A Unifying Framework for Spectrum-Preserving Graph Sparsification and Coarsening (1902.09702v5)

Abstract: How might one "reduce" a graph? That is, generate a smaller graph that preserves the global structure at the expense of discarding local details? There has been extensive work on both graph sparsification (removing edges) and graph coarsening (merging nodes, often by edge contraction); however, these operations are currently treated separately. Interestingly, for a planar graph, edge deletion corresponds to edge contraction in its planar dual (and more generally, for a graphical matroid and its dual). Moreover, with respect to the dynamics induced by the graph Laplacian (e.g., diffusion), deletion and contraction are physical manifestations of two reciprocal limits: edge weights of $0$ and $\infty$, respectively. In this work, we provide a unifying framework that captures both of these operations, allowing one to simultaneously sparsify and coarsen a graph while preserving its large-scale structure. The limit of infinite edge weight is rarely considered, as many classical notions of graph similarity diverge. However, its algebraic, geometric, and physical interpretations are reflected in the Laplacian pseudoinverse $\mathbf{\mathit{L}}^{\dagger}$, which remains finite in this limit. Motivated by this insight, we provide a probabilistic algorithm that reduces graphs while preserving $\mathbf{\mathit{L}}^{\dagger}$, using an unbiased procedure that minimizes its variance. We compare our algorithm with several existing sparsification and coarsening algorithms using real-world datasets, and demonstrate that it more accurately preserves the large-scale structure.

• The paper unifies spectrum-preserving graph sparsification and coarsening into a single framework using the Laplacian pseudoinverse.
• A new probabilistic algorithm effectively reduces graphs by preserving the Laplacian pseudoinverse, showing superior structural preservation over traditional methods.
• The framework has wide practical applications in large-scale network analysis and potential use in machine learning on graph-structured data.

Insights and Implications of Spectrum-Preserving Graph Sparsification and Coarsening

The paper presents a novel unifying framework for graph reduction through spectrum-preserving sparsification and coarsening techniques. It addresses the need to generate smaller graphs that maintain the global structure while shedding local details. The authors introduce a probabilistic algorithm that leverages Laplacian pseudoinverse preservation to achieve graph reduction efficiently.

Central Contributions

1. Unified Approach: Traditionally, graph sparsification and coarsening are treated independently. Sparsification generally involves edge removal, while coarsening entails node merging. The authors propose a unification by recognizing that edge deletion in a graph corresponds to edge contraction in its dual, leading to a comprehensive framework.
2. Laplacian Pseudoinverse: The paper emphasizes preserving the Laplacian pseudoinverse, denoted as $\lapinv$, a matrix critical in maintaining the structural essence of the graph in terms of its spectrum. The pseudoinverse remains robust even in the limit of infinite edge weights typically associated with node contractions, thus facilitating robust graph reduction.
3. Probabilistic Algorithm: A probabilistic algorithm is devised that enables graph reduction while minimizing changes to the $\lapinv$. This method probabilistically chooses between edge deletion, contraction, or reweighting, guided by constraints that ensure expectation preservation of the pseudoinverse.
4. Comparative Analysis: The paper provides evidence that the proposed algorithm surpasses existing sparsification and coarsening methods concerning large-scale structure preservation, evaluated through synthetic and real-world datasets.

Strong Numerical Results

The implementation of this framework showcases superior preservation of global structures in reduced graphs compared to traditional methods. On various datasets, including social networks and complex mesh simulations, the algorithm demonstrates enhanced fidelity to the original structural properties, especially those evident in eigenvectors associated with sparse graphs' larger scales.

Implications and Future Directions

Theoretical Impact: The integration of sparsification and coarsening within a singular framework opens avenues for more holistic theories in graph reduction processes, facilitating better understanding and manipulation of network structures in theoretical graph studies.

Practical Applications: This framework has extensive applications in fields dealing with large-scale networks, such as computational neuroscience, social network analysis, and infrastructure mapping. Its probabilistic nature and focus on spectrum preservation suggest utility in areas requiring frequent updates and modifications of large network data.

Machine Learning Integration: The methodology could potentially enhance machine learning models relying on graph-structured data. As graphs become integral to machine learning tasks, efficient reduction strategies can significantly improve computational feasibility and model accuracy.

Future Prospects: Considering the strong empirical backing, future studies might focus on extending the scalability of the algorithm for larger networks and improving algorithmic efficiency. Additionally, exploring more profound implications in dynamic, time-evolving graphs could provide insightful advancements.

Conclusion

The paper presents a compelling advancement in graph reduction techniques through the innovative use of the Laplacian pseudoinverse. By blending sparsification and coarsening into a unified process, it not only refines theoretical approaches but also sets a precedent for handling large network data efficiently. The implications are vast, with promising directions in theoretical exploration and practical applications across diverse, graph-heavy domains. The proposed algorithm is a robust tool dedicated to preserving the structural integrity of reduced graph representations, offering significant improvements over traditional methodologies in this space.