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\title{Graph Self-supervised Learning via Proximity Distribution Minimization\\(Supplementary Material)}

% The standard author block has changed for UAI 2023 to provide
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%
% All author information is authomatically removed by the class for the
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\author[1]{\href{mailto:<tz21@rice.edu>}{Tianyi Zhang}{}}
\author[2]{Zhenwei Dai}
\author[1]{Zhaozhuo Xu}
\author[1]{Anshumali Shrivastava}
% Add affiliations after the authors
\affil[1]{%
    Department of Computer Science\\
    Rice University\\
    Houston, Texas, USA
}
\affil[2]{%
    Amazon\\
    Palo Alto, California, USA
}
  
  \begin{document}
  
\onecolumn %% Turn this off if single column is desired for the supplement
\maketitle

% This Supplementary Material should be submitted as a separate file. Please do not append the Supplementary Material to the main paper. 

% Fig. \ref{fig:pitt} and Eq \ref{eq:example} in the main paper can be cross referenced using \texttt{xr}. 
\appendix


% \subsection{Statistics of datasets} Table~\ref{tab:data_stat} presents the statistics of the 6 graph datasets used in our evaluations. 

% \begin{table}[ht]
% \centering
% \small
% \caption{Statistics of the graphs used in the experiments.}
% \setlength\tabcolsep{4pt}
% \begin{tabular}{l|llllll}
% \toprule
% Dataset           & Cora         & Citeseer & PubMed   & ogbn-arxiv & ogbn-proteins               & ogbn-products          \\
% \midrule
% \# of Nodes   & 2,708        & 3,327    & 19,717   & 169,343    & 132,534                     & 2,449,029            \\
% \# of Edges   & 5,429        & 4,732    & 44,338   & 1,166,243  & 39,561,252                  & 61,859,140         \\
% Metric       & Accuracy     & Accuracy & Accuracy & Accuracy   & ROC-AUC                     & Accuracy             \\
% \# of Classes & 7            & 6        & 3        & 40         & 112    & 47  \\
% \bottomrule
% \end{tabular}
% \label{tab:data_stat}
% \end{table}

% \subsection{Testbed}
% \label{sec:testbed}
% We implement our proposed method with the Deep Graph Library~\citep{wang2019deep}. Our experiments are conducted on a machine with 1 NVIDIA Tesla V100 32GB GPU, 2 24-core/48-thread Intel Xeon Gold 5220R CPUs, and 1.5TB of RAM. 

% \subsection{Ablation Study}
% \label{sec:ablation}

% We study the sensitivity of our method to hyper-parameter changes. A robust SSL method should not be sensitive to hyper-parameters. This has been a weakness of prior SSL methods, which require vastly different corruption parameters for different datasets~\citep{you2021graph, thakoor2022largescale, ccassg}. We vary the hyper-parameters in computing the diffusion matrix for both types of diffusion matrix (diffusion time $t$ for heat kernel and teleport probability $\beta$ for PPR) and present the test accuracy on Cora, Citeseer and PubMed in Table~\ref{tab:sensitivity}. Our method is not sensitive to hyper-parameters of the diffusion target, since the accuracy drops at most $1.0\%$ from the best accuracy for all diffusion parameters and diffusion types considered. Therefore, our method is more robust than previous SSL approaches which are sensitive to hyper-parameter changes.

% \begin{table}[ht]
% \centering
% \small
% \caption{Evaluate performance of \method by varying the diffusion hyper-parameters.}
% \begin{tabular}{l|lllll|lllll}
% \toprule
%        & \multicolumn{5}{c|}{Heat ($t$)}         & \multicolumn{5}{c}{PPR ($\beta$)}          \\
%        & 3    & 4    & 5    & 6    & 7    & 0.02 & 0.04 & 0.06 & 0.08 & 0.10 \\
% \midrule
% Cora & 83.9 & 84.1 & 84.0 & 84.0 & \textbf{84.4} & 84.1 & 84.0 & 84.3 & 84.1 & 84.0 \\
% Citeseer & 73.3 & 74.1 & 74.3 & 74.1 & 74.1 & 74.0 & \textbf{74.6} & 74.2 & 74.0 & 74.2 \\
% PubMed & 82.8 & 83.0 & 83.6 & 83.1 & 82.7 & 83.6 & 83.2 & \textbf{83.8} & 82.8 & 83.4 \\
% \bottomrule
% \end{tabular}
% \label{tab:sensitivity}
% \end{table}
% \vspace{-1cm}
\section{Test Accuracy, Training Time, and Training Details}

