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，边对应于友谊。在基因网络中，节点代表基因，其连接对应于它们的共同表达。在生态网络中，节点是动物物种，边根据谁吃谁来确定。 网络或图形数据的研究的主要焦点是识别节点的社区成员身份。 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性质和演变，例如个体基因特征的变化作为某种未知因素的函数，如疾病。 对使用节点和边的其他测量特征的关注可以以决定性的方式增加从观察到的边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，或社交网络中用户的人口统计信息。 统计推断在这样的模型，尽管其重要性，才刚刚开始被研究。 由于拟合模型的复杂性和数据集的大小，理论和计算都面临挑战。 这项研究将导致拟合模型和置信度统计测量算法的发展，并在许多领域具有潜在的应用。研究的重点是块模型的图，当节点或边的协变量。当公式化时，这些模型不再是块模型，而是其成员资格概率取决于协变量并且其连接概率取决于块成员资格和单个协变量的模型。 拟合算法涉及拟合块和协变量参数之间的交替。 变分（平均场）的方法，有效地导致半参数模型拟合与nK成员的“滋扰”参数，与n代表节点的数量和K的社区的数量，进行检查。 由于PI发现这些方法对于大的n是不稳定的，PI已经开始研究分治算法的理论和实践方面，其中许多子图是独立适合的。 PI将研究统计特性，无论是渐进的，并通过模拟，并制定切实可行的和计算稳定的方法，大型，相对稀疏的图形。

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.