This paper surveys recent theoretical advances in convex optimization approaches for community detection. We introduce some important theoretical techniques and results for establishing the consistency of convex community detection under various statistical models. In particular, we discuss the basic techniques based on the primal and dual analysis. We also present results that demonstrate several distinctive advantages of convex community detection, including robustness against outlier nodes, consistency under weak assortativity, and adaptivity to heterogeneous degrees.

This survey is not intended to be a complete overview of the vast literature on this fast-growing topic. Instead, we aim to provide a big picture of the remarkable recent development in this area and to make the survey accessible to a broad audience. We hope that this expository article can serve as an introductory guide for readers who are interested in using, designing, and analyzing convex relaxation methods in network analysis.

This paper surveys some recent developments in fundamental limits and optimal algorithms for network analysis. We focus on minimax optimal rates in three fundamental problems of network analysis: graphon estimation, community detection and hypothesis testing. For each problem, we review state-of-the-art results in the literature followed by general principles behind the optimal procedures that lead to minimax estimation and testing. This allows us to connect problems in network analysis to other statistical inference problems from a general perspective.

Network datasets typically exhibit certain types of statistical patterns, such as within-dyad correlation, degree heterogeneity, and triadic patterns such as transitivity and clustering. The first two of these can be well represented with a social relations model, a type of additive effects model originally developed for continuous dyadic data. Higher-order patterns can be represented with multiplicative effects models, which are related to matrix decompositions that are commonly used for matrix-variate data analysis. Additionally, these multiplicative effects models generalize other popular latent feature network models, such as the stochastic blockmodel and the latent space model. In this article, we review a general regression framework for the analysis of network data that combines these two types of effects, and accommodates a variety of network data types, including continuous, binary and ordinal network relations.

We discuss how sampling design, units, the observation mechanism and other basic statistical notions figure into modern network data analysis. These considerations pose several new challenges that cannot be adequately addressed by merely extending or generalizing classical methods. Such challenges stem from fundamental differences between the domains in which network data emerge and those for which classical tools were developed. By revisiting these basic statistical considerations, we suggest a framework in which to develop theory and methods for network analysis in a way that accounts for both conceptual and practical challenges of network science. We then discuss how some well-known model classes fit within this framework.

We define a latent structure random graph as a random dot product graph (RDPG) in which the latent position distribution incorporates both probabilistic and geometric constraints, delineated by a family of underlying distributions on some fixed Euclidean space, and a structural support submanifold from which are drawn the latent positions for the graph. For a one-dimensional latent structure model with known structural support, we extend existing results on the consistency of spectral estimates in RDPGs to demonstrate that the parameters of the underlying distribution can be efficiently estimated. We describe how to estimate or learn the structural support in cases where it is unknown, with a focus on graphs with latent positions along the Hardy–Weinberg curve. Finally, we use the latent structural model formulation to address a hitherto-open question in neuroscience on the bilateral homology of the Drosophila left and right hemisphere connectome.

The rise of network data in many different domains has offered researchers new insights into the problem of modeling complex systems and propelled the development of numerous innovative statistical methodologies and computational tools. In this paper, we primarily focus on two types of biological networks, gene networks and brain networks, where statistical network modeling has found both fruitful and challenging applications. Unlike other network examples such as social networks where network edges can be directly observed, both gene and brain networks require careful estimation of edges using measured data as a first step. We provide a discussion on existing statistical and computational methods for edge estimation and subsequent statistical inference problems in these two types of biological networks.

Statistical methods to evaluate the effectiveness of interventions are increasingly challenged by the inherent interconnectedness of units. Specifically, a recent flurry of methods research has addressed the problem of interference between observations, which arises when one observational unit’s outcome depends not only on its treatment but also the treatment assigned to other units. We introduce the setting of bipartite causal inference with interference, which arises when (1) treatments are defined on observational units that are distinct from those at which outcomes are measured and (2) there is interference between units in the sense that outcomes for some units depend on the treatments assigned to many other units. The focus of this work is to formulate definitions and several possible causal estimands for this setting, highlighting similarities and differences with more commonly considered settings of causal inference with interference. Toward an empirical illustration, an inverse probability of treatment weighted estimator is adapted from existing literature to estimate a subset of simplified, but interesting, estimands. The estimators are deployed to evaluate how interventions to reduce air pollution from 473 power plants in the U.S. causally affect cardiovascular hospitalization among Medicare beneficiaries residing at 18,807 zip code locations.

