Information Limits for Community Detection in Hypergraph with Label Information

Abstract

In network data mining, community detection refers to the problem of partitioning the nodes of a network into clusters (communities). This is equivalent to identifying the cluster label of each node. A label estimator is said to be an exact recovery of the true labels (communities) if it coincides with the true labels with a probability convergent to one. In this work, we consider the effect of label information on the exact recovery of communities in an m-uniform Hypergraph Stochastic Block Model (HSBM). We investigate two scenarios of label information: $\left(1\right)$ a noisy label for each node is observed independently, with $1-{\alpha }_n$ as the probability that the noisy label will match the true label; $\left(2\right)$ the true label of each node is observed independently, with the probability of $1-{\alpha }_n$. We derive sharp boundaries for exact recovery under both scenarios from an information-theoretical point of view. The label information improves the sharp detection boundary if and only if ${\alpha }_n=n^{-\beta +o\left(1\right)}$ for a constant $\beta >0$.

1. Introduction

A graph or network consists of a set of nodes (vertices) and an edge set. Graphs have been used extensively to model a variety of systems in many fields [1,2,3,4,5]. Due to the widespread application, network data analysis has drawn a lot of attention in both statistical and machine learning communities [6,7,8,9,10,11,12,13]. Real-world networks are usually more complex than ordinary graphs, and in this case, a hypergraph is a popular alternative model [8,14,15,16,17,18,19,20,21]. Given a positive integer n, let $\mathcal{V}=\left[n\right]:=\{1,2,\dots ,n\}$. An undirected m-uniform hypergraph on $\mathcal{V}$ is a pair ${\mathcal{H}}_m=(\mathcal{V},\mathcal{E})$ in which $\mathcal{E}$ is a set of subsets of $\mathcal{V}$ such that $\left|e\right|=m$ for every $e\in \mathcal{E}$, and each element in $\mathcal{E}$ is called a hyperedge. That is, in ${\mathcal{H}}_m$, each hyperedge consists of exactly m distinct nodes. For $i_1<i_2<\cdots <i_m$, we denote $A_{i_1{i}_2\dots i_m}=1$ if $\{i_1,i_2,\dots ,i_m\}$ is a hyperedge, and $A_{i_1{i}_2\dots i_m}=0$ if otherwise. Suppose $A_{i_1{i}_2\dots i_m}=A_{j_1{j}_2\dots j_m}$ if $\{i_1,i_2,\dots ,i_m\}=\{j_1,j_2,\dots ,j_m\}$, and $A_{i_1{i}_2\dots i_m}=0$ if $\left|\right\{i_1,i_2,\dots ,i_m\left\}\right|\ne m$. That is, a hypergraph that is symmetric and a self-loop is not allowed. Then, the m-dimensional symmetric array $A=\left(A_{i_1,\dots ,i_m}\right)\in {\{0,1\}}^{\otimes n^m}$ is called the adjacency tensor of hypergraph ${\mathcal{H}}_m$. When $m=2$, ${\mathcal{H}}_2$ is just the usual graph that has been widely used in community detection problems [9].

A hypergraph is random if $A_{i_1{i}_2\cdots i_m}$ is a random variable for $i_1<i_2<\cdots <i_m$. The binary m-uniform Hypergraph Stochastic Block Model (HSBM) ${\mathcal{H}}_{m,n,p,q}$ is defined as follows: Each node $i\in \left[n\right]$ is independently assigned a community label ${\sigma }_i$, and randomly with $\mathbb{P}({\sigma }_i=1)=\mathbb{P}({\sigma }_i=-1)=\frac{1}{2}$. We denote $\sigma =({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_n)$ and ${\mathrm{I}}_{+}={\mathrm{I}}_{+}\left(\sigma \right)=\left\{i\right|{\sigma }_i=+1\}$, ${\mathrm{I}}_{-}={\mathrm{I}}_{-}\left(\sigma \right)=\left\{i\right|{\sigma }_i=-1\}$. Then for $1\le i_1<i_2<\dots <i_m\le n$,

$$A_{i_1,i_2,\dots ,i_m}\sim \left\{\begin{array}{cc}Bern\left(p\right),\hfill & \mathrm{if}\{i_1,i_2,\dots ,i_m\}\subset {\mathrm{I}}_{+}\mathrm{or}{\mathrm{I}}_{-},\hfill \\ Bern\left(q\right),\hfill & \mathrm{o}.\mathrm{w}..\hfill \end{array}\right.$$

In addition, $A_{i_1,i_2\dots ,i_m}(i_1<i_2<\cdots <i_m)$ are assumed to be independent and conditional on $\sigma$. Here, $+1,-1$ represent two communities and ${\mathrm{I}}_{+}$ and ${\mathrm{I}}_{-}$ denote the nodes that belong to the $+1$ and $-1$ communities, respectively. The subset $\{i_1,i_2,\dots ,i_m\}$ forms a hyperedge with a probability p if the distinct nodes $i_1,i_2,\dots ,i_m$ are in the same community. Otherwise, $\{i_1,i_2,\dots ,i_m\}$ forms a hyperedge with a probability q. Throughout this paper, the community structure is assumed to be balanced; that is, the number of nodes in each community is $\frac{n}{2}$, as in [6,22,23]. Moreover, we focus on the case $p=\frac{alogn}{n^{m-1}}$ and $q=\frac{blogn}{n^{m-1}}$ with $a\ge b>0$, since this is the smallest order of hyperedge probability where exact recovery is possible. We denote the binary m-uniform Hypergraph of Stochastic Block Model (HSBM) as ${\mathcal{H}}_{m,n,a,b}={\mathcal{H}}_m(n,\frac{alog\left(n\right)}{n^{m-1}},\frac{blog\left(n\right)}{n^{m-1}})$.

Community detection refers to the problem of identifying the true label $\sigma$ based on an observation of a hypergraph A. Let $\widehat{\sigma }$ be an estimator of $\sigma$. We say $\widehat{\sigma }$ is an exact recovery of $\sigma$ or $\widehat{\sigma }$ exactly recovers $\sigma$ if:

$$\mathbb{P}(\exists s\in \{\pm 1\}:\widehat{\sigma }=s\sigma )=1-o\left(1\right).$$

In words, exact recovery means that the estimated label $\widehat{\sigma }$ is equal to the true label $\sigma$, with a probability convergent to one as the number of nodes goes to infinity. We say exact recovery is possible if there is an estimator $\widehat{\sigma }$ that exactly recovers $\sigma$, and exact recovery is impossible if any estimator $\widehat{\sigma }$ does not exactly recover $\sigma$.

