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Since the publication of Breiman (2001), Random Forests (RF) have been widely used in both regression and classification. Later on, other forests are also proposed and studied in literature and Mondrian Forests are notable examples built on the Mondrian process; see Lakshminarayanan et al. (2014). In this paper, we propose an ensemble estimator in general statistical learning based on Mondrian Forests, which can be regarded as an extension of RF. This general framework includes many common learning problems, such as least squared regression, least $\ell_1$ regression, quantile regression and classification. Under mild conditions of loss functions, we give the upper bound of the regret function of our estimator and show that such estimator is statistically consistent.

Modeling and estimating mixed memberships for overlapping unipartite un-weighted networks has been well studied in recent years. However, to our knowledge, there is no model for a more general case, the overlapping bipartite weighted networks. To close this gap, we introduce a novel model, the Bipartite Mixed Membership Distribution-Free (BiMMDF) model. Our model allows an adjacency matrix to follow any distribution as long as its expectation has a block structure related to node membership. In particular, BiMMDF can model overlapping bipartite signed networks and it is an extension of many previous models, including the popular mixed membership stochastic blcokmodels. An efficient algorithm with a theoretical guarantee of consistent estimation is applied to fit BiMMDF. We then obtain the separation conditions of BiMMDF for different distributions. Furthermore, we also consider missing edges for sparse networks. The advantage of BiMMDF is demonstrated in extensive synthetic networks and eight real-world networks.

We consider the problem of community detection in overlapping weighted networks, where nodes can belong to multiple communities and edge weights can be finite real numbers. To model such complex networks, we propose a general framework - the mixed membership distribution-free (MMDF) model. MMDF has no distribution constraints of edge weights and can be viewed as generalizations of some previous models, including the well-known mixed membership stochastic blockmodels. Especially, overlapping signed networks with latent community structures can also be generated from our model. We use an efficient spectral algorithm with a theoretical guarantee of convergence rate to estimate community memberships under the model. We also propose the fuzzy weighted modularity to evaluate the quality of community detection for overlapping weighted networks with positive and negative edge weights. We then provide a method to determine the number of communities for weighted networks by taking advantage of our fuzzy weighted modularity. Numerical simulations and real data applications are carried out to demonstrate the usefulness of our mixed membership distribution-free model and our fuzzy weighted modularity.

The bipartite network appears in various areas, such as biology, sociology, physiology, and computer science. \cite{rohe2016co} proposed Stochastic co-Blockmodel (ScBM) as a tool for detecting community structure of binary bipartite graph data in network studies. However, ScBM completely ignores edge weight and is unable to explain the block structure of a weighted bipartite network. Here, to model a weighted bipartite network, we introduce a Bipartite Distribution-Free model by releasing ScBM's distribution restriction. We also build an extension of the proposed model by considering the variation of node degree. Our models do not require a specific distribution on generating elements of the adjacency matrix but only a block structure on the expected adjacency matrix. Spectral algorithms with theoretical guarantees on the consistent estimation of node labels are presented to identify communities. Our proposed methods are illustrated by simulated and empirical examples., Comment: 27pages

Mixed membership community detection is a challenge problem in network analysis. To estimate the memberships and study the impact of regularized spectral clustering under the mixed membership stochastic block (MMSB) model, this article proposes two efficient spectral clustering approaches based on regularized Laplacian matrix, Simplex Regularized Spectral Clustering (SRSC) and Cone Regularized Spectral Clustering (CRSC). SRSC and CRSC methods are designed based on the ideal simplex structure and the ideal cone structure in the variants of the eigen-decomposition of the population regularized Laplacian matrix. We show that these two approaches SRSC and CRSC are asymptotically consistent under mild conditions by providing error bounds for the inferred membership vector of each node under MMSB. Through the theoretical analysis, we give the upper and lower bound for the regularizer $\tau$. By introducing a parametric convergence probability, we can directly see that when $\tau$ is large these two methods may still have low error rates but with a smaller probability. Thus we give an empirical optimal choice of $\tau$ is $O(log(n))$ with $n$ the number of nodes to detect sparse networks. The proposed two approaches are successfully applied to synthetic and empirical networks with encouraging results compared with some benchmark methods., Comment: arXiv admin note: text overlap with arXiv:2011.12239

Charge-based field-effect transistors (FETs) greatly suffer from unavoidable carrier scattering and heat dissipation. In analogy to valley degree of freedom in semiconductors, chiral anomaly current in Weyl/Dirac semimetals is theoretically predicted to be nearly non-dissipative over long distances, but still lacks experimental ways to efficiently control its transport. Here we demonstrate field-effect chirality devices with Dirac semimetal PtSe2, in which its Fermi level is close to the Dirac point in conduction band owing to intrinsic defects. The chiral anomaly is further corroborated with nonlocal valley transport measurement, which can also be effectively modulated by external fields, showing robust nonlocal valley transport with micrometer diffusion length. Similar to charge-based FETs, the chiral conductivity in PtSe2 devices can be modulated by electrostatic gating with an ON/OFF ratio more than 103. We also demonstrate basic logic functions in the devices with electric and magnetic fields as input signals.

