[Paper Review] Graph partitioning and a componentwise PageRank algorithm

Christopher Engström, Sergei Silvestrov|arXiv (Cornell University)|Sep 28, 2016

Data Management and Algorithms6 references3 citations

TL;DR

This paper proposes a novel graph partitioning method that divides directed graphs into strongly connected components (SCCs) and connected acyclic components (CACs), enabling parallel and component-specific PageRank computation. By exploiting the hierarchical structure of the partitioned graph, the method reduces total iterations and accelerates convergence, especially for large graphs and low error tolerances, outperforming standard power iteration in both serial and parallel settings.

ABSTRACT

In this article we will present a graph partitioning algorithm which partitions a graph into two different types of components: the well-known `strongly connected components' as well as another type of components we call `connected acyclic component'. We will give an algorithm based on Tarjan's algorithm for finding strongly connected components used to find such a partitioning. We will also show that the partitioning given by the algorithm is unique and that the underlying graph can be represented as a directed acyclic graph (similar to a pure strongly connected component partitioning). In the second part we will show how such an partitioning of a graph can be used to calculate PageRank of a graph effectively by calculating PageRank for different components on the same `level' in parallel as well as allowing for the use of different types of PageRank algorithms for different types of components. To evaluate the method we have calculated PageRank on four large example graphs and compared it with a basic approach, as well as our algorithm in a serial as well as parallel implementation.

Motivation & Objective

• To develop a graph partitioning algorithm that decomposes directed graphs into strongly connected components (SCCs) and connected acyclic components (CACs) for improved PageRank computation.
• To enable parallel and component-specific PageRank computation by leveraging the hierarchical structure of the partitioned graph.
• To reduce the number of power iterations required for convergence by isolating components with different structural properties.
• To evaluate the performance of the proposed method against standard PageRank using large real-world and synthetic graphs.
• To explore the potential of the method for memory-constrained environments and incremental updates.

Proposed method

• Uses a modified Tarjan’s algorithm to identify strongly connected components (SCCs) and derive connected acyclic components (CACs) from the condensation of the graph.
• Assigns each component a level based on its position in the directed acyclic graph (DAG) of components, enabling hierarchical processing from top to bottom.
• Applies a CAC-PageRank algorithm that computes PageRank in a single iteration per level due to the acyclic nature of CACs.
• Applies an SCC-PageRank algorithm that uses standard power iteration within each SCC, exploiting their strong connectivity.
• Combines results from CAC and SCC components using a level-wise propagation model, ensuring consistency across the graph.
• Implements both serial and parallel versions of the algorithm, with parallelization applied across components at the same level.

Experimental results

Research questions

• RQ1 Can a graph partitioning strategy based on SCCs and CACs reduce the number of power iterations required for PageRank convergence?
• RQ2 How does the performance of the componentwise PageRank algorithm compare to standard power iteration in terms of runtime and convergence speed?
• RQ3 To what extent can parallelization across components at the same level improve performance, especially for large graphs?
• RQ4 How does the method scale with decreasing error tolerance, and does it maintain efficiency at high precision?
• RQ5 Can the component-specific approach enable faster updates to PageRank after graph modifications?

Key findings

• The proposed partitioning algorithm produces a unique decomposition of the graph into SCCs and CACs, forming a directed acyclic graph (DAG) of components.
• The method significantly reduces the number of power iterations required for convergence, especially in large graphs with high connectivity.
• For the Barabási-Albert model graph, PageRank computation time remained nearly constant across different error tolerances, indicating robustness and efficiency.
• The serial implementation outperformed the basic power iteration method as graph size increased and error tolerance decreased.
• The parallel implementation showed higher overhead than expected, suggesting room for optimization in low-level memory and task management.
• The component-specific approach enabled faster convergence on graphs with few SCCs, such as scale-free networks, due to reduced iteration counts in CACs.