In practice, along with the hypergraph A, side information about node labels is usually available [23,24,25,26,27,28,29,30]. For example, in a co-authorship or co-citation network, the cluster labels of some authors are known [28]. In a Web query network, some queries are labeled [30]. In student relational networks, the dorms in which students live can serve as label information [29]. Various algorithms have been developed to incorporate label information in community recovery in hypergraphs [28,29,30], and incorporating side information has been shown to improve clustering performance [23,24,25,26,27,28,29,30]. The sharp recovery boundary with a label or side information were given by [23,27] in the graph case. However, to the best of our knowledge, the sharp recovery boundary for hypergraphs is still unknown. In this paper, we study the effect of label information on the boundary of exact recovery for hypergraphs and consider two types of label information: $\left(1\right)$ a noisy label for each node is observed independently, with $1-{\alpha }_n$ as the probability that the noisy label will match the true label; $\left(2\right)$ the true label of each node is observed independently, with the probability of $1-{\alpha }_n$. Let ${\alpha }_n=n^{-\beta +o\left(1\right)}$ with a constant $\beta \ge 0$. From an information-theoretical point of view, we derive sharp boundaries of exact recovery in terms of $m,a,b,\beta$. Interestingly, label information is useful if and only if $\beta >0$. The main result is summarized in

Table 1. Regions for the exact recovery of community structure in a hypergraph with label information.

Region Where Noisy Labels Are Observed	Recovery
$\left(i\right)$${\eta }_{m,a,b}\left(\beta \right)<1$ and $\beta <C_{m,a,b}(a-b)$	Exact recovery is impossible
$\left(ii\right)$$\beta <1$ and $\beta >C_{m,a,b}(a-b)$	Exact recovery is impossible
$\left(i\right)$${\eta }_{m,a,b}\left(\beta \right)>1$ and ${\beta }_m<C_{m,a,b}(a-b)$	Exact recovery is possible
$\left(ii\right)$$\beta >1$ and $\beta >C_{m,a,b}(a-b)$	Exact recovery is possible
Region Where True Labels Are Partially Observed	Recovery
$\frac{{\left(\sqrt{a}-\sqrt{b}\right)}^2}{2^{m-1}(m-1)!}+\beta <1$	Exact recovery is impossible
$\frac{{\left(\sqrt{a}-\sqrt{b}\right)}^2}{2^{m-1}(m-1)!}+\beta >1$	Exact recovery is possible

, where ${\eta }_{m,a,b}\left(\beta \right),C_{m,a,b}$ are defined in Equations (1) and (2). In both cases, for a fixed m, the region (in terms of $a,b$) where exact recovery is impossible shrinks as $\beta$ gets larger. The label information is helpful if and and only if $\beta >0$; that is, ${\alpha }_n$ has to converge to zero at the rate of $n^{-\beta }$ for $\beta >0$. The visualization of the regions in Table 1 can be found in Figures 1 and 2 in Section 2.

2. Main Result

In this section, we consider community detection in hypergraphs through an observation of noisy labels or a proportion of the true labels from an information-theoretical point of view. We derive sharp boundaries of exact recovery, which provides a benchmark for developing practical community detection algorithms.

2.1. Detection with Noisy Label Information

In this subsection, we consider community detection in hypergraphs through a noisy version of node labels available. In the graph regime, community detection with noisy label information was proposed in [27] and extensively studied in [23]. Here, we focus on an m-uniform hypergraph with an arbitrary fixed $m\ge 2$. Given a true label vector $\sigma =({\sigma }_1,\dots ,{\sigma }_n)$, for each node i, a noisy label $Y_i$ is independently observed and $Y_i$ coincides with the true label ${\sigma }_i$ with a probability of $1-{\alpha }_n$. More specifically, $\mathbb{P}(Y_i={\sigma }_i|{\sigma }_i)=1-{\alpha }_n$ and $\mathbb{P}(Y_i=-{\sigma }_i|{\sigma }_i)={\alpha }_n$, ${\alpha }_n\in [0,\frac{1}{2}]$ and $Y_i(1\le i\le n)$ are independent and conditional on $\sigma$. If ${\alpha }_n=0$, the true label for each node is fully known. If ${\alpha }_n=\frac{1}{2}$, the noisy labels $Y=(Y_1,Y_2,\dots ,Y_n)$ do not provide any information about the true labels. The hypergraph A and Y are assumed to be independent and conditional on $\sigma$. In this subsection, we focus on the effect of a noisy label Y on community detection in a hypergraph.

Assume ${\alpha }_n=n^{-\beta +o\left(1\right)}$ with a constant $\beta \ge 0$ and define the following:

$$\begin{array}{cc}\hfill {\eta }_{m,a,b}\left(\beta \right)& =\frac{1}{2^{m-1}(m-1)!}\left\{a+b-\frac{{\gamma }_{m,a,b}\left(\beta \right)}{C_{m,a,b}}+\frac{\beta }{2C_{m,a,b}}log\left(\frac{{\gamma }_{m,a,b}\left(\beta \right)+\beta }{{\gamma }_{m,a,b}\left(\beta \right)-\beta }\right)\right\}+\frac{\beta }{2},\hfill \end{array}$$

(1)

where

$$C_{m,a,b}=\frac{log\left(a\right)-log\left(b\right)}{2^{m-1}(m-1)!},{\gamma }_{m,a,b}\left(\beta \right)=\sqrt{{\beta }^2+4ab{C}_{m,a,b}^2}.$$

(2)

Here, $C_{m,a,b}$ and ${\gamma }_{m,a,b}$ are defined just for notation convenience without any practical meaning. The quantity ${\eta }_{m,a,b}\left(\beta \right)$ can be considered the signal contained in the model. The larger ${\eta }_{m,a,b}\left(\beta \right)$ is, the easier exact recovery becomes. It is clearer to see this in the special case of $\beta =0$:

$${\eta }_{m,a,b}\left(0\right)=\frac{{\left(\sqrt{a}-\sqrt{b}\right)}^2}{2^{m-1}(m-1)!}.$$

(3)

For a fixed m, a large ${\eta }_{m,a,b}\left(0\right)$ implies that the difference of $\sqrt{a}-\sqrt{b}$ is large. The within-community nodes are also more densely connected than between-community nodes. Hence, it gets easier to cluster the nodes into groups. Note that ${\eta }_{m,a,b}\left(0\right)$ was used to characterize the sharp detection boundary in [6]. For an arbitrary $\beta \ge 0$, we provide the necessary and sufficient conditions for the exact recovery of the community structure. To this end, we firstly investigate the maximum likelihood estimator (MLE) of true labels. The region where exact recovery is impossible corresponds to the region where MLE fails. Then, based on the noisy labels, we construct an estimator that exactly recovers the community structure. The result is summarized in the following theorem.

Theorem 1.