Mixed membership problem for undirected network has been well studied in network analysis recent years. However, the more general case of mixed membership for directed network in which nodes can belong to multiple communities remains a challenge. Here, we propose an interpretable and identifiable model: directed mixed membership stochastic blockmodel (DiMMSB) for directed mixed membership networks. DiMMSB allows that row nodes and column nodes of the adjacency matrix can be different and these nodes may have distinct community structure in a directed network. We also develop an efficient spectral algorithm called DiSP designed based on simplex structures inherent in the left and right singular vectors of the population adjacency matrix to estimate the mixed memberships for both row nodes and column nodes in a directed network. We show that DiSP is asymptotically consistent under mild conditions by providing error bounds for the inferred membership vectors of each row node and each column node using delicate spectral analysis. Numerical results on computer-generated directed mixed membership networks support our theoretical findings and show that our DiSP outperforms its competitor in both error rates and run-time. Applications of DiSP to real-world directed networks demonstrate the advantages of DiSP in studying the asymmetric structure of directed networks., Comment: 36 pages, 8 figures, 1 table

Mixed membership community detection is a challenging problem. In this paper, to detect mixed memberships, we propose a new method Mixed-SLIM which is a spectral clustering method on the symmetrized Laplacian inverse matrix under the degree-corrected mixed membership model. We provide theoretical bounds for the estimation error on the proposed algorithm and its regularized version under mild conditions. Meanwhile, we provide some extensions of the proposed method to deal with large networks in practice. These Mixed-SLIM methods outperform state-of-art methods in simulations and substantial empirical datasets for both community detection and mixed membership community detection problems.

Community detection has been well studied recent years, but the more realistic case of mixed membership community detection remains a challenge. Here, we develop an efficient spectral algorithm Mixed-ISC based on applying more than K eigenvectors for clustering given K communities for estimating the community memberships under the degree-corrected mixed membership (DCMM) model. We show that the algorithm is asymptotically consistent. Numerical experiments on both simulated networks and many empirical networks demonstrate that Mixed-ISC performs well compared to a number of benchmark methods for mixed membership community detection. Especially, Mixed-ISC provides satisfactory performances on weak signal networks., Comment: 24 pages, 2 figures, 14 tables. arXiv admin note: substantial text overlap with arXiv:2011.12239

Mixed-SCORE is a recent approach for mixed membership community detection proposed by Jin et al. (2017) which is an extension of SCORE (Jin, 2015). In the note Jin et al. (2018), the authors propose SCORE+ as an improvement of SCORE to handle with weak signal networks. In this paper, we propose a method called Mixed-SCORE+ designed based on the Mixed-SCORE and SCORE+, therefore Mixed-SCORE+ inherits nice properties of both Mixed-SCORE and SCORE+. In the proposed method, we consider K+1 eigenvectors when there are K communities to detect weak signal networks. And we also construct vertices hunting and membership reconstruction steps to solve the problem of mixed membership community detection. Compared with several benchmark methods, numerical results show that Mixed-SCORE+ provides a significant improvement on the Polblogs network and two weak signal networks Simmons and Caltech, with error rates 54/1222, 125/1137 and 94/590, respectively. Furthermore, Mixed-SCORE+ enjoys excellent performances on the SNAP ego-networks., Comment: 17 pages, 4 figures, 2 tables

Community detection in network analysis is an attractive research area recently. Here, under the degree-corrected mixed membership (DCMM) model, we propose an efficient approach called mixed regularized spectral clustering (Mixed-RSC for short) based on the regularized Laplacian matrix. Mixed-RSC is designed based on an ideal cone structure of the variant for the eigen-decomposition of the population regularized Laplacian matrix. We show that the algorithm is asymptotically consistent under mild conditions by providing error bounds for the inferred membership vector of each node. As a byproduct of our bound, we provide the theoretical optimal choice for the regularization parameter {\tau}. To demonstrate the performance of our method, we apply it with previous benchmark methods on both simulated and real-world networks. To our knowledge, this is the first work to design spectral clustering algorithm for mixed membership community detection problem under DCMM model based on the application of regularized Laplacian matrix.

