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Learning Backbones: Sparsifying Graphs through Zero Forcing for Effective Graph-Based Learning

Obaid Ullah Ahmad, Anwar Said, Mudassir Shabbir, Xenofon Koutsoukos, Waseem Abbas

TL;DR

The paper tackles the challenge of sparsifying graphs for learning without sacrificing predictive performance by introducing learning backbones based on zero-forcing and controllability principles. It develops a ZFS-based backbone that yields a tree-like sparse structure preserving key controllability properties, and extends this with a distance-based backbone to maintain critical graph distances. Across eight datasets and six GNN models, the method often matches or improves performance while reducing edge counts, and outperforms random spanning trees in many cases. This approach connects controllability concepts to practical graph learning, offering a scalable framework with potential extensions via graph distances and the Graph Lottery Ticket hypothesis.

Abstract

This paper introduces a novel framework for graph sparsification that preserves the essential learning attributes of original graphs, improving computational efficiency and reducing complexity in learning algorithms. We refer to these sparse graphs as "learning backbones". Our approach leverages the zero-forcing (ZF) phenomenon, a dynamic process on graphs with applications in network control. The key idea is to generate a tree from the original graph that retains critical dynamical properties. By correlating these properties with learning attributes, we construct effective learning backbones. We evaluate the performance of our ZF-based backbones in graph classification tasks across eight datasets and six baseline models. The results demonstrate that our method outperforms existing techniques. Additionally, we explore extensions using node distance metrics to further enhance the framework's utility.

Learning Backbones: Sparsifying Graphs through Zero Forcing for Effective Graph-Based Learning

TL;DR

The paper tackles the challenge of sparsifying graphs for learning without sacrificing predictive performance by introducing learning backbones based on zero-forcing and controllability principles. It develops a ZFS-based backbone that yields a tree-like sparse structure preserving key controllability properties, and extends this with a distance-based backbone to maintain critical graph distances. Across eight datasets and six GNN models, the method often matches or improves performance while reducing edge counts, and outperforms random spanning trees in many cases. This approach connects controllability concepts to practical graph learning, offering a scalable framework with potential extensions via graph distances and the Graph Lottery Ticket hypothesis.

Abstract

This paper introduces a novel framework for graph sparsification that preserves the essential learning attributes of original graphs, improving computational efficiency and reducing complexity in learning algorithms. We refer to these sparse graphs as "learning backbones". Our approach leverages the zero-forcing (ZF) phenomenon, a dynamic process on graphs with applications in network control. The key idea is to generate a tree from the original graph that retains critical dynamical properties. By correlating these properties with learning attributes, we construct effective learning backbones. We evaluate the performance of our ZF-based backbones in graph classification tasks across eight datasets and six baseline models. The results demonstrate that our method outperforms existing techniques. Additionally, we explore extensions using node distance metrics to further enhance the framework's utility.

Paper Structure

This paper contains 15 sections, 1 theorem, 7 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Preliminaries
  4. Problem Formulation
  5. Learning Backbone
  6. Controllability Framework
  7. Zero Forcing for Controllability Backbone
  8. Controllability Backbone as Learning Backbone
  9. Generalized Learning Backbone
  10. Experimental Results
  11. Datasets
  12. Experimental Setup
  13. Results Analysis
  14. Conclusion and Future Work
  15. Acknolwdgements

Key Result

theorem thmcountertheorem

Given a graph $G=(V,E)$, Algorithm alg:backbone_zfs returns a learning backbone, a connected tree, that is strong structurally controllable for the computed leader set $V_\ell$.

Figures (4)

  • Figure 1: Main Idea: Sparsify the graph while maintaining the critical learning backbone for downstream machine learning tasks such as graph classification. The predicted label $\phi(\hat{G}, \bm{\theta}) = \tilde{y}$ should be close to the true label $y$.
  • Figure 2: $V_\ell = \{v_1, v_2, v_5, v_6\}$ is the input set. After the ZF process, $dset(G, V_\ell) = V$, as indicated by the black vertices. Hence, $V_\ell$ is a ZFS.
  • Figure 3: Illustration of the proposed framework: The process begins by identifying a leader vertex subset within the graph. Using network control theory, a graph sparsification framework is then applied to derive a tree-like structure, called the 'learning backbone', from the original graph.
  • Figure 4: Comparing the Efficacy of Network Backbone Structures for Graph Classification. The backbone represents the best-performing structure between $B_z$ and $B_d$. The results are compared against the original graphs and random spanning tree subgraphs of the original graphs.

Theorems & Definitions (8)

  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition: Zero Forcing (ZF) Process
  • definition thmcounterdefinition: Derived Set
  • definition thmcounterdefinition: Zero Forcing Set (ZFS)
  • definition thmcounterdefinition
  • theorem thmcountertheorem
  • proof