Same Stats, Different Graphs: Exploring the Space of Graphs in Terms of Graph Properties

TL;DR

This paper explores the relationships between various graph properties using visual analytics and sampling methods, highlighting how different graphs can share similar statistics yet differ significantly.

Contribution

It introduces a visual analytics system for analyzing graph property correlations and investigates sampling methods for higher-order graphs, including generating graphs with identical properties but different structures.

Findings

Correlation patterns between graph properties identified

Sampling methods impact distribution of graph properties

Graphs with identical statistics can be structurally distinct

Abstract

Data analysts commonly utilize statistics to summarize large datasets. While it is often sufficient to explore only the summary statistics of a dataset (e.g., min/mean/max), Anscombe's Quartet demonstrates how such statistics can be misleading. We consider a similar problem in the context of graph mining. To study the relationships between different graph properties, we examine low-order non-isomorphic graphs and provide a simple visual analytics system to explore correlations across multiple graph properties. However, for larger graphs, studying the entire space quickly becomes intractable. We use different random graph generation methods to further look into the distribution of graph properties for higher order graphs and investigate the impact of various sampling methodologies. We also describe a method for generating many graphs that are identical over a number of graph properties and statistics yet are clearly different and identifiably distinct.