Statistical Network Analysis: Model Selection, Differential Privacy, and Dynamic Structures

在本文中，我们解决了统计网络分析中以下三个方面的一些具有挑战性的问题。选择网络模型的抖动重采样在统计建模中，第一步也是最重要的一步是为给定的数据集选择合适的模型。虽然统计中通常存在许多数据驱动的模型选择方法，包括那些基于数据重用的方法（即，bootstrap重采样，交叉验证），但它们在网络数据中的应用存在问题。因此，主观地选择网络模型仍然是常见的。网络数据重用的主要困难是模拟潜在的概率机制。一些现有的尝试包括在某些特定设置下的交叉验证。我们提出了一种新的“自举抖动”或“抖动重采样”方法来选择合适的网络模型。该方法不强加任何特定的形式/条件，因此为网络模型选择提供了一个通用的工具。网络数据的边缘差分隐私网络数据个体通常用节点表示，它们之间的相互关系用边表示。因此，网络数据通常包含敏感的个人/个人信息。另一方面，应该保留数据中感兴趣的信息。因此，对数据隐私的主要关注是双重的：(a)仅发布原始网络数据的经过处理的版本以保护隐私；(b)经过处理的数据应保留感兴趣的信息，以便基于经过处理的数据进行的分析仍然有意义。这是一个充满活力的研究领域，因为在这个信息时代，数据隐私变得越来越敏感和重要，并且可以获得大量的数字形式的个人信息，尽管统计学的贡献仍处于初级阶段。我们将采用所谓的dyadwise随机反应方法。虽然这种方案是不同的私有的，但基于发布数据的推断在很大程度上是未知的。我们的初步调查揭示了这种方法的一些有吸引力的特点，表明比基于其他数据发布机制的统计推断更有效。我们将进一步开发该方案，以处理具有附加节点特征/属性的网络（例如，具有年龄、性别、爱好、职业等附加信息的社交网络）。建模和预测动态网络大多数现有的网络统计推理方法都局限于静态网络数据，尽管实际网络中有很大一部分是动态的。理解并能够预测随时间的变化对于以下方面非常重要，例如，监测互联网流量网络中的异常情况，预测电力供应网络的需求和定价，管理传感器网络中环境读数中的自然资源，以及理解新闻和意见如何在在线社交网络中传播。不幸的是，动态网络基础的发展仍处于起步阶段，可用的建模和推理工具很少。对于处理网络的动态变化，大多数可用的技术都是基于快照网络随时间的演化分析，而不是真正的动态建模。虽然这反映了大多数网络随时间缓慢变化的事实，但它并没有提供任何关于变化背后的动态的见解，而且对于未来的预测几乎无能为力，因此建立适当的随机模型来明确地捕捉动态依赖和动态变化是必不可少的。结合张量分解和因子驱动降维的最新发展，以及指数平滑和卡尔曼滤波等有效的时间序列工具，我们将接受这一挑战，建立一些新的动态模型。

In this proposal we tackle some challenging problems in the following three aspects of statistical network analysis.1. Jittered resampling for selecting network modelsThe first and arguably the most important step in statistical modelling is to choose an appropriate model for a given data set. While there exist many data-driven model-selection methods in statistics in general, including those based on data reuse (i.e., bootstrap resampling, cross-validation), their application to network data is problematic. Therefore it remains common to choose a network model subjectively. The major difficulty in the reuse of network data is to mimic the underlying probability mechanisms. A few existing attempts include cross-validation under some specific settings. We propose a new `bootstrap jittering' or `jittered resampling' method for selecting an appropriate network model. The method does not impose any specific forms/conditions, therefore providing a generic tool for network model selection.2. Edge differential privacy for network dataIn network data individuals are typically represented by nodes and their inter-relationships are represented by edges. Therefore network data often contain sensitive individual/personal information. On the other hand the information of interest in the data should be perserved. Hence the primary concern for data privacy is two-folded: (a) to release only a sanitized version of the original network data to protect privacy, and (b) the sanitized data should preserves the information of interest such that the analysis based on the sanitized data is still meaningful. This is a vibrant research area now as data privacy becomes ever increasingly sensitive and important with available abundant personal information in digital format in this information age, though the contribution from statistics is still at a preliminary stage. We will adopt the so-called dyadwise randomized response approach. While such a scheme is differentially private, the inference based on the released data is largely unknown. Our initial investigation reveals some attractive features of this approach, suggesting more efficient statistical inference than those based on other data release mechanism. We will further develop this scheme to handle networks with additional node features/attributes (e.g., social networks with additional information on age, gender, hobby, occupation etc).3. Modelling and forecasting dynamic networksMost existing statistical inference methods for networks are confined to static network data, though a substantial proportion of real networks are dynamic in nature. Understanding and being able to forecast the changes over time are of immense importance for, e.g., monitoring anomalies in internet traffic networks, predicting demand and setting pricing in electricity supply networks, managing natural resources in environmental readings in sensor networks, and understanding how news and opinion propagates in online social networks. Unfortunately the development of the foundation for dynamic networks is still in its infancy, and the available modelling and inference tools are sparse. As for dealing with dynamic changes of networks, most available techniques are based on the evolution analysis of snapshot networks over time without really modelling the changes dynamically. Although this reflects the fact that most networks change slowly over time, it does not provides any insight on the dynamics underlying the changes and is almost powerless for future prediction for which it is essential to build appropriate stochastic models to capture dynamic dependence and dynamic changes explicitly. Combining recent developments on tensor decomposition and factor-driven dimension reduction with the efficient time series tools such as exponential smoothing and Kalman filters, we will take on this challenge to build some new dynamic models.