Assume ${\alpha }_n=n^{-\beta +o\left(1\right)}$ with a constant $\beta \ge 0$, then exact recovery in HSBM ${\mathcal{H}}_{m,n,a,b}$ is impossible if

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\eta }_{m,a,b}\left(\beta \right)<1,\hfill & \mathrm{when}\beta <C_{m,a,b}(a-b),\hfill \\ \beta <1,\hfill & \mathrm{when}\beta >C_{m,a,b}(a-b).\hfill \end{array}\right.\end{array}$$

(4)

Exact recovery is possible if

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\eta }_{m,a,b}\left(\beta \right)>1,\hfill & \mathrm{when}\beta <C_{m,a,b}(a-b),\hfill \\ \beta >1,\hfill & \mathrm{when}\beta >C_{m,a,b}(a-b).\hfill \end{array}\right.\end{array}$$

(5)

Here, ${\eta }_{m,a,b}\left(\beta \right)$ and $C_{m,a,b}$ are defined in Equations (1) and (2).

Based on Theorem 1, there is a phase transition phenomenon for exact recovery in ${\mathcal{H}}_{m,n,a,b}$. In the region $\beta <C_{m,a,b}(a-b)$, exact recovery is possible if ${\eta }_{m,a,b}\left(\beta \right)>1$, and not possible if ${\eta }_{m,a,b}\left(\beta \right)<1$. In the region $\beta >C_{m,a,b}(a-b)$, exact recovery is possible if $\beta >1$, and exact recovery is impossible if $\beta <1$. In this sense, phase transition occurs at 1, and 1 is the sharp boundary for exact recovery. When ${\alpha }_n$ is bounded away from zero, $\beta =0$ and $C_{m,a,b}(a-b)>0$ trivially hold. Then, Theorem 1 recovers Theorem 4 and Theorem 5 in [6]. This shows that the noisy label is useful if and only if ${\alpha }_n$ converges to zero at a rate of $n^{-\beta }$ for $\beta >0$. Furthermore, exact recovery with a fixed $m,a,b$ becomes easier as $\beta$ increases, since ${\eta }_{m,a,b}\left(\beta \right)$ is increasing in $\beta$ (see Lemma A1). When $m=2$, Theorem 1 is reduced to Theorem 1 and Theorem 2 in [23]. Note that $C_{m,a,b}$ is decreasing in m. Then, given a fixed $\beta$, the region of exact recovery for $m=2$ contains the region of $m\ge 3$ as a proper subset. These findings can be summarized in Figure 1, where we visualize the regions characterized by (4) and (5) with $m=2,3$ and $\beta =0,0.4,0.8$. In Figure 1, the red (green) region represents exact recovery as impossible (possible). We point out that the time complexity of our estimator for exact recovery is $O\left(n^m\right)$. Since our focus in this paper is to derive the sharp boundary of exact recovery as in [23,27] and not to propose algorithms with optimal performance, our estimator may or may not outperform existing algorithms.

2.2. Detection with Partially Observed Labels

In this subsection, we consider the community detection problem with true labels partially observed. This type of side information was considered in [23,24,25,26] in the context of graphs, and in [28,29,30] for hypergraphs. Here, we focus on an m-uniform hypergraph with an arbitrary fixed $m\ge 2$. Given the true labels $\sigma =({\sigma }_1,\dots ,{\sigma }_n)$, for each node i, the true label is independently observed with a probability of $1-{\alpha }_n$. More specifically, we define a random variable $Y_i$ with $\mathbb{P}(Y_i={\sigma }_i|{\sigma }_i)=1-{\alpha }_n$ and $\mathbb{P}(Y_i=0|{\sigma }_i)={\alpha }_n$, ${\alpha }_n\in [0,1]$. Here, $Y_i=0$ indicates that the true label for node i is not observed. If ${\alpha }_n=1$, no label information is observed. If ${\alpha }_n=0$, all the true labels are observed and community detection is not necessary. We study how ${\alpha }_n$ changes the sharp detection boundary from the information-theoretical point of view. To this end, we investigate the maximum likelihood estimator (MLE) of true labels. The region where exact recovery is impossible corresponds to the region where MLE fails. The exact recovery estimator is constructed based on the partially observed labels. The result is summarized in the following theorem.

Theorem 2.

Assume ${\alpha }_n=n^{-\beta +o\left(1\right)}$ with a constant $\beta \ge 0$, then exact recovery in HSBM ${\mathcal{H}}_{m,n,a,b}$ is impossible if

$$\begin{array}{c}\hfill \frac{{\left(\sqrt{a}-\sqrt{b}\right)}^2}{2^{m-1}(m-1)!}+\beta <1.\end{array}$$

(6)

Exact recovery is possible if

$$\begin{array}{c}\hfill \frac{{\left(\sqrt{a}-\sqrt{b}\right)}^2}{2^{m-1}(m-1)!}+\beta >1.\end{array}$$

(7)

Theorem 2 clearly characterizes how the partially observed labels affect the boundary for exact recovery. A phase transition phenomenon of exact recovery exists at 1, since exact recovery is possible if ${\eta }_{m,a,b}\left(0\right)+\beta >1$, but not possible if ${\eta }_{m,a,b}\left(0\right)+\beta <1$. When $\beta =0$, Theorem 2 recovers Theorem 4 and Theorem 5 in [6]. If $\beta >0$, the region (6) where exact recovery is impossible is smaller than that in [6]. The side information of partially known labels makes exact recovery easier if and only if $\beta >0$. When $m=2$, Theorem 2 is reduced to Theorem 1 and Theorem 2 in [23]. For a fixed $\beta$, the region of exact recovery of $m\ge 3$ is smaller than that of $m=2$. These findings can be verified in Figure 2, where we visualize the regions characterized by (6) and (7) with $m=2,3$ and $\beta =0,0.4,0.8$. In Figure 2, the red (green) region represents exact recovery as impossible (possible). Finally, we point out that the time complexity of our estimator achieving exact recovery is $O\left(n^m\right)$. Again, our focus in this paper is to derive the sharp boundary of exact recovery as in [23,27], not to propose algorithms with optimal performance; hence, our estimator may or may not outperform existing algorithms.

3. Proof of Main Result

In this section, we provide detailed proof of Theorems 1 and 2.

