Sequential locality of graphs and its hypothesis testing

TL;DR

This paper introduces a hypothesis testing framework to evaluate local connectivity patterns in graphs based on vertex sequences, aiding in understanding graph structures and optimizing vertex arrangements.

Contribution

It presents a novel statistical testing approach for assessing local vertex connectivity in graphs considering vertex orderings, grounded in combinatorial and block model methods.

Findings

Effective for small to moderate data sizes

Provides a statistical basis for envelope reduction

Applicable to graphs with intrinsic vertex ordering

Abstract

The adjacency matrix is the most fundamental and intuitive object in graph analysis that is useful not only mathematically but also for visualizing the structures of graphs. Because the appearance of an adjacency matrix is critically affected by the ordering of rows and columns, or vertex ordering, statistical assessment of graphs together with their vertex sequences is important in identifying the characteristic structures of graphs. In this paper, we propose a hypothesis testing framework that assesses how locally vertices are connected to each other along a specified vertex sequence, which provides a statistical foundation for an optimization problem called envelope reduction or minimum linear arrangement. The proposed tests are particularly suitable for moderately small data and formulated based on a combinatorial approach and a block model with intrinsic vertex ordering.