The test accuracy of PDM with different proximity measures are presented in Table \ref{tab:prox_measures}. The training time comparison of PDM with DGI \cite{velickovic2019deep} is presented in Table \ref{tab:training_time}. The training details and hyperparameters are shown in Table \ref{tab:hyperparam}.

\begin{table}[ht]
\centering
% \small
\caption{Test accuracy of PDM with different proximity measures}
\label{tab:prox_measures}
\begin{tabular}{l|rrr}
\toprule
                  & Cora & Citeseer & PubMed \\
                  \midrule
PDM (heat kernel) & 84.4 & 74.3     & 83.6   \\
PDM (PPR)         & 84.1 & 74.6     & 83.8   \\
PDM (SimRank)     & 82.2 & 74.6     & 82.3  \\
\bottomrule
\end{tabular}
\end{table}

\begin{table}[ht]
\centering
% \small
\caption{Training Time Comparison between DGI and PDM}
\label{tab:training_time}
\begin{tabular}{l|rrrrrr}
\toprule
    & \multicolumn{3}{c}{Cora}                                       & \multicolumn{3}{c}{Citeseer}                                   \\
    & \makecell{Total Training\\Time (ms)} & \makecell{Time Per\\Epoch (ms)} & Test Accuracy & \makecell{Total Training\\Time (ms)} & \makecell{Time Per\\Epoch (ms)} & Test Accuracy \\
\midrule
DGI & 2980                     & 14.9                & 82.3          & 4060                     & 20.3                & 71.8          \\
PDM & 7340                     & 36.7                & 84.4          & 10180                    & 50.9                & 74.6          \\
\bottomrule
\end{tabular}
\end{table}

\begin{table}[ht]
\centering
% \small
\caption{Training details and hyper-parameters for PDM on all datasets}
\label{tab:hyperparam}
\begin{tabular}{l|rrrrrr}
\toprule
               & Cora       & Citeseer & PubMed & ogbn-arxiv & ognb-proteins & ogbn-products \\
               \midrule
Architecture   & GCN        & GCN      & GCN    & GCN        & GCN           & GCN           \\
Depth          & 1          & 1        & 2      & 3          & 3             & 3             \\
Hidden Size    & 2048        & 2048      & 32     & 2048       & 2048          & 1024          \\
Activation     & Leaky ReLU & tanh     & tanh   & Leaky ReLU & Leaky ReLU    & tanh          \\
Learning Rate  & 5e-3       & 5e-4     & 1e-2   & 1e-4       & 1e-4          & 1e-4          \\
Diffusion Type & Heat       & Heat      & PPR    & PPR        & PPR           & PPR           \\
Epochs         & 100        & 100      & 100    & 100        & 200           & 10            \\
Optimizer      & \multicolumn{6}{c}{AdamW~\citep{adamw}} \\
\bottomrule
\end{tabular}
\end{table}

\section{Comparison with Augmentation-free Methods}

We compare the performance of PDM with the augmentation-free method AFGRL \citep{augmentationfree}, and the results are shown in Table \ref{tab:aug_free}. For AFGRL, we consider the best test accuracy for number of neighbors $k \in \{4, 8, 16\}$. PDM offers significant improvements over AFGRL, up to 5.1\% on citeseer.

\begin{table}[ht]
\centering
\caption{Test Accuracy of AFGRL and PDM}
\label{tab:aug_free}
\begin{tabular}{l|rrr}
\toprule
      & Cora & Citeseer & PubMed \\
\midrule
AFGRL & 81.1 & 69.5     & 79.2   \\
PDM   & 84.4 & 74.6     & 83.8   \\
\bottomrule
\end{tabular}
\end{table}

% \section{Additional simulation results}
% Table~\ref{tab:supp-data} lists additional simulation results; see also \citet{einstein} for a comparison. 

% \begin{table}[!h]
%     \centering
%     \caption{An Interesting Table.} \label{tab:supp-data}
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%       \toprule % from booktabs package
%       \bfseries Dataset & \bfseries Result\\
%       \midrule % from booktabs package
%       Data1 & 0.12345\\
%       Data2 & 0.67890\\
%       Data3 & 0.54321\\
%       Data4 & 0.09876\\
%       \bottomrule % from booktabs package
%     \end{tabular}
% \end{table}

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