To start with, we derive the explicit expression of the likelihood function. The likelihood function of hypergraph A given the node label $\sigma$ is:

$$\begin{array}{ccc}& & \mathbb{P}\left(A\right|\sigma )\hfill \\ & =& \prod _{1\le i_1<\dots <i_m\le n}{\left(p^{A_{i_1,\dots ,i_m}}{(1-p)}^{1-A_{i_1,\dots ,i_m}}\right)}^{\mathrm{I}[{\sigma }_{i_1}=\cdots ={\sigma }_{i_m}]}\hfill \\ & & \prod _{1\le i_1<\dots <i_m\le n}{\left(q^{A_{i_1,\dots ,i_m}}{(1-q)}^{1-A_{i_1,\dots ,i_m}}\right)}^{1-\mathrm{I}[{\sigma }_{i_1}=\cdots ={\sigma }_{i_m}]}\hfill \\ & =& \prod _{1\le i_1<\dots <i_m\le n}{\left(\frac{p(1-q)}{q(1-p)}\right)}^{A_{i_1,\dots ,i_m}\mathrm{I}[{\sigma }_{i_1}=\cdots ={\sigma }_{i_m}]}{\left(\frac{q}{1-q}\right)}^{A_{i_1,\dots ,i_m}}{\left(\frac{1-p}{1-q}\right)}^{\mathrm{I}[{\sigma }_{i_1}=\cdots ={\sigma }_{i_m}]}(1-q).\hfill \end{array}$$

Then, the log-likelihood function can be written as follows:

$$\begin{array}{ccc}\hfill log\left(\mathbb{P}\right(A\left|\sigma \right))& =& log\left(\frac{p(1-q)}{q(1-p)}\right)\sum _{1\le i_1<\dots <i_m\le n}{A}_{i_1,\dots ,i_m}\mathrm{I}[{\sigma }_{i_1}=\cdots ={\sigma }_{i_m}]\hfill \\ & +& log\left(\frac{q}{1-q}\right)\sum _{1\le i_1<\dots <i_m\le n}{A}_{i_1,\dots ,i_m}\hfill \\ & +& log\left(\frac{1-p}{1-q}\right)\sum _{1\le i_1<\dots <i_m\le n}\mathrm{I}[{\sigma }_{i_1}=\cdots ={\sigma }_{i_m}]\hfill \\ & +& log(1-q)\sum _{1\le i_1<\dots <i_m\le n}1\hfill \\ & :=& \mathbb{I}+\mathbb{II},\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill \mathbb{I}& =& log\left(\frac{p(1-q)}{q(1-p)}\right)\sum _{1\le i_1<\dots <i_m\le n}{A}_{i_1,\dots ,i_m}\mathrm{I}[{\sigma }_{i_1}=\cdots ={\sigma }_{i_m}]\hfill \\ & =& \left(log\left(\frac{p}{q}\right)+log\left(\frac{1-q}{1-p}\right)\right)\sum _{1\le i_1<\dots <i_m\le n}{A}_{i_1,\dots ,i_m}\mathrm{I}[{\sigma }_{i_1}=\cdots ={\sigma }_{i_m}]\hfill \\ & :=& \left(C_{a,b}+o\left(1\right)\right)\left(e_{+}+e_{-}\right)\hfill \\ & \asymp & C_{a,b}\left(e_{+}+e_{-}\right),\hfill \\ \hfill C_{a,b}& =& log\left(a\right)-log\left(b\right),\hfill \\ \hfill e_{+}& =& \sum _{1\le i_1<\dots <i_m\le \left|{\mathrm{I}}_{+}\right|}{A}_{i_1,\dots ,i_m}\mathrm{I}[{\sigma }_{i_1}=\cdots ={\sigma }_{i_m}=+1],\hfill \\ \hfill e_{-}& =& \sum _{1\le i_1<\dots <i_m\le \left|{\mathrm{I}}_{-}\right|}{A}_{i_1,\dots ,i_m}\mathrm{I}[{\sigma }_{i_1}=\cdots ={\sigma }_{i_m}=-1],\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \mathbb{II}& =& log\left(\frac{q}{1-q}\right)\sum _{1\le i_1<\dots <i_m\le n}{A}_{i_1,\dots ,i_m}\hfill \\ & +& log\left(\frac{1-p}{1-q}\right)\sum _{1\le i_1<\dots <i_m\le n}\mathrm{I}[{\sigma }_{i_1}=\cdots ={\sigma }_{i_m}]\hfill \\ & +& log(1-q)\sum _{1\le i_1<\dots <i_m\le n}1\hfill \\ & =& e_{\left[n\right]}log\left(\frac{q}{1-q}\right)+2\left(\genfrac{}{}{0pt}{}{\frac{n}{2}}{m}\right)log\left(\frac{1-p}{1-q}\right)+\left(\genfrac{}{}{0pt}{}{n}{m}\right)log(1-q)\hfill \\ \hfill e_{\left[n\right]}& =& \sum _{1\le i_1<\dots <i_m\le n}{A}_{i_1,\dots ,i_m}.\hfill \end{array}$$

Note that $\mathbb{II}$ is independent of $\sigma$.

Proof of Theorem 1

The likelihood function of the vector of noisy labels Y given node lable $\sigma$ is the following:

$$\begin{array}{ccc}\hfill \mathbb{P}\left(Y\right|\sigma )& =& \prod _{i=1}^n{(1-{\alpha }_n)}^{\mathrm{I}[y_i{\sigma }_i=+1]}{\alpha }_n^{\mathrm{I}[y_i{\sigma }_i=-1]}=\prod _{i=1}^n{(1-{\alpha }_n)}^{\mathrm{I}[y_i{\sigma }_i=+1]}{\alpha }_n^{1-\mathrm{I}[y_i{\sigma }_i=+1]}\hfill \end{array}$$

Then, the log-likelihood function can be written as follows:

$$\begin{array}{ccc}\hfill log\left(\mathbb{P}\right(Y\left|\sigma \right))& =& log\left(\frac{1-{\alpha }_n}{{\alpha }_n}\right)\sum _{i=1}^n\mathrm{I}[y_i{\sigma }_i=1]+log\left({\alpha }_n\right)\sum _{i=1}^{n}1\hfill \\ & :=& {\mathbb{I}}_s+{\mathbb{II}}_s,\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill {\mathbb{I}}_s& =& log\left(\frac{1-{\alpha }_n}{{\alpha }_n}\right)\sum _{i=1}^n\mathrm{I}[y_i{\sigma }_i=1]\hfill \\ & =& \left(log\left(\frac{1}{{\alpha }_n}\right)+log\left(1-{\alpha }_n\right)\right)\sum _{i=1}^n\mathrm{I}[y_i{\sigma }_i=1]\hfill \\ & \asymp & C_{{\alpha }_n}(s_{+}+s_{-}),\hfill \\ \hfill C_{{\alpha }_n}& =& log\left(\frac{1-{\alpha }_n}{{\alpha }_n}\right),\hfill \\ \hfill s_{+}& =& \sum _{i=1}^n\mathrm{I}[y_i=+1,{\sigma }_i=+1],\hfill \\ \hfill s_{-}& =& \sum _{i=1}^n\mathrm{I}[y_i=-1,{\sigma }_i=-1],\hfill \end{array}$$

and ${\mathbb{II}}_s=nlog\left({\alpha }_n\right).$

Noting that A and Y are independent given $\sigma$, the joint log-likelihood of $A,Y$ given $\sigma$ is the following:

$$\begin{array}{cc}\hfill log\left(\mathbb{P}\right(A,Y\left|\sigma \right))& =log\left(\mathbb{P}\right(A\left|\sigma \right))+log(\mathbb{P}\left(Y\right|\sigma \left)\right)\hfill \\ \hfill & =\mathbb{I}+\mathbb{II}+{\mathbb{I}}_s+{\mathbb{II}}_s\hfill \\ \hfill & \asymp C_{a,b}(e_{+}+e_{-})+C_{{\alpha }_n}(s_{+}+s_{-})+\mathbb{II}+{\mathbb{II}}_s.\hfill \end{array}$$

(8)

where $C_{a,b}=log\left(a\right)-log\left(b\right)$, $C_{{\alpha }_n}=log(1-{\alpha }_n)-log\left({\alpha }_n\right)$, and $\mathbb{II}+{\mathbb{II}}_s$ consist of the terms that are independent of $\sigma$.

