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Challenges of Generating Structurally Diverse Graphs
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Formulates generating structurally diverse graph sets, argues Energy is a suitable diversity measure, and compares greedy, genetic, local optimization, and iterative DiGress-based generative approaches against random graph baselines.
Problem
Generating a set of graphs that is maximally structurally diverse, for testing graph algorithms and neural approximators, which has not been studied; existing diversity measures (Average, Bottleneck, Vendi Score, #Circles) fail desirable properties.
Approach
Defines diversity via an Energy measure over pairwise graph distances (monotonicity and uniqueness properties), and proposes several optimization algorithms (greedy with theoretical bound, genetic, local edge-edit search, and iterative DiGress generative modeling) that optimize it for arbitrary graph distances.
Key results
- Combined Greedy→Genetic→LocalOpt achieves the best non-neural results, e.g. GCD Energy 0.135 vs 0.553 for random graph generators (lower is better).
- IGGM[1M] achieves the best GCD Energy (0.120) and Portrait-div Energy (1.213), and the highest Average for GCD (8.687).
- Greedy is provably within factor 2^gamma of optimal Energy for a given candidate set, and Average/Bottleneck bounds are 1/2.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use IGGM instead of ER-mix | Graph generation · GCD | Energy: 0.281 -> 0.120 | Helped |
| Use IGGM instead of Random Graph Generators | Graph generation · Portrait-div | Energy: 6.009 -> 1.213 | Helped |
| Use Greedy → Genetic → LocalOpt instead of GraphWorld | Graph generation · NetLSD-heat | Energy: 5.108 -> 0.673 | Helped |
| Use ER-mix → Genetic instead of ER-mix | Graph generation · NetLSD-wave | Energy: 0.583 -> 0.117 | Helped |
| Use IGGM instead of ER-mix | Graph generation · GCD | Average pairwise distance: 4.350 -> 8.687 | Helped |
| Use IGGM instead of ER-mix | Graph generation · Portrait-div | Average pairwise distance: 0.607 -> 0.854 | Helped |
| Use Greedy → Genetic → LocalOpt instead of GraphWorld | Graph generation · NetLSD-heat | Average pairwise distance: 1.270 -> 3.073 | Helped |
| Use ER-mix → Genetic instead of ER-mix | Graph generation · NetLSD-wave | Average pairwise distance: 6.302 -> 10.625 | Helped |
| Use larger Energy parameter γ (ablation) | Energy optimization · Points distributed in a square | Coverage | Helped |
| Optimize diversity using NetLSD instead of GCD or Portrait-div | IGGM · Generated graph structures | Graph density | Mixed |
Surprising findings
- NetLSD-based diversity produces many extremely sparse graphs, often with zero clustering coefficient.
- The Vendi Score violates monotonicity and uniqueness, and the average pairwise distance can lead to degenerate solutions.
Limitations
- Scalability to large node counts n is a concern since the space of graphs grows rapidly and LocalOpt single-edge moves may be infeasible.
- Energy is unbounded when elements coincide and no computationally feasible measure satisfying all desirable properties is known.
Experimental setup
Datasets: ER-mix random graphs random graph generator samples GraphWorld-generated graphs
Benchmarks: Energy diversity on GCD, Portrait-div, NetLSD-heat, NetLSD-wave for 100 graphs with 16 nodes Energy diversity for 100 graphs with 64 nodes
Metrics: Energy Average pairwise distance
Baselines: Erdős-Rényi mixture (ER-mix) GraphWorld Random Graph Generators Average pairwise distance
Builds on: DiGress NetLSD Graphlet Correlation Distance Portrait Divergence Vendi Score
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