\begin{algorithm}
    \caption{Node subsampling/resampling bootstrap for two-sample inference}
    \label{algorithm::node-resample}
    {\bf Input:} Networks $A,B$; bootstrap repetition $N_{\rm boot}$; if subsampling: subsample sizes $m_{\rm sub},n_{\rm sub}$\\
    {\bf Output:} Bootstrapped studentized empirical moment discrepancies $\{\hat T_{m,n}^{(b)}\}_{b=1,\ldots,N_{\rm boot}}$\\
    {\bf Steps:} For $b=1,\ldots,N_{\rm boot}$, do
    \begin{enumerate}
        \item Node subsample/resample $A,B$, obtain $A^{(b)}, B^{(b)}$. If resampling, randomly sample $m$ nodes ${\cal J}_A$ from $[1:m]$ with replacement; if subsampling, randomly sample $m_{\rm sub}$ nodes ${\cal J}_A$ from $[1:m]$ without replacement; In either case, set $A^{(b)}\leftarrow A_{{\cal J}_A}$; do the same for $B$
        \item Compute $\hat T_{m,n}^{(b)}$ using \eqref{def::S_m,n} and \eqref{def::T_m,n}, with $A^{(b)}, B^{(b)}$ as the input
    \end{enumerate}
\end{algorithm}

\begin{algorithm}
\caption{Node subsampling/resampling bootstrap for two-sample inference}
{\bf Input:} Networks $A,B$; bootstrap repetition $N_{\rm boot}$; if subsampling: subsample sizes $m_{\rm sub},n_{\rm sub}$\\
{\bf Output:} Bootstrapped studentized empirical moment discrepancies $\{\hat T_{m,n}^{(b)}\}_{b=1,\ldots,N_{\rm boot}}$\\
{\bf Steps:} For $b=1,\ldots,N_{\rm boot}$, do
\begin{enumerate}
\item Node subsample/resample $A,B$, obtain $A^{(b)}, B^{(b)}$. If resampling, randomly sample $m$ nodes ${\cal J}_A$ from $[1:m]$ with replacement; if subsampling, randomly sample $m_{\rm sub}$ nodes ${\cal J}_A$ from $[1:m]$ without replacement; In either case, set $A^{(b)}\leftarrow A_{{\cal J}_A}$; do the same for $B$
\item Compute $\hat T_{m,n}^{(b)}$ using \eqref{def::S_m,n} and \eqref{def::T_m,n}, with $A^{(b)}, B^{(b)}$ as the input
\end{enumerate}
\end{algorithm}
{
    "title": "higher-order accurate two-sample network inference and network hashing",
    "id": "2208.07573",
    "abstract": "two-sample hypothesis testing for network comparison presents many significant challenges, including: leveraging repeated network observations and known node registration, but without requiring them to operate; relaxing strong structural assumptions; achieving finite-sample higher-order accuracy; handling different network sizes and sparsity levels; fast computation and memory parsimony; controlling false discovery rate (fdr) in multiple testing; and theoretical understandings, particularly regarding finite-sample accuracy and minimax optimality. in this paper, we develop a comprehensive toolbox, featuring a novel main method and its variants, all accompanied by strong theoretical guarantees, to address these challenges. our method outperforms existing tools in speed and accuracy, and it is proved power-optimal. our algorithms are user-friendly and versatile in handling various data structures (single or repeated network observations; known or unknown node registration). we also develop an innovative framework for offline hashing and fast querying as a very useful tool for large network databases. we showcase the effectiveness of our method through comprehensive simulations and applications to two real-world datasets, which revealed intriguing new structures.",
    "categories": "stat.me math.st stat.ml stat.th",
    "doi": "",
    "created": "2022-08-16",
    "updated": "2024-02-02",
    "authors": [
        "meijia shao",
        "dong xia",
        "yuan zhang",
        "qiong wu",
        "shuo chen"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2208.07573"
}
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