For community detection problem, spectral clustering is a widely used method for detecting clusters in networks. In this paper, we propose an improved spectral clustering (ISC) approach under the degree corrected stochastic block model (DCSBM). ISC is designed based on the k-means clustering algorithm on the weighted leading K + 1 eigenvectors of a regularized Laplacian matrix where the weights are their corresponding eigenvalues. Theoretical analysis of ISC shows that under mild conditions the ISC yields stable consistent community detection. Numerical results show that ISC outperforms classical spectral clustering methods for community detection on both simulated and eight empirical networks. Especially, ISC provides a significant improvement on two weak signal networks Simmons and Caltech, with error rates of 121/1137 and 96/590, respectively.

Spectral clustering methods are widely used for detecting clusters in networks for community detection, while a small change on the graph Laplacian matrix could bring a dramatic improvement. In this paper, we propose a dual regularized graph Laplacian matrix and then employ it to three classical spectral clustering approaches under the degree-corrected stochastic block model. If the number of communities is known as $K$, we consider more than $K$ leading eigenvectors and weight them by their corresponding eigenvalues in the spectral clustering procedure to improve the performance. Three improved spectral clustering methods are dual regularized spectral clustering (DRSC) method, dual regularized spectral clustering on Ratios-of-eigenvectors (DRSCORE) method, and dual regularized symmetrized Laplacian inverse matrix (DRSLIM) method. Theoretical analysis of DRSC and DRSLIM show that under mild conditions DRSC and DRSLIM yield stable consistent community detection, moreover, DRSCORE returns perfect clustering under the ideal case. We compare the performances of DRSC, DRSCORE and DRSLIM with several spectral methods by substantial simulated networks and eight real-world networks., Comment: 43 pages, 10 figures, 5 tables

Based on the classical Degree Corrected Stochastic Blockmodel (DCSBM) model for network community detection problem, we propose two novel approaches: principal component clustering (PCC) and normalized principal component clustering (NPCC). Without any parameters to be estimated, the PCC method is simple to be implemented. Under mild conditions, we show that PCC yields consistent community detection. NPCC is designed based on the combination of the PCC and the RSC method (Qin & Rohe 2013). Population analysis for NPCC shows that NPCC returns perfect clustering for the ideal case under DCSBM. PCC and NPCC is illustrated through synthetic and real-world datasets. Numerical results show that NPCC provides a significant improvement compare with PCC and RSC. Moreover, NPCC inherits nice properties of PCC and RSC such that NPCC is insensitive to the number of eigenvectors to be clustered and the choosing of the tuning parameter. When dealing with two weak signal networks Simmons and Caltech, by considering one more eigenvectors for clustering, we provide two refinements PCC+ and NPCC+ of PCC and NPCC, respectively. Both two refinements algorithms provide improvement performances compared with their original algorithms. Especially, NPCC+ provides satisfactory performances on Simmons and Caltech, with error rates of 121/1137 and 96/590, respectively., Comment: 33 pages, 6 figures, 3 tables

Varying coefficient model is often used in statistical modeling since it is more flexible than the parametric model. However, model detection and variable selection of varying coefficient model are poorly understood in mode regression. Existing methods in the literature for these problems often based on mean regression and quantile regression. In this paper, we propose a novel method to solve these problems for mode varying coefficient model based on the B-spline approximation and SCAD penalty. Moreover, we present a new algorithm to estimate the parameters of interest, and discuss the parameters selection for the tuning parameters and bandwidth. We also establish the asymptotic properties of estimated coefficients under some regular conditions. Finally, we illustrate the proposed method by some simulation studies and an empirical example., Comment: 13 pages

Nano-thick metallic transition metal dichalcogenides such as VS$_{2}$ are essential building blocks for constructing next-generation electronic and energy-storage applications, as well as for exploring unique physical issues associated with the dimensionality effect. However, such 2D layered materials have yet to be achieved through either mechanical exfoliation or bottom-up synthesis. Herein, we report a facile chemical vapor deposition route for direct production of crystalline VS$_{2}$ nanosheets with sub-10 nm thicknesses and domain sizes of tens of micrometers. The obtained nanosheets feature spontaneous superlattice periodicities and excellent electrical conductivities (~3$\times$10$^{3}$ S cm$^{-1}$), which has enabled a variety of applications such as contact electrodes for monolayer MoS$_{2}$ with contact resistances of ~1/4 to that of Ni/Au metals, and as supercapacitor electrodes in aqueous electrolytes showing specific capacitances as high as 8.6$\times$10$^{2}$ F g$^{-1}$. This work provides fresh insights into the delicate structure-property relationship and the broad application prospects of such metallic 2D materials., Comment: 23 pages, 5 figues