \Xgen^{(\ell, j)};  \phi_{\text{KS}}) = \phi_{\text{KS}}(\Xtest^{(\ell, j)},\cG_{\sgen,W}^{(\ell,j)};h_{\ell})$ - For our specific case: when $\cG_{1,W1} = \Xtest^{(\ell, j)} $ and $\cG_{2,W2} = \cG_{\sgen,W}^{(\ell,j)}$
    \item $\cG_{\textnormal{memo},W_m}^{(\ell,j)} = \{\{\Gbar_i \sim EmpDist(\Xtrain)\}\}$ and let $\cG_{\sgen,W}^{(\ell,j)} = \cG_{\textnormal{memo},W}^{(\ell,j)}$ for when the model is a model that memorizes the data \mai{revise notation}
    \item $\bar{\phi}_{\text{KS}}$ - A normalized version of $\phi_{\text{KS}}$ where $\bar{\phi}_{\text{KS}}(\cG_{\textnormal{memo},W_m}^{(\ell,j)} , \Xtest^{(\ell, j)},\cG_{\textnormal{gen},W}^{(\ell,j)} ; h_{\ell})  = \frac{\phi_{\text{KS}}(\Xtest^{(\ell, j)}, \cG_{\textnormal{gen},W}^{(\ell,j)}; h_{\ell})}{\phi_{\text{KS}}(\Xtest^{(\ell, j)}, \cG_{\textnormal{memo},W_m}^{(\ell,j)} ; h_{\ell})}$
    
\end{enumerate}

