Graph Sparsification by Effective Resistances (0803.0929v4)

Abstract: We present a nearly-linear time algorithm that produces high-quality sparsifiers of weighted graphs. Given as input a weighted graph $G=(V,E,w)$ and a parameter $\epsilon>0$, we produce a weighted subgraph $H=(V,\tilde{E},\tilde{w})$ of $G$ such that $|\tilde{E}|=O(n\log n/\epsilon^2)$ and for all vectors $x\in\R^V$ $(1-\epsilon)\sum_{uv\in E}(x(u)-x(v))^2w_{uv}\le \sum_{uv\in\tilde{E}}(x(u)-x(v))^2\tilde{w}_{uv} \le (1+\epsilon)\sum_{uv\in E}(x(u)-x(v))^2w_{uv}. ()$ This improves upon the sparsifiers constructed by Spielman and Teng, which had $O(n\log^c n)$ edges for some large constant $c$, and upon those of Bencz\'ur and Karger, which only satisfied () for $x\in{0,1}^V$. A key ingredient in our algorithm is a subroutine of independent interest: a nearly-linear time algorithm that builds a data structure from which we can query the approximate effective resistance between any two vertices in a graph in $O(\log n)$ time.

• The paper presents a nearly-linear time algorithm that constructs spectral sparsifiers with O(n log n/ε²) edges while preserving the quadratic form of the graph Laplacian.
• It employs effective resistance to define edge sampling probabilities, significantly reducing computational complexity for graph processing.
• The findings offer practical benefits for numerical simulations, graph partitioning, and solving linear systems in large-scale network analysis.

Overview of "Graph Sparsification by Effective Resistances" by Daniel A. Spielman and Nikhil Srivastava

Introduction

Graph sparsification is a technique used to approximate a dense graph $G$ with a sparser graph $H$ on the same set of vertices, while preserving certain properties of $G$. This enables more efficient computations on the graph $H$ without significantly compromising accuracy. The concept was initially motivated by the notion of cut sparsification introduced by Benczur and Karger, which targets the approximation of cut properties. Spielman and Teng extended this to spectral sparsification, which addresses a broader range of properties connected to the Laplacian matrix of the graph.

Contributions

The paper by Spielman and Srivastava presents a nearly-linear time algorithm that constructs high-quality spectral sparsifiers. Given a weighted graph $G = (V, E, w)$ and a parameter $\epsilon > 0$, the algorithm produces a weighted subgraph $H = (V, \tilde{E}, \tilde{w})$ containing $O(n \log n / \epsilon^2)$ edges. This is a significant improvement over previous approaches, particularly those by Spielman and Teng, which had more edges.

Theoretical Foundations

The main result guarantees that the sparsified graph $H$ preserves the quadratic form of the Laplacian of $G$ within a factor of $1 \pm \epsilon$. More formally, for all vectors $x \in \mathbb{R}^V$,

$$(1 - \epsilon) x^T L x \le x^T \tilde{L} x \le (1 + \epsilon) x^T L x,$$

where $L$ and $\tilde{L}$ are the Laplacians of $G$ and $H$, respectively.

Algorithm and Key Techniques

The algorithm employs a novel approach based on the concept of effective resistance. An edge's effective resistance is proportional to its likelihood of appearing in a random spanning tree and is related to the commute time between its endpoints. This effective resistance serves as the basis for the sampling probabilities used to construct the sparsifier.

A key component of the method is the efficient approximation of effective resistances. The authors present a subroutine that builds a data structure in nearly-linear time, enabling quick queries of approximate effective resistances between any two vertices.

Main Theorem and Proof

The core theorem posits that including each edge $e \in E$ of $G$ in $H$ with a probability proportional to its effective resistance $R_e$ ensures that the resulting subgraph $H$ has the desired sparsity and approximates the original graph $G$ spectrally. The proof utilizes properties of the incidence matrix, the Laplacian, and their pseudoinverses, combined with concentration inequalities and the Johnson-Lindenstrauss Lemma for dimensionality reduction.

Numerical Results

The algorithm's performance is quantified by the number of edges $O(n \log n / \epsilon^2)$ in the sparsifier, significantly lower than earlier results with larger constant factors. The paper provides empirical evidence supporting the theoretical guarantees.

Implications

1. Practical Applications: Graph sparsification aids in various computational tasks, such as solving linear systems, graph partitioning, and simulation of network flows. The reduced computational complexity and storage requirements are beneficial for large-scale data analysis.
2. Theoretical Significance: The results contribute to the broader understanding of graph spectral properties and pave the way for future research on efficient graph algorithms.

Future Work

The discussion concludes by highlighting potential future developments:

• Exploration of adaptive sparsification techniques.
• Extension to other forms of graph approximations.
• Applications in dynamic graph settings where the graph structure evolves over time.

Conclusion

The paper by Spielman and Srivastava on graph sparsification via effective resistances offers a robust and efficient method for constructing spectral sparsifiers. This work marks an important advance in the field, providing both theoretical insights and practical tools for handling large graph datasets. The approach's simplicity and efficiency open new avenues for further research and application in various domains of computer science and applied mathematics.