Accurate statistical inference on large, complex networks is a vitally important, inter-disciplinary research area that has witnessed exponential growth over the last several years, ranging from the construction of a plethora of random graph models themselves to a host of approaches for inference of graph model parameters Nevertheless, many graph estimation techniques are somewhat ad-hoc: maximum likelihood estimates for certain exponential random graph models, for instance or spectral methods for combinatorial graph analysis. But a mere regression coefficient here, a parametric estimate there, a clustering here, and an upper bound there do not constitute a unified, parsimonious approach to random graph inference. Thus the synthesis of disparate models and methods into a more comprehensive and familiar paradigm for graph inference is both necessary and welcome. This proposal address the need for such foundational approach to graph inference. We focus on the development of a unified spectral framework for mathematical statistics on graphs, itself inspired by cornerstones of classical Euclidean inference. In particular, for random graphs with independent edges, we use low-rank approximation of their adjacency matrices to build estimates of underlying model parameters. We then systematically address the graph-inferential analogues of the central tenets of Euclidean inference: consistency of estimators; asymptotic normality or appropriate limit distributions of estimators; asymptotic relative efficiency and optimality; one-, two- and multi-sample graph hypothesis testing; and robustness.

This proposal aims to develop methodologies for automated inference in high-dimensional and complex data. The proposal is part of the D3M (Data Driven Discovery of Models) program in which we have just raw data as input and we need to discover primitives -- simple yet robust and agile procedures that can be easily combined to form sophisticated framework/methodologies -- and generate models for presentation to domain experts for feedback & selection, all of this done without a data scientist assistance. As an example of the applicability such a framework, consider our experience with linear regression where there is a well-understood pipeline to take multivariate linear regression data and automatically generate plots and diagnostics that assist the non-expert user. For thir proposal we will consider datasets such as (a) multivariate time series together with event-of-interest time points (t1,t2,��������������������������� ,tn), (b) multispectral imagery together with event-of-interest locations (x1, x2,��������������������������� , xn), and (c) a relational network together with event-of-interest nodes (v1, v2,��������������������������� , vn). We will first develop methodologies to automatically discover primitives for these type of data. We will then develop methodologies to automatically compose these discovered primitives into a collection of models for performing subsequent inference. The final result is a discoverable archive of data modeling primitives, procedures for automatic selection of primitives, and frameworks for composition of primitives into complex modeling pipelines.