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We generalize the sign covariance introduced by Bergsma & Dassios (2014) to multivariate random variables and beyond. The new interpoint-ranking sign covariance is applicable to general types of random objects as long as a meaningful similarity measure can be defined, and it is shown to be zero if and only if the two random variables are independent. The test statistic is a |$U$| -statistic, whose large-sample behaviour guarantees that the proposed test is consistent against general types of alternatives. Numerical experiments and data analyses demonstrate the superior empirical performance of the proposed method. [ABSTRACT FROM AUTHOR]

We consider multi-layer network data where the relationships between pairs of elements are reflected in multiple modalities, and may be described by multivariate or even high-dimensional vectors. Under the multi-layer stochastic block model framework we derive consistency results for a least squares estimation of memberships. Our theorems show that, as compared to single-layer community detection, a multi-layer network provides much richer information that allows for consistent community detection from a much sparser network, with required edge density reduced by a factor of the square root of the number of layers. Moreover, the multi-layer framework can detect cohesive community structure across layers, which might be hard to detect by any single-layer or simple aggregation. Simulations and a data example are provided to support the theoretical results. [ABSTRACT FROM AUTHOR]