Denote $e_{S_1,S_2}$ as the number of edges between two sets of nodes, say $S_1$ and $S_2$. Then, define the following events as follows:

$$\begin{array}{cc}\hfill \mathrm{F}& =\left\{\mathrm{maximum}\mathrm{likelihood}\mathrm{fails}\right\},\hfill \\ \hfill {\mathrm{F}}_{i,+}& =\{i\in {\mathrm{I}}_{+}:e_{i,{\mathrm{I}}_{-}}\ge e_{i,{\mathrm{I}}_{+}}+\frac{C_{{\alpha }_n}}{C_{a,b}}{y}_i+1\},\hfill \\ \hfill {\mathrm{F}}_{i,-}& =\{i\in {\mathrm{I}}_{-}:e_{i,{\mathrm{I}}_{+}}\ge e_{i,{\mathrm{I}}_{-}}-\frac{C_{{\alpha }_n}}{C_{a,b}}{y}_i+1\},\hfill \\ \hfill {\mathrm{F}}_{+}& ={\cup }_{i\in {\mathrm{I}}_{+}}{\mathrm{F}}_{i,+},\hfill \\ \hfill {\mathrm{F}}_{-}& ={\cup }_{i\in {\mathrm{I}}_{-}}{\mathrm{F}}_{i,-}.\hfill \end{array}$$

(9)

Lemma 1.

$\mathbb{P}\left(\mathrm{F}\right)=1-o\left(1\right)$, if ${\mathrm{F}}_{+}\ne \varnothing$ and ${\mathrm{F}}_{-}\ne \varnothing$.

Proof. 

Take $i_0\in {\mathrm{F}}_{+}$ and $j_0\in {\mathrm{F}}_{-}$. Then, define the following:

$${\tilde{\mathrm{I}}}_{+}={\mathrm{I}}_{+}\backslash \left\{i_0\right\}\cup \left\{j_0\right\},{\tilde{\mathrm{I}}}_{-}={\mathrm{I}}_{-}\backslash \left\{j_0\right\}\cup \left\{i_0\right\}.$$

Denote $\Lambda =\frac{\mathbb{P}(A,Y|\tilde{\sigma })}{\mathbb{P}(A,Y|\sigma )}$, and we need to show that $\mathbb{P}(\Lambda >1)=1-o\left(1\right)$.

By (8), we have the following:

$$log(\Lambda )\asymp C_{a,b}\left\{(e_{{\tilde{\mathrm{I}}}_{+}}+e_{{\tilde{\mathrm{I}}}_{-}})-(e_{+}+e_{-})\right\}+C_{{\alpha }_n}\left\{(s_{{\tilde{\mathrm{I}}}_{+}}+s_{{\tilde{\mathrm{I}}}_{-}})-(s_{+}+s_{-})\right\}.$$

(10)

It is clear that

$$\begin{array}{ccc}\hfill e_{{\tilde{\mathrm{I}}}_{+}}-e_{+}& =& -e_{i_0,{\mathrm{I}}_{+}}+e_{j_0,{\mathrm{I}}_{+}}-e_{j_0,{\mathrm{I}}_{+}\backslash \left\{i_0\right\}},\hfill \\ \hfill e_{{\tilde{\mathrm{I}}}_{-}}-e_{-}& =& -e_{j_0,{\mathrm{I}}_{-}}+e_{i_0,{\mathrm{I}}_{-}}-e_{i_0,{\mathrm{I}}_{-}\backslash \left\{j_0\right\}},\hfill \\ \hfill s_{{\tilde{\mathrm{I}}}_{+}}-s_{+}& =& -\mathrm{I}[y_{i_0}=+1]+\mathrm{I}[y_{j_0}=+1],\hfill \\ \hfill s_{{\tilde{\mathrm{I}}}_{-}}-s_{-}& =& -\mathrm{I}[y_{j_0}=-1]+\mathrm{I}[y_{i_0}=-1].\hfill \end{array}$$

Plugging into (10) yields

$$\begin{array}{ccc}\hfill log(\Lambda )& \asymp & C_{a,b}\left\{(e_{j_0,{\mathrm{I}}_{+}}-e_{j_0,{\mathrm{I}}_{-}})-(e_{i_0,{\mathrm{I}}_{+}}-e_{i_0,{\mathrm{I}}_{-}})-e_{j_0,{\mathrm{I}}_{+}\backslash \left\{i_0\right\}}-e_{i_0,{\mathrm{I}}_{-}\backslash \left\{j_0\right\}}\right\}+C_{{\alpha }_n}\{y_{j_0}-y_{i_0}\}\hfill \\ & =& \left\{C_{a,b}(e_{j_0,{\mathrm{I}}_{+}}-e_{j_0,{\mathrm{I}}_{-}})+C_{{\alpha }_n}{y}_{j_0}\right\}+\left\{C_{a,b}(e_{i_0,{\mathrm{I}}_{-}}-e_{i_0,{\mathrm{I}}_{+}})-C_{{\alpha }_n}{y}_{i_0}\right\}\hfill \\ & -& C_{a,b}{e}_{i_0,{\mathrm{I}}_{-}\backslash \left\{j_0\right\}}-C_{a,b}{e}_{j_0,{\mathrm{I}}_{+}\backslash \left\{i_0\right\}}\hfill \\ & \ge & 2C_{a,b}(1-e_{i_0,{\mathrm{I}}_{-}\backslash \left\{j_0\right\}}).\hfill \end{array}$$

In the last inequality, we assumed that $e_{i_0,{\mathrm{I}}_{-}\backslash \left\{j_0\right\}}\ge e_{j_0,{\mathrm{I}}_{+}\backslash \left\{i_0\right\}}$ without loss of generality.

