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Two-sample hypothesis testing is a classical problem in statistics. Given i.i.d. samples X1,…,Xm∼P and Y1,…,Yn∼Q, the two-sample testing is to test whether P is significantly different from Q or no...

Community detection is a fundamental task in network data mining. Various algorithms have been proposed to detect the communities of a network. However, the output of these algorithms are meaningful only if community structure exists in the network. It is necessary to statistically test the presence of community structure before applying any community detection algorithms. The existing algorithms or testing procedures mainly focus on unweighted graph, that is, the edge presence or absence is coded as a binary variable. However, most real-world networks have weights. Recently, several algorithms have been devised to detect communities in weighted networks. In this paper, we consider the fundamental problem whether community structure exists in a weighted network. Specifically, we propose a test statistic based on the number of weighted triangles and edges, derive its limiting distribution under the null hypothesis and analyze its power. The simulation results and real data application show that the proposed test can achieve high power.

Network (graph) data analysis is a popular research topic in statistics and machine learning. In application, one is frequently confronted with graph two-sample hypothesis testing where the goal is to test the difference between two graph populations. Several statistical tests have been devised for this purpose in the context of binary graphs. However, many of the practical networks are weighted and existing procedures can't be directly applied to weighted graphs. In this paper, we study the weighted graph two-sample hypothesis testing problem and propose a practical test statistic. We prove that the proposed test statistic converges in distribution to the standard normal distribution under the null hypothesis and analyze its power theoretically. The simulation study shows that the proposed test has satisfactory performance and it substantially outperforms the existing counterpart in the binary graph case. A real data application is provided to illustrate the method., to appear in Journal of Applied Statistics

In network data analysis, one of the fundamental data mining tasks is to find dense subgraphs, which has wide applications in biology, finance, graph compression and so on. In practice, given netwo...

Community detection refers to the problem of clustering the nodes of a network (either graph or hypergrah) into groups. Various algorithms are available for community detection and all these methods apply to uncensored networks. In practice, a network may has censored (or missing) values and it is shown that censored values have non-negligible effect on the structural properties of a network. In this paper, we study community detection in censored $m$-uniform hypergraph from information-theoretic point of view. We derive the information-theoretic threshold for exact recovery of the community structure. Besides, we propose a polynomial-time algorithm to exactly recover the community structure up to the threshold. The proposed algorithm consists of a spectral algorithm plus a refinement step. It is also interesting to study whether a single spectral algorithm without refinement achieves the threshold. To this end, we also explore the semi-definite relaxation algorithm and analyze its performance.

This paper considers both approximate and exact designs for estimating the total effects under one crossover and two interference models. They are different from the traditional block designs in the sense that the assigned treatments also affect their neighboring plots, hence a design is understood as a collection of sequences of treatments. A notable result in literature is that the circular neighbor balanced design (CNBD) is optimal among designs, which do not allow treatments to be neighbors of themselves. However, we find it necessary to allow self-neighboring, and further show that it is the best to allocate each treatment in a subblock of adjacent plots with equal or almost equal numbers of replications. This explains why the efficiency of CNBD drops down to 50% as the sequence length, say k, increases. Unlike CNBD or the designs for direct effects, our proposed designs do not try to put as many treatments in a sequence as possible. The optimal number of distinct treatments in a sequence is around 2k for crossover designs and k/0.96 for interference models, whenever they are smaller than the total number of treatments under consideration. We systematically study necessary and sufficient conditions for any design to be universally optimal under the approximate design framework, based on which algorithms for deriving optimal or efficient exact designs are proposed. This hybrid nature of cohesively combining theories with algorithms makes our method more flexible than existing ones in the following aspects. (i) Not only symmetric designs are studied, general procedures for producing asymmetric designs are also provided. (ii) Our method applies to any form of within-block covariance matrix instead of specific forms. (iii) We cover all configurations of the numbers of treatments and sequence lengths, especially for large values of them when purely computational methods are not applicable. (iv) On top of the latter, we cover a continuous spectrum of the number of sequences instead of special numbers decided by combinatorial constraints.

In this paper, we investigate the hypothesis test of the parametric component in partial functional linear regression model based on B-spline. The test statistic is constructed by using the residua...

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: (1) a noisy label for each node is observed independently, with 1−αn as the probability that the noisy label will match the true label, (2) the true label of each node is observed independently, with the probability of 1−αn. 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 αn=n−β+o(1) for a constant β&gt, 0.

We study conformal vector fields on a Finsler manifold whose metric is defined by a Riemannian metric, a $1$-form and its norm. We find PDEs characterizing conformal vector fields. Then we obtain the explicit expressions of conformal vector fields for certain spherically symmetric metrics on $\mathbb R^n$.

In this paper, we study the stochastic block model (SBM) with growing number of clusters and bounded degree. Specifically, for SBM G s n ( n , a n , b n ) with diverging s n blocks and fixed a and b ( a > b > 0 ) , we prove that if ( a − b ) 2 b , then it is impossible to distinguish G s n ( n , a n , b n ) from the corresponding Erdos–Renyi model G n , a + ( s n − 1 ) b n s n .

A fundamental problem in network data analysis is to test Erd\"{o}s-R\'{e}nyi model $\mathcal{G}\left(n,\frac{a+b}{2n}\right)$ versus a bisection stochastic block model $\mathcal{G}\left(n,\frac{a}{n},\frac{b}{n}\right)$, where $a,b>0$ are constants that represent the expected degrees of the graphs and $n$ denotes the number of nodes. This problem serves as the foundation of many other problems such as testing-based methods for determining the number of communities (\cite{BS16,L16}) and community detection (\cite{MS16}). Existing work has been focusing on growing-degree regime $a,b\to\infty$ (\cite{BS16,L16,MS16,BM17,B18,GL17a,GL17b}) while leaving the bounded-degree regime untreated. In this paper, we propose a likelihood-ratio (LR) type procedure based on regularization to test stochastic block models with bounded degrees. We derive the limit distributions as power Poisson laws under both null and alternative hypotheses, based on which the limit power of the test is carefully analyzed. We also examine a Monte-Carlo method that partly resolves the computational cost issue. The proposed procedures are examined by both simulated and real-world data. The proof depends on a contiguity theory developed by Janson \cite{J95}., Comment: In this new submission, we add a comment in introduction stating that > the classic test based on counting the $k_n$-cycles with > $k_n=\log^{1/4}{n}$ is unrealistic in practice, which is also the > motivation of our regularized LR test

In this paper, we study a significant non-Riemannian quantity Ξ-curvature, which is defined by S-curvature. Firstly, we obtain the formula of Ξ-curvature for (α, β)-metrics. Based on it, we show that the Ξ-curvature vanishes for a class of (α, β)-metrics. In the end, we get the relation of Ξ-curvature for conformally related Finsler metrics, and classify conformally flat (α, β)-metrics with almost vanishing Ξ-curvature.

In this paper, we apply empirical likelihood method to infer for the regression parameters in the partial functional linear regression models based on B-spline. We prove that the empirical log-likelihood ratio for the regression parameters converges in law to a weighted sum of independent chi-square distributions. Our simulation shows that the proposed empirical likelihood method produces more accurate confidence regions in terms of coverage probability than the asymptotic normality method.

In this paper, we study a class of Finsler metrics defined by a Riemannian metric and 1-form. We classify those metrics which are projectively flat with weakly isotropic flag curvature.