Collaborative Research: Inference for Network Models with Covariates: Leveraging Local Information for Statistically and Computationally Efficient Estimation of Global Parameters

大型数据集，自然地被建模为网络或图，几乎出现在人类努力的每个领域。例如，Facebook是一个社交网络，其中节点是用户，边缘对应友谊。在基因网络中，节点表示具有与其共表达相对应的连接的基因。在生态网络中，节点是动物物种，其边缘是根据谁吃谁来确定的。网络或图数据研究的一个主要焦点是识别节点的社区成员。然而，对于科学目的来说，更重要的是检查边缘和成员概率的本质和进化，例如，个体基因特征的变化作为某种未知因素（如疾病）的函数。将重点放在使用节点和边缘的其他测量特征上，可以以决定性的方式增加从观察到的边缘或节点之间的相互作用中获得的信息。这些可能是疾病症状或测试结果，或社交网络用户的人口统计信息。这些模型中的统计推断，尽管很重要，但研究才刚刚开始。由于模型拟合的复杂性和数据集的规模，这在理论上和计算上都存在挑战。这项研究将导致拟合模型和置信度统计措施的算法的发展，在许多领域具有潜在的应用。当节点或边协变量存在时，研究的重点是图的块模型。当制定这些模型时，这些模型不再是块模型，而是成员概率依赖于协变量的模型，其连接概率依赖于块成员和单个协变量的模型。拟合算法涉及在拟合块和协变量参数之间交替进行。变分（平均场）方法，有效地导致半参数模型拟合与nK成员“讨厌”参数，其中n代表节点的数量和K的社区的数量，进行了检查。由于pi已经发现这些方法对于大n是不稳定的，pi已经开始研究分而治之算法的理论和实践方面，其中许多子图是独立拟合的。pi将通过渐近和模拟研究统计性质，并为大型相对稀疏的图开发实用且计算稳定的方法。

Large datasets, which are naturally modeled as a network or graph, arise in almost every field of human endeavor. For example, Facebook is a social network, where nodes are users, with edges corresponding to friendships. In gene networks, nodes represent genes with connections corresponding to their co-expression. In ecological networks, the nodes are animal species, with edges determined according to who eats whom. A major focus of research for network or graph data has been on identifying community membership of the nodes. However, what is often more important for scientific purposes is examining the nature and evolution of edge and membership probabilities, for instance changes in gene features of individuals as a function of some unknown factor, like a disease. The focus on using other measured features of nodes and edges could add, in decisive ways, to the information available from observed edges or interactions between nodes. These could be disease symptoms or test results, or demographic information of users in social networks. Statistical inference in such models, despite its importance, has only just begun to be studied. There are both theoretical and computational challenges, due both to the complexity of models fitted, and the size of data sets. The research will lead to the development of algorithms for fitting models and statistical measures of confidence, with potential applications to many fields. The research is focused on block models for graphs, when node or edge covariates are present. When formulated, these models are no longer block models, but models whose membership probabilities depend upon covariates and whose connection probabilities depend both on block membership and individual covariates. Fitting algorithms involve alternating between fitting block and covariate parameters. Variational (mean field) approaches which effectively lead to semi-parametric model fitting with nK membership "nuisance" parameters, with n representing the number of nodes and K the number of communities, are examined. As these approaches have been found by the PIs to be unstable for large n, the PIs have already begun to investigate the theoretical and practical aspects of divide and conquer algorithms where many subgraphs are independently fit. The PIs will study the statistical properties, both asymptotically and through simulations, and develop practicable and computationally stable methods for large, relatively sparse graphs.