[Paper Review] Enumerating k-Vertex Connected Components in Large Graphs

Dong Wen, Lu Qin+3 more|arXiv (Cornell University)|Mar 25, 2017

Complex Network Analysis Techniques33 references4 citations

TL;DR

This paper proposes a polynomial-time algorithm to enumerate all k-vertex connected components (k-VCCs) in large graphs, using recursive graph partitioning and optimized connectivity testing via neighbor sweep and group sweep strategies. The method effectively eliminates the free-rider effect by ensuring subgraphs remain connected after removal of any k−1 vertices, while maintaining bounded diameter, high cohesiveness, and low overlapping, with experiments showing high efficiency on seven real-world datasets.

ABSTRACT

Cohesive subgraph detection is an important graph problem that is widely applied in many application domains, such as social community detection, network visualization, and network topology analysis. Most of existing cohesive subgraph metrics can guarantee good structural properties but may cause the free-rider effect. Here, by free-rider effect, we mean that some irrelevant subgraphs are combined as one subgraph if they only share a small number of vertices and edges. In this paper, we study k-vertex connected component (k-VCC) which can effectively eliminate the free-rider effect but less studied in the literature. A k-VCC is a connected subgraph in which the removal of any k-1 vertices will not disconnect the subgraph. In addition to eliminating the free-rider effect, k-VCC also has other advantages such as bounded diameter, high cohesiveness, bounded graph overlapping, and bounded subgraph number. We propose a polynomial time algorithm to enumerate all k-VCCs of a graph by recursively partitioning the graph into overlapped subgraphs. We find that the key to improving the algorithm is reducing the number of local connectivity testings. Therefore, we propose two effective optimization strategies, namely neighbor sweep and group sweep, to largely reduce the number of local connectivity testings. We conduct extensive performance studies using seven large real datasets to demonstrate the effectiveness of this model as well as the efficiency of our proposed algorithms.

Motivation & Objective

• To address the free-rider effect in cohesive subgraph detection, where loosely connected subgraphs are merged into a single component.
• To develop an efficient, polynomial-time algorithm for enumerating all k-vertex connected components (k-VCCs) in large graphs.
• To ensure structural properties such as bounded diameter, high cohesiveness, and limited subgraph overlapping in detected components.
• To reduce the number of expensive local connectivity tests through novel optimization strategies in the enumeration pipeline.

Proposed method

• The algorithm recursively partitions the graph into overlapping subgraphs to identify maximal k-vertex connected components.
• It employs a neighbor sweep strategy to minimize redundant connectivity tests by focusing on vertex neighborhoods.
• A group sweep optimization further reduces testing by processing vertices in batches based on shared connectivity patterns.
• The method leverages vertex connectivity as a formal measure of cohesion, ensuring that k-VCCs remain connected after removal of any k−1 vertices.
• The algorithm operates in polynomial time and is designed to scale efficiently on large real-world graphs.
• Connectivity testing is the performance bottleneck, and optimization strategies are specifically targeted at reducing its frequency.

Experimental results

Research questions

• RQ1 Can k-vertex connected components effectively eliminate the free-rider effect in cohesive subgraph detection?
• RQ2 How can all k-vertex connected components be enumerated efficiently in large graphs?
• RQ3 What optimization strategies can significantly reduce the number of local connectivity tests in k-VCC enumeration?
• RQ4 How do k-VCCs compare to existing models in terms of structural properties like diameter, cohesiveness, and overlapping?
• RQ5 What is the scalability and performance of the proposed algorithm on real-world large-scale graphs?

Key findings

• The k-VCC model effectively eliminates the free-rider effect by requiring that no k−1 vertices can disconnect the component, ensuring only strongly cohesive subgraphs are identified.
• Each k-VCC has a bounded diameter, with the maximum diameter of a k-VCC being at most ⌊(|V′|−2)/κ(G′)⌋+1, where κ(G′) is the vertex connectivity of the component.
• k-VCCs are nested within both k-cores and k-edge connected components, ensuring high cohesiveness and structural robustness.
• The proposed neighbor sweep and group sweep optimizations significantly reduce the number of local connectivity tests, improving algorithm efficiency.
• Experiments on seven large real-world datasets demonstrate the algorithm's high efficiency and scalability, with strong performance in detecting meaningful, non-overlapping cohesive subgraphs.
• The method achieves polynomial-time enumeration of all k-VCCs, making it practical for large-scale graph analysis.