A survey of discrete methods in (algebraic) statistics for networks

TL;DR

This survey reviews open problems at the intersection of discrete mathematics, algebraic statistics, and network analysis, emphasizing the importance of algebraic and combinatorial methods in understanding network models.

Contribution

It highlights recent work connecting algebraic and graph-theoretic concepts to statistical models for networks, proposing open problems to unify these fields.

Findings

Identifies key open problems in algebraic statistics for networks.

Connects concepts from commutative algebra, graph theory, and statistics.

Suggests research directions to improve statistical analysis of network data.

Abstract

Sampling algorithms, hypergraph degree sequences, and polytopes play a crucial role in statistical analysis of network data. This article offers a brief overview of open problems in this area of discrete mathematics from the point of view of a particular family of statistical models for networks called exponential random graph models. The problems and underlying constructions are also related to well-known concepts in commutative algebra and graph-theoretic concepts in computer science. We outline a few lines of recent work that highlight the natural connection between these fields and unify them into some open problems. While these problems are often relevant in discrete mathematics in their own right, the emphasis here is on statistical relevance with the hope that these lines of research do not remain disjoint. Suggested specific open problems and general research questions should advance algebraic statistics theory as well as applied statistical tools for rigorous statistical analysis of networks.