Gaussian processes (GPs) are commonly used as models for functions, time series, and spatial fields, but they are computationally infeasible for large datasets. Focusing on the typical setting of modeling data as a GP plus an additive noise term, we propose a generalization of the Vecchia (J. Roy. Statist. Soc. Ser. B 50 (1988) 297–312) approach as a framework for GP approximations. We show that our general Vecchia approach contains many popular existing GP approximations as special cases, allowing for comparisons among the different methods within a unified framework. Representing the models by directed acyclic graphs, we determine the sparsity of the matrices necessary for inference, which leads to new insights regarding the computational properties. Based on these results, we propose a novel sparse general Vecchia approximation, which ensures computational feasibility for large spatial datasets but can lead to considerable improvements in approximation accuracy over Vecchia’s original approach. We provide several theoretical results and conduct numerical comparisons. We conclude with guidelines for the use of Vecchia approximations in spatial statistics.

Depth notions in location have generated tremendous attention in the literature. In fact, data depth and its applications remain as one of the most active research topics in statistics over the last three decades. Most favored notions of depth in location include Tukey (In Proceedings of the International Congress of Mathematicians $($Vancouver, B.C., 1974$)$, Vol. 2 (1975) 523–531) half-space depth (HD), Liu (Ann. Statist. 18 (1990) 405–414) simplicial depth and projection depth (PD) (Stahel (1981) and Donoho (1982), Liu (In $L_{1}$-Statistical Analysis and Related Methods $($Neuchâtel, 1992$)$ (1992) 279–294 North-Holland), Zuo and Serfling (Ann. Statist. 28 (2000) 461–482) and (ZS00) and Zuo (Ann. Statist. 31 (2003) 1460–1490)), among others.

Depth notions in regression have also been proposed sporadically, nevertheless. The regression depth (RD) of Rousseeuw and Hubert (J. Amer. Statist. Assoc. 94 (1999) 388–433) (RH99), the most famous, exemplifies a direct extension of Tukey HD to regression. Other notions include Carrizosa (J. Multivariate Anal. 58 (1996) 21–26) and the ones proposed in this article via modifying a functional in Maronna and Yohai (Ann. Statist. 21 (1993) 965–990) (MY93). Is there any relationship between Carrizosa depth and the RD of RH99? Do these depth notions possess desirable properties? What are the desirable properties? Can existing notions serve well as depth notions in regression? These questions remain open.

The major objectives of the article include (i) revealing the connection between Carrizosa depth and RD of RH99; (ii) expanding location depth evaluating criteria in ZS00 for regression depth notions; (iii) examining the existing regression notions with respect to the gauges; and (iv) proposing the regression counterpart of the eminent location projection depth.

This conversation began in June 2015 in the Department of Statistics at Columbia University during Lai’s visit to his alma mater where he celebrated his seventieth birthday. It continued in the subsequent years at Columbia and Stanford. Lai was born on June 28, 1945, in Hong Kong, where he grew up and attended The University of Hong Kong, receiving his B.A. degree (First Class Honors) in Mathematics in 1967. He went to Columbia University in 1968 for graduate study in statistics and received his Ph.D. degree in 1971. He stayed on the faculty at Columbia and was appointed Higgins Professor of Mathematical Statistics in 1986. A year later he moved to Stanford, where he is currently Ray Lyman Wilbur Professor of Statistics, and by courtesy, also of Biomedical Data Science and Computational and Mathematical Engineering. He is a fellow of the Institute of Mathematical Statistics, the American Statistical Association and an elected member of Academia Sinica in Taiwan. He was the third recipient of the COPSS Award which he won in 1983. He has been married to Letitia Chow since 1975, and they have two sons and two grandchildren.