Basic models and questions in statistical network analysis

TL;DR

This paper reviews fundamental statistical questions in network analysis, focusing on community detection, graph embedding, and vertex identification across three canonical models, integrating advanced probability theory concepts.

Contribution

It introduces a comprehensive overview of key statistical challenges and methods in analyzing canonical network models, connecting them with advanced probability tools.

Findings

Analysis of community detection in stochastic block models

Embedding techniques for random geometric graphs

Vertex identification in preferential attachment trees

Abstract

Extracting information from large graphs has become an important statistical problem since network data is now common in various fields. In this minicourse we will investigate the most natural statistical questions for three canonical probabilistic models of networks: (i) community detection in the stochastic block model, (ii) finding the embedding of a random geometric graph, and (iii) finding the original vertex in a preferential attachment tree. Along the way we will cover many interesting topics in probability theory such as P\'olya urns, large deviation theory, concentration of measure in high dimension, entropic central limit theorems, and more.