Next, we will show that $\mathbb{E}\left(e_{i_0,{\mathrm{I}}_{-}\backslash \left\{j_0\right\}}\right)=o\left(1\right)$. Rewrite $e_{i_0,{\mathrm{I}}_{-}\backslash \left\{j_0\right\}}$ as:

$$e_{i_0,{\mathrm{I}}_{-}\backslash \left\{j_0\right\}}=\sum _{\begin{array}{c}1\le i_3<\dots <i_m\le \frac{n}{2}-1\\ \{i_3,\dots ,i_m\}\in {\mathrm{I}}_{-}\backslash \left\{j_0\right\}\end{array}}{A}_{i_0{j}_0{i}_3\dots i_m}=\sum _{k=1}^{\left(\genfrac{}{}{0pt}{}{\frac{n}{2}-1}{m-2}\right)}{Z}_k,$$

where $Z_k{\sim }_{i.i.d.}Z=Bern\left(q\right)$ and $q=\frac{blog\left(n\right)}{n^{m-1}}$. Then,

$$\mathbb{E}\left(e_{i_0,{\mathrm{I}}_{-}\backslash \left\{j_0\right\}}\right)=\left(\genfrac{}{}{0pt}{}{\frac{n}{2}-1}{m-2}\right)\frac{blog\left(n\right)}{n^{m-1}}=\frac{log\left(n\right)}{n}O\left(1\right)=o\left(1\right).$$

Applying Markov’s inequality, we have

$$\begin{array}{ccc}\hfill \mathbb{P}(\Lambda >1)& =& \mathbb{P}(log(\Lambda )>0)\ge \mathbb{P}(2C_{a,b}(1-e_{i_0,{\mathrm{I}}_{-}\backslash \left\{j_0\right\}})>0)=\mathbb{P}(e_{i_0,{\mathrm{I}}_{-}\backslash \left\{j_0\right\}}<1)\hfill \\ & =& 1-\mathbb{P}(e_{i_0,{\mathrm{I}}_{-}\backslash \left\{j_0\right\}}\ge 1)\ge 1-\mathbb{E}\left(e_{i_0,{\mathrm{I}}_{-}\backslash \left\{j_0\right\}}\right)=1-o\left(1\right),\hfill \end{array}$$

which complete the proof.    □

A. Proof of the impossible part of Theorem 1

Let $\mathrm{H}$ be a fixed subset of ${\mathrm{I}}_{+}$ of size $\left|\mathrm{H}\right|=\frac{n}{{log}^{\tau }\left(n\right)}$, with a constant $\tau >\frac{1}{m-1}$ and ${\theta }_n=\frac{log\left(n\right)}{loglog\left(n\right)}$. For any $i\in \mathrm{H}$, define the following events:

$$\begin{array}{ccc}\hfill {\Delta }_{i,\mathrm{H}}& =& \{e_{i,\mathrm{H}}\ge {\theta }_n\},\hfill \\ \hfill {\mathrm{F}}_{i,\mathrm{H}}& =& \{e_{i,{\mathrm{I}}_{-}}\ge e_{i,{\mathrm{I}}_{+}\backslash \mathrm{H}}+{\theta }_n+\frac{C_{{\alpha }_n}}{C_{a,b}}{y}_i+1\},\hfill \\ \hfill {\Delta }_{\mathrm{H}}& =& {\cup }_{i\in \mathrm{H}}{\Delta }_{i,\mathrm{H}},\hfill \\ \hfill {\mathrm{F}}_{\mathrm{H}}& =& {\cup }_{i\in \mathrm{H}}{\mathrm{F}}_{i,\mathrm{H}}.\hfill \end{array}$$

Lemma 2.

$\mathbb{P}\left({\Delta }_{\mathrm{H}}\right)=o\left(1\right)$.

Proof. 

Let $W_k{\sim }_{i.i.d.}Bern\left(p\right)$. By definition, for any $i\in \mathrm{H}$,

$$\begin{array}{ccc}\hfill e_{i,\mathrm{H}}& =& \sum _{\begin{array}{c}1\le i_2<\dots <i_m\le \left|\mathrm{H}\right|-1\\ \{i_2,\dots ,i_m\}\subset \mathrm{H}\backslash \left\{i\right\}\end{array}}{A}_{i{i}_2\dots i_m}=\sum _{k=1}^{\left(\genfrac{}{}{0pt}{}{\left|\mathrm{H}\right|-1}{m-1}\right)}{W}_k,\hfill \\ \hfill \mathbb{E}\left(e_{i,\mathrm{H}}\right)& =& \left(\genfrac{}{}{0pt}{}{\left|\mathrm{H}\right|-1}{m-1}\right)\frac{alog\left(n\right)}{n^{m-1}}.\hfill \end{array}$$

Then, the multiplicative Chernoff bound (see $\left(iii\right)$ in Lemma A1) gives the following:

$$\begin{array}{ccc}\hfill \mathbb{P}\left({\Delta }_{i,\mathrm{H}}\right)& =& \mathbb{P}\left(e_{i,\mathrm{H}}\ge {\theta }_n\right)=\mathbb{P}\left(\sum _{k=1}^{\left(\genfrac{}{}{0pt}{}{\left|\mathrm{H}\right|-1}{m-1}\right)}{W}_k\ge {\theta }_n\right)\le {\left(\frac{n^{m-1}loglog\left(n\right)}{ea\left(\genfrac{}{}{0pt}{}{\left|\mathrm{H}\right|-1}{m-1}\right)}\right)}^{-{\theta }_n}.\hfill \end{array}$$

By union bound we have the following:

$$\begin{array}{c}\hfill \mathbb{P}\left({\Delta }_{\mathrm{H}}\right)\le \left|\mathrm{H}\right|{\left(\frac{n^{m-1}loglog\left(n\right)}{ea\left(\genfrac{}{}{0pt}{}{\left|\mathrm{H}\right|-1}{m-1}\right)}\right)}^{-{\theta }_n}=n^{1-(m-1)\tau +o\left(1\right)}=o\left(1\right),\end{array}$$

since $\tau >\frac{1}{m-1}$ by the assumption on $\left|\mathrm{H}\right|$.    □

Lemma 3.

$\mathbb{P}\left({\mathrm{F}}_{i,\mathrm{H}}\right)\gtrsim \frac{1}{\left|\mathrm{H}\right|}log\left(\frac{1}{\delta }\right)$ under the condition in (4), for any $\delta \in (0,1)$.

Proof. 

See Appendices Appendix A.1 and Appendix A.2.    □

Lemma 4.

$\mathbb{P}\left({\mathrm{F}}_{\mathrm{H}}\right)=1-o\left(1\right)$ under the condition in (4).

Proof. 

