Statistical inference on random dot product graphs: a survey (1709.05454v1)

Abstract: The random dot product graph (RDPG) is an independent-edge random graph that is analytically tractable and, simultaneously, either encompasses or can successfully approximate a wide range of random graphs, from relatively simple stochastic block models to complex latent position graphs. In this survey paper, we describe a comprehensive paradigm for statistical inference on random dot product graphs, a paradigm centered on spectral embeddings of adjacency and Laplacian matrices. We examine the analogues, in graph inference, of several canonical tenets of classical Euclidean inference: in particular, we summarize a body of existing results on the consistency and asymptotic normality of the adjacency and Laplacian spectral embeddings, and the role these spectral embeddings can play in the construction of single- and multi-sample hypothesis tests for graph data. We investigate several real-world applications, including community detection and classification in large social networks and the determination of functional and biologically relevant network properties from an exploratory data analysis of the Drosophila connectome. We outline requisite background and current open problems in spectral graph inference.

• The paper introduces a comprehensive framework for estimating latent positions and testing structural equivalence in random dot product graphs using spectral embedding methods.
• It establishes the consistency and asymptotic normality of these embeddings, ensuring reliable recovery of latent positions as graph size increases.
• The paper validates its methods through real-world applications, including community detection and network analysis in connectomics and social networks.

Statistical Inference on Random Dot Product Graphs: A Survey

The paper "Statistical inference on random dot product graphs: a survey" provides a comprehensive framework for statistical inference involving Random Dot Product Graphs (RDPGs). RDPGs form a versatile model in graph theory, acting as a generalization over simpler models such as Stochastic Block Models (SBMs), while also allowing for efficient spectral methods. This survey extensively covers methods to utilize RDPGs for estimating latent positions and performing hypothesis tests, using methodologies deeply rooted in linear algebra and probability theory.

Summary of Core Concepts and Methods

1. RDPG Model: An RDPG defines a graph in terms of a latent position space, where the vertices of the graph are associated with positions in a high-dimensional Euclidean space. The probability of an edge existing between two vertices is given by the dot product of their associated latent positions. This mechanism allows RDPGs to model complex relational data.
2. Spectral Embedding Techniques: The survey provides a thorough investigation into spectral embedding techniques, particularly focusing on the Adjacency Spectral Embedding (ASE) and Laplacian Spectral Embedding (LSE). These embeddings are pivotal for transforming graph data into Euclidean spaces where classical statistical methods can be applied.
3. Consistency and Asymptotic Normality: A significant focus is given to proving the consistency of these embeddings, demonstrating that as graph size increases, the estimated latent positions converge to true latent positions, up to orthogonal transformations. Additionally, asymptotic distributional properties are derived, showing that the estimated latent positions follow a multivariate normal distribution under certain conditions.
4. Hypothesis Testing: The paper introduces techniques for testing equivalence of latent position structures across multiple sample graphs, both semiparametric and nonparametric. The main hypothesis tested is whether two RDPGs could have originated from the same or similarly transformed latent positions. These tests are crucial in applications for comparing structures of different networks or sampled graph instances.
5. Real-world Applications: Practical applications are discussed, such as community detection in networks and exploratory data analysis in connectomics. The paper highlights the efficacy of spectral methods in uncovering meaningful structures even within noisy datasets.

Implications and Future Directions

The application of RDPGs and spectral embeddings in various fields such as neuroscience and social network analysis illustrates the versatility and robustness of these methods. RDPGs can succinctly capture complex interaction patterns, which simplifies visualization and analysis of substantial datasets typically found in these fields.

Furthermore, the survey lays the groundwork for future exploration in several areas:

• Robustness to Model Mis-specification: While foundational results are established, there are opportunities to improve the methods' applicability in scenarios with misspecified models or noise contamination.
• Handling Large-scale Data: Further studies could explore scalability and computational efficiency, especially when dealing with exceptionally large or dynamic datasets.
• Joint Graph Inference: There is potential for developing more generalized techniques that can simultaneously handle multiple graphs with shared properties, thus facilitating holistic analyses of interconnected systems.

Overall, this paper serves as both a foundation and an inspiration for ongoing research in statistical inference and network analysis. Through careful analytic techniques and practical demonstrations, it showcases the profound impact that well-rounded statistical methods can have on understanding graph-structured data across disciplines.