We mainly consider three problems in this thesis. In the first part, we focus on the community detection for directed networks, which could be regarded as the second-order tensor. Most of the existing research ignores the dynamic pattern of an important network feature called node popularity and restricts node popularities for nodes in the same community to behave identically or change uniformly in all communities. Motivated by the variability of node popularity in empirical networks, we propose a novel probabilistic framework for directed network community detection called the two-way node popularity model (TNPM). For the model fitting and community structure uncovering, we develop the Rank One Approximation Algorithm (ROA) and Two-Stage Divided Cosine Algorithm (TSDC), and the last on...[ Read more ]

We mainly consider three problems in this thesis. In the first part, we focus on the community detection for directed networks, which could be regarded as the second-order tensor. Most of the existing research ignores the dynamic pattern of an important network feature called node popularity and restricts node popularities for nodes in the same community to behave identically or change uniformly in all communities. Motivated by the variability of node popularity in empirical networks, we propose a novel probabilistic framework for directed network community detection called the two-way node popularity model (TNPM). For the model fitting and community structure uncovering, we develop the Rank One Approximation Algorithm (ROA) and Two-Stage Divided Cosine Algorithm (TSDC), and the last one is more computationally efficient to fit for large-scale networks. In addition, we establish the consistency of ROA for community detection. Extensive numerical studies demonstrate the advantages of our proposed method in terms of both estimation accuracy and computation efficiency. The method is also applied to the Worldwide Trading Networks, yielding some interesting findings.

In the second part, we consider the problem of detecting volatility structure change points for tensor sequence data. The majority of approaches to the problem of change point detection focus only on the univariate or multivariate case. Tensor sequence data has not been considered so far. To address this, we propose a new method, which preserves the multi-dimensional data structure and overcomes the curse of dimensionality for covariance parameter estimation. Furthermore, we prove consistency under general conditions. More precisely, the consistency still holds even when the data has non-Gaussian distribution. Extensive numerical studies show that our proposed method improves the estimation accuracy substantially. The detected changes for two real data examples coincide well with both economic growth and recession periods.

Thirdly, we study the problem of measuring the dependence between variables. Most of the existing correlation coefficients are designed either for testing independence or for measuring the strength of the relationship between the variables. Besides, they could be easily affected by the oscillation and some local noise. To address this deficiency, we define a new dependence measure called the Local Adjusted Chatterjee Correlation Coefficient (LAC3), which takes on its extreme values precisely at independence and almost surely functional dependence respectively. Furthermore, we establish the corresponding consistency and asymptotic theories. In addition, we propose a new procedure for local signal identification. Extensive numerical studies and the real data application demonstrate the advantages of LAC3 in terms of local signal detection, generality property, and equitability property.