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A Hierarchical Scale-free Graph Generator under Limited Resources

Xiaorui Qi, Yanlong Wen, Xiaojie Yuan

TL;DR

The paper tackles graph generation under limited or no access to training data by leveraging invariant scale-free properties. It introduces a two-stage hierarchical generator: first, anchor-based substructure formation to enforce scale-free structure, and second, a degree-mixing edge-generation process with two thresholds to control exotic structures, underpinned by theoretical guarantees. Across 12 datasets from three categories, the method generally achieves closer alignment to ground truth distributions (measured by MMD on degree, clustering, and orbit) than traditional baselines and several deep-generative models, though some highly clustered or lattice-like graphs remain challenging. The approach offers a practical non-learning-based path for generating diverse graphs under resource constraints, with potential impact on domains where data are restricted or costly to obtain.

Abstract

Graph generation is one of the most challenging tasks in recent years, and its core is to learn the ground truth distribution hiding in the training data. However, training data may not be available due to security concerns or unaffordable costs, which severely blows the learning models, especially the deep generative models. The dilemma leads us to rethink non-learned generation methods based on graph invariant features. Based on the observation of scale-free property, we propose a hierarchical scale-free graph generation algorithm. Specifically, we design a two-stage generation strategy. In the first stage, we sample multiple anchor nodes to further guide the formation of substructures, splitting the initial node set into multiple ones. Next, we progressively generate edges by sampling nodes through a degree mixing distribution, adjusting the tolerance towards exotic structures via two thresholds. We provide theoretical guarantees for hierarchical generation and verify the effectiveness of our method under 12 datasets of three categories. Experimental results show that our method fits the ground truth distribution better than various generation strategies and other distribution observations.

A Hierarchical Scale-free Graph Generator under Limited Resources

TL;DR

The paper tackles graph generation under limited or no access to training data by leveraging invariant scale-free properties. It introduces a two-stage hierarchical generator: first, anchor-based substructure formation to enforce scale-free structure, and second, a degree-mixing edge-generation process with two thresholds to control exotic structures, underpinned by theoretical guarantees. Across 12 datasets from three categories, the method generally achieves closer alignment to ground truth distributions (measured by MMD on degree, clustering, and orbit) than traditional baselines and several deep-generative models, though some highly clustered or lattice-like graphs remain challenging. The approach offers a practical non-learning-based path for generating diverse graphs under resource constraints, with potential impact on domains where data are restricted or costly to obtain.

Abstract

Graph generation is one of the most challenging tasks in recent years, and its core is to learn the ground truth distribution hiding in the training data. However, training data may not be available due to security concerns or unaffordable costs, which severely blows the learning models, especially the deep generative models. The dilemma leads us to rethink non-learned generation methods based on graph invariant features. Based on the observation of scale-free property, we propose a hierarchical scale-free graph generation algorithm. Specifically, we design a two-stage generation strategy. In the first stage, we sample multiple anchor nodes to further guide the formation of substructures, splitting the initial node set into multiple ones. Next, we progressively generate edges by sampling nodes through a degree mixing distribution, adjusting the tolerance towards exotic structures via two thresholds. We provide theoretical guarantees for hierarchical generation and verify the effectiveness of our method under 12 datasets of three categories. Experimental results show that our method fits the ground truth distribution better than various generation strategies and other distribution observations.

Paper Structure

This paper contains 21 sections, 2 theorems, 6 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Traditional Methods
  4. Deep Generative Models
  5. Preliminary
  6. Methodology
  7. Overview
  8. First Step of Generation
  9. Hierarchical Scale-free Generation Strategy
  10. Discussion
  11. Experiments
  12. Experimental Settings
  13. Datasets
  14. Baselines
  15. Evaluation Metrics
  16. ...and 6 more sections

Key Result

theorem thmcountertheorem

The more connections between nodes, the more likely it is to receive new connections. Consequently, there are central nodes or hubs in networks linking to abundant new nodes.

Figures (6)

  • Figure 1: An illustration of deep learning models with limited/no training data failing to generate graphs due to safety propose or high price (in red). However, with invariant features dragging from observation on universal data available, we can generate graphs fitting ground truth distribution well without learning (in blue).
  • Figure 2: Generation strategy comparison between several baselines and our method.
  • Figure 3: Illustration of the edge generation. (A) connection to plain nodes. (B) connection to anchor nodes.
  • Figure 4: Generation steps with/without $d_{max}$ limitation. Anchor nodes and edges to add in each step are in red.
  • Figure 5: Degree distribution comparisons between generated and real-world graphs.
  • ...and 1 more figures

Theorems & Definitions (6)

  • definition thmcounterdefinition: Graph
  • definition thmcounterdefinition: Problem Statement
  • definition thmcounterdefinition: Substructure
  • definition thmcounterdefinition: Entry
  • theorem thmcountertheorem: Preferential Attachment
  • theorem thmcountertheorem: Desired Degree