A Local Clustering Algorithm for Massive Graphs and its Application to Nearly-Linear Time Graph Partitioning (0809.3232v1)

Abstract: We study the design of local algorithms for massive graphs. A local algorithm is one that finds a solution containing or near a given vertex without looking at the whole graph. We present a local clustering algorithm. Our algorithm finds a good cluster--a subset of vertices whose internal connections are significantly richer than its external connections--near a given vertex. The running time of our algorithm, when it finds a non-empty local cluster, is nearly linear in the size of the cluster it outputs. Our clustering algorithm could be a useful primitive for handling massive graphs, such as social networks and web-graphs. As an application of this clustering algorithm, we present a partitioning algorithm that finds an approximate sparsest cut with nearly optimal balance. Our algorithm takes time nearly linear in the number edges of the graph. Using the partitioning algorithm of this paper, we have designed a nearly-linear time algorithm for constructing spectral sparsifiers of graphs, which we in turn use in a nearly-linear time algorithm for solving linear systems in symmetric, diagonally-dominant matrices. The linear system solver also leads to a nearly linear-time algorithm for approximating the second-smallest eigenvalue and corresponding eigenvector of the Laplacian matrix of a graph. These other results are presented in two companion papers.

• The paper's main contribution is the Nibble algorithm, which uses truncated random walks to identify high-quality clusters in massive graphs.
• It achieves nearly-linear time performance by exploring only a fraction of the graph, ensuring scalability for large-scale network analysis.
• The methodology provides robust probabilistic guarantees for low-conductance cuts while paving the way for efficient graph partitioning techniques.

Analysis of "A Local Clustering Algorithm for Massive Graphs and its Application to Nearly-Linear Time Graph Partitioning"

The paper by Daniel A. Spielman and Shang-Hua Teng presents a significant contribution to the field of graph algorithms, specifically focusing on local clustering and its implications for graph partitioning. Central to their discussion is a novel local clustering algorithm, which they introduce as a method capable of effectively finding clusters around a specified vertex in large-scale graphs through a primarily local exploration.

Local Clustering Algorithm

The proposed local clustering algorithm is distinguished by its capacity to determine a "good" cluster—a subset of vertices heavily interconnected internally compared to their external connections—while examining only a fraction of the overall graph. A key innovation in this method is the use of random walks, specifically leveraging the concept of truncated random walks to maintain computational efficiency and ensure the operation is sensitive to output size. The algorithm is designed to run in time proportional to the size of the output cluster, thus making it particularly scalable for massive graphs encountered in applications like social network analysis and web graph processing.

The authors introduce the "Nibble" algorithm as the core process within this framework. This algorithm, with a careful set of stopping criteria, ensures the generation of clusters with provably low conductance—a measure of the cluster's quality in terms of its edge-cut balance. A significant theoretical result provided is that for sufficiently small target conductance and clusters, "Nibble" can successfully identify a subset within the expected quality bounds with high probability.

Nearly Linear-Time Graph Partitioning

Building on this local clustering mechanism, Spielman and Teng proceed to demonstrate how it can be applied to create a graph partitioning algorithm that operates in nearly linear time. This partitioning approach strives to identify approximate sparsest cuts with near-optimal balance. The ability to perform such partitioning tasks in nearly-linear time addresses the critical need for efficiency as graph sizes continue to expand rapidly, particularly in scenarios like circuit simulation and web data processing, where traditional O(n^1.5) algorithms are no longer viable.

The significance of the partitioning algorithm lies in its probabilistic guarantee to either find a well-balanced cut or decompose the graph in a manner that touches a significant portion of any preexisting low-conductance subgraph. This is achieved by systematically applying the "Random Nibble" procedure to randomly selected vertices of the graph. The expected running time remains low due to the careful control over which parts of the graph are explored, relying heavily on the properties of the initial Nibble algorithm.

Theoretical and Practical Implications

From a theoretical standpoint, the results presented in this paper contribute to the foundational understanding of graph decomposition via local processes, challenging traditional global approaches that often suffer from scalability issues. Practically, the techniques discussed open avenues for efficient processing of large graphs in various domains, improving upon existing methods in terms of both speed and quality of output.

Looking forward, the insights and tools developed in this work could influence future algorithmic strategies in the sphere of approximation algorithms for partitioning and beyond. They underline a paradigm where local approximations play a pivotal role in solving global optimization problems in large-scale graphs. This work not only enriches the graph theory algorithmic toolkit but also sets the stage for further refinement and application of local clustering concepts to diverse computational contexts in the field of big data and network analysis.