Challenges of Generating Structurally Diverse Graphs

TL;DR

This work addresses the problem of generating a set of graphs that are structurally diverse by formalizing diversity through pairwise graph distances and proposing an Energy-based measure that satisfies key properties like monotonicity and uniqueness. It compares multiple optimization strategies—Greedy, Genetic, LocalOpt, and Iterative Graph Generative Modeling (IGGM)—and demonstrates that these approaches can significantly improve diversity beyond basic random graph generators. The study also analyzes how the choice of graph distance influences the structural properties of the generated graphs, shedding light on the properties captured by different distances. The results suggest practical pathways for evaluating graph algorithms and neural approximations on a representative, diverse set of graph instances, while highlighting scalability and distance-selection challenges for future research.

Abstract

For many graph-related problems, it can be essential to have a set of structurally diverse graphs. For instance, such graphs can be used for testing graph algorithms or their neural approximations. However, to the best of our knowledge, the problem of generating structurally diverse graphs has not been explored in the literature. In this paper, we fill this gap. First, we discuss how to define diversity for a set of graphs, why this task is non-trivial, and how one can choose a proper diversity measure. Then, for a given diversity measure, we propose and compare several algorithms optimizing it: we consider approaches based on standard random graph models, local graph optimization, genetic algorithms, and neural generative models. We show that it is possible to significantly improve diversity over basic random graph generators. Additionally, our analysis of generated graphs allows us to better understand the properties of graph distances: depending on which diversity measure is used for optimization, the obtained graphs may possess very different structural properties which gives a better understanding of the graph distance underlying the diversity measure.

Figures (12)

• Figure 1: A sample of generated graphs
• Figure 2: Average node degree and average clustering coefficient in the Erdős-Rényi model with $n=16$ and $p=0.5$ (ER-0.5) or varying $p$ (ER-mix)
• Figure 3: Optimized Average or Energy on a line segment
• Figure 4: Joint distribution of graph characteristics for GCD, Portrait-div, NetLSD-heat
• Figure 5: Joint distribution of graph characteristics for graphs from IGGM: comparing graph distances

Theorems & Definitions (11)

• Proposition 2.1
• Theorem A.1
• proof
• Definition A.2: Vendi Score, friedman2023vendi
• Theorem C.1
• proof
• Proposition C.2
• proof
• Proposition C.3
• proof