Under the condition in (4), Lemma 3 holds. That is,

$$\mathbb{P}\left({\mathrm{F}}_{i,\mathrm{H}}\right)>\frac{1}{\left|\mathrm{H}\right|}log\left(\frac{1}{\delta }\right)$$

(11)

for any $\delta \in (0,1)$ and for a sufficiently large n. By union bound, we have the following:

$$\begin{array}{ccc}\hfill \mathbb{P}\left({\mathrm{F}}_{\mathrm{H}}\right)& =& \mathbb{P}\left({\cup }_{i\in \mathrm{H}}{\mathrm{F}}_{i,\mathrm{H}}\right)=1-\mathbb{P}\left({\cap }_{i\in \mathrm{H}}{\left({\mathrm{F}}_{i,\mathrm{H}}\right)}^c\right)=1-{(1-\mathbb{P}\left({\mathrm{F}}_{i,\mathrm{H}}\right))}^{\left|\mathrm{H}\right|}\hfill \\ & >& 1-{(1-\mathbb{P}\left({\mathrm{F}}_{i,\mathrm{H}}\right))}^{-log\left(\delta \right)\frac{1}{\mathbb{P}\left({\mathrm{F}}_{i,\mathrm{H}}\right)}}\hfill \\ & =& 1-{\delta }^{-\frac{1}{\mathbb{P}\left({\mathrm{F}}_{i,\mathrm{H}}\right)}log(1-\mathbb{P}\left({\mathrm{F}}_{i,\mathrm{H}}\right))}\hfill \\ & \ge & 1-\delta .\hfill \end{array}$$

Here, we used (11) in the first inequality.    □

Lemma 5.

$\mathbb{P}\left(\mathrm{F}\right)=1-o\left(1\right)$ under the condition in (4).

Proof. 

By Lemmas 2 and 4, there exists a sufficiently small $\delta >0$, such that $\mathbb{P}\left({\Delta }^c\right)\ge 1-\delta$ and $\mathbb{P}\left({\mathrm{F}}_{\mathrm{H}}\right)\ge 1-\delta$ simultaneously hold. Clearly, ${\Delta }^c\cap {\mathrm{F}}_{\mathrm{H}}\Rightarrow {\mathrm{F}}_{+}$. Hence,

$$\mathbb{P}\left({\mathrm{F}}_{+}\right)\ge \mathbb{P}\left({\Delta }^c\right)+\mathbb{P}\left({\mathrm{F}}_{\mathrm{H}}\right)-1\ge 1-2\delta .$$

By symmetry, we have

$$\mathbb{P}\left({\mathrm{F}}_{-}\right)\ge 1-2\delta .$$

Then, it follows from Lemma 1 that

$$\mathbb{P}\left(\mathrm{F}\right)\ge \mathbb{P}\left({\mathrm{F}}_{{\mathrm{I}}_{+}}\right)+\mathbb{P}\left({\mathrm{F}}_{{\mathrm{I}}_{-}}\right)-1\ge 1-4\delta ,$$

which completes the proof.    □

The impossible part of Theorem 1 follows directly from Lemma 5.

B. Proof of the possible part of Theorem 1

Let ${\mathrm{H}}_1$ be an independently generated random hypergraph, built on the same node set of $\mathcal{V}=\left[n\right]$, with an edge probability of $\frac{d_n}{log\left(n\right)}$. Denote its complement by ${\mathrm{H}}_2={\mathrm{H}}_1^c$ and

$${\mathcal{H}}_1=\mathcal{H}\cap {\mathrm{H}}_1,{\mathcal{H}}_2=\mathcal{H}\cap {\mathrm{H}}_2.$$

A weak recovery algorithm [12] is applied to ${\mathcal{H}}_1$ to return a partition of two communities ${\tilde{\mathrm{I}}}_{+}$ and ${\tilde{\mathrm{I}}}_{-}$, which agree with the true communities ${\mathrm{I}}_{+}$ and ${\mathrm{I}}_{-}$ on at least $(1-\delta )n$ nodes. Here, $\delta =\delta \left(d_n\right)$ depends on $d_n$ such that $\delta \to 0$ as $d_n\to \infty$. $d_n$ can be taken as $O(loglog(n\left)\right)$ [12]. In the next step, we will use ${\mathcal{H}}_2$ to decide whether to flip a node’s membership or not. More specifically, for a node $i\in {\tilde{\mathrm{I}}}_{+}$, if it has more edges in ${\mathcal{H}}_2$ going to ${\tilde{\mathrm{I}}}_{-}$, plus the scaled side information $\frac{C_{{\alpha }_n}}{C_{a,b}}{y}_i$, then we reset $i\in {\tilde{\mathrm{I}}}_{-}$. Similarly, for $i\in {\tilde{\mathrm{I}}}_{-}$, if it has more edges in ${\mathcal{H}}_2$ going to ${\tilde{\mathrm{I}}}_{+}$, minus the scaled side information $\frac{C_{{\alpha }_n}}{C_{a,b}}{y}_i$, then we reset $i\in {\tilde{\mathrm{I}}}_{+}$. If the number of flips in each community is not the same, then keep the discard change. This algorithm has been summarized in Algorithm 1.

Algorithm 1: Algorithm for exact recovery of community structure in hypergraphs with label information.

1. Input: Hypergraph $\mathcal{H}$ and label information y; 2. Partition $\mathcal{H}={\mathcal{H}}_1\bigsqcup {\mathcal{H}}_2$, where ${\mathcal{H}}_1=\mathcal{H}\cap {\mathrm{H}}_1$, ${\mathcal{H}}_2=\mathcal{H}\cap {\mathrm{H}}_1^c$, and ${\mathrm{H}}_1$ make an     Erd$\ddot{\mathrm{o}}$s–Renyi graph generated with an edge probability of $\frac{d_n}{log\left(n\right)}$; 3. Apply weak recovery algorithm [12] to ${\mathcal{H}}_1$ to return a partition ${\mathrm{I}}_{+,0}\bigsqcup {\mathrm{I}}_{-,0}$; 4. Initialize ${\tilde{\mathrm{I}}}_{+}\leftarrow {\mathrm{I}}_{+,0}$, and ${\tilde{\mathrm{I}}}_{-}\leftarrow {\mathrm{I}}_{-,0}$; 5. Flip membership if      $i\in {\tilde{\mathrm{I}}}_{+}$ and $e_{i,{\tilde{\mathrm{I}}}_{-}}\ge e_{i,{\tilde{\mathrm{I}}}_{+}}+\frac{C_{{\alpha }_n}}{C_{a,b}}{y}_i$ in ${\mathcal{H}}_2$, or      $i\in {\tilde{\mathrm{I}}}_{-}$ and $e_{i,{\tilde{\mathrm{I}}}_{+}}\ge e_{i,{\tilde{\mathrm{I}}}_{-}}-\frac{C_{{\alpha }_n}}{C_{a,b}}{y}_i$ in ${\mathcal{H}}_2$; 6. If $|{\tilde{\mathrm{I}}}_{+}|\ne |{\mathrm{I}}_{+,0}|$, then keep ${\mathrm{I}}_{+,0}$ and ${\mathrm{I}}_{-,0}$ unchanged.

To show (5), the possible part of Theorem 1, we first introduce the following definitions. For any node $i\in \left[n\right]$, it is mis-classified if and only if it belongs to the following:

$$\begin{array}{c}\hfill \left\{i\in {\tilde{\mathrm{I}}}_{+}:{\sigma }_i=-1\right\}\cup \left\{i\in {\tilde{\mathrm{I}}}_{-}:{\sigma }_i=+1\right\}.\end{array}$$

With WLOG, we assume that $i\in {\tilde{\mathrm{I}}}_{+}$. Then, the mis-classification probability of node i is given by the following:

$$\begin{array}{cc}\hfill {\mathrm{M}}_i& =\mathbb{P}\left(e_{i,{\tilde{\mathrm{I}}}_{-}}\ge e_{i,{\tilde{\mathrm{I}}}_{+}}+\frac{C_{{\alpha }_n}}{C_{a,b}}{y}_i\right)\hfill \\ \hfill & =\mathbb{P}\left(\sum _{k=1}^{\left(\genfrac{}{}{0pt}{}{\frac{n}{2}}{m-1}\right)-\left(\genfrac{}{}{0pt}{}{\frac{n}{2}\delta }{m-1}\right)}{Z}_k+\sum _{k=1}^{\left(\genfrac{}{}{0pt}{}{\frac{n}{2}\delta }{m-1}\right)}{W}_k\ge \sum _{k=1}^{\left(\genfrac{}{}{0pt}{}{\frac{n}{2}}{m-1}\right)-\left(\genfrac{}{}{0pt}{}{\frac{n}{2}\delta }{m-1}\right)}{W}_k+\sum _{k=1}^{\left(\genfrac{}{}{0pt}{}{\frac{n}{2}\delta }{m-1}\right)}{Z}_k+\frac{C_{{\alpha }_n}}{C_{a,b}}{y}_i\right).\hfill \end{array}$$

(12)

In the last equation, we assumed ${\mathrm{H}}_2$ to be a complete graph. Then, the probability of the existence of a mis-classified label is:

$$\begin{array}{c}\hfill \mathrm{M}\le n{\mathrm{M}}_i,\end{array}$$

by union bound on all nodes.

Consider a node $i\in {\mathrm{H}}_1$. Its degree is given by the following:

$$deg\left(i\right)=\sum _{\begin{array}{c}1\le i_2<\dots <i_m\le n-1\\ \{i_2,\dots ,i_m\}\in \left[n\right]\backslash \left\{i\right\}\end{array}}{A}_{i{i}_2\dots i_m}.$$

Lemma 6.

${max}_{i\in {\mathrm{H}}_1}\{deg\left(i\right)\}\lesssim 2\left(\genfrac{}{}{0pt}{}{n-1}{m-1}\right)\frac{d_n}{log\left(n\right)}$.

Proof. 

Note that $deg\left(i\right)\sim Binom\left(\left(\genfrac{}{}{0pt}{}{n-1}{m-1}\right),\frac{d_n}{log\left(n\right)}\right)$. Then, the multiplicative Chernoff bound (see $\left(iii\right)$ in Lemma A1) gives the following:

$$\begin{array}{ccc}\hfill \mathbb{P}\left(deg\left(i\right)\ge 2\mu \right)& \le & {\left(\frac{e}{4}\right)}^{\mu }<e^{-\frac{1}{4}\mu },\hfill \end{array}$$

where $\mu =\left(\genfrac{}{}{0pt}{}{n-1}{m-1}\right)\frac{d_n}{log\left(n\right)}$. By union bound, we have the following:

$$\begin{array}{c}\hfill \mathbb{P}\left(\underset{i\in {\mathrm{H}}_1}{max}\{deg\left(i\right)\}\ge 2\mu \right)\le n\mathbb{P}\left(deg\left(i\right)\ge 2\mu \right)<n{e}^{-\frac{1}{4}\mu }=o\left(1\right).\end{array}$$

□

The Lemma above suggests that:

$$\underset{i\in {\mathrm{H}}_2}{min}\{deg\left(i\right)\}\gtrsim \left(\genfrac{}{}{0pt}{}{n-1}{m-1}\right)\left(1-\frac{2d_n}{log\left(n\right)}\right).$$

Therefore, taking into account the incompleteness of ${\mathrm{H}}_2$, we will loose the upper bound (12) by removing $2\left(\genfrac{}{}{0pt}{}{n-1}{m-1}\right)\frac{d_n}{log\left(n\right)}$ terms from both of the summations on the right-hand side of (12). That is,

$$\begin{array}{ccc}\hfill {\mathrm{M}}_i& =& \mathbb{P}\left(\sum _{k=1}^{\left(\genfrac{}{}{0pt}{}{\frac{n}{2}}{m-1}\right)-\left(\genfrac{}{}{0pt}{}{\frac{n}{2}\delta }{m-1}\right)}{Z}_k+\sum _{k=1}^{\left(\genfrac{}{}{0pt}{}{\frac{n}{2}\delta }{m-1}\right)}{W}_k\ge \sum _{k=1}^{\left(\genfrac{}{}{0pt}{}{\frac{n}{2}}{m-1}\right)-\left(\genfrac{}{}{0pt}{}{\frac{n}{2}\delta }{m-1}\right)-2\left(\genfrac{}{}{0pt}{}{n-1}{m-1}\right)\frac{d_n}{log\left(n\right)}}{W}_k+\sum _{k=1}^{\left(\genfrac{}{}{0pt}{}{\frac{n}{2}\delta }{m-1}\right)-2\left(\genfrac{}{}{0pt}{}{n-1}{m-1}\right)\frac{d_n}{log\left(n\right)}}{Z}_k+\frac{C_{{\alpha }_n}}{C_{a,b}}{y}_i\right).\hfill \end{array}$$

Lemma 7.

$$\mathrm{M}\le \left\{\begin{array}{cc}\hfill & n^{1-{\eta }_{m,a,b}\left(\beta \right)+o\left(1\right)},\beta <C_{m,a,b}(a-b),\hfill \\ \hfill & n^{1-\beta +o\left(1\right)},\beta >C_{m,a,b}(a-b).\hfill \end{array}\right.$$

Proof. 

See Appendix A.3. □

The possible part of Theorem 1 follows directly from Lemma 7.

4. Conclusions

In this paper, we studied the effect of label information on the exact recovery of communities in uniform hypergraphs from an information-theoretical point of view. Specifically, we considered two types of label information: a noisy label for each node was observed independently, with $1-{\alpha }_n$ as the probability that the noisy label would match the true label, and the true label of each node was observed independently, with a probability of $1-{\alpha }_n$. We used the maximum likelihood method to derive a lower bound for exact recovery and then constructed an estimator that could exactly recover the communities above the lower bound. In this way, we obtained sharp boundaries for exact recovery under both scenarios. We found that the label information improved the sharp detection boundary if and only if ${\alpha }_n$ converges´d to zero at a rate of $n^{-\beta }$ for some positive constant $\beta$.

There are several possible future research directions: (I) The sharp recovery boundary for general HSBM with label information is still unknown. Characterizing the boundary in this case is an important problem. (II) In this paper, we focused on the label information. It is important to consider other side information, such as the covariates observed for each node.