Efficient Enumeration of Large Maximal k-Plexes in Large Graphs

Core Concepts

This paper proposes an efficient branch-and-bound algorithm and its parallel version to enumerate all maximal k-plexes with at least q vertices in large graphs.

Abstract

The paper addresses the problem of efficiently enumerating all maximal k-plexes (a relaxed clique model) with at least q vertices in large graphs. The key contributions are: A method for search space partitioning to create independent searching tasks, with a time complexity analysis. A new pivot vertex selection approach that maximizes the number of "saturated" vertices in the k-plex, effectively reducing the candidate vertex set. An effective upper-bounding technique to prune useless search branches. Three novel pruning techniques by vertex pairs to further reduce the search space. A task-based parallel computing approach with a timeout mechanism to eliminate straggler tasks and maximize cache locality while ensuring load balancing. The proposed sequential and parallel algorithms are shown to outperform the state-of-the-art solutions, with up to 5x and 18.9x speedups respectively. Ablation studies demonstrate the effectiveness of the pruning techniques, bringing up to 7x speedup.

Stats

The maximum degree in the graph is denoted as Δ. The degeneracy of the graph is denoted as D, which is typically much smaller than the number of vertices n. The time complexity of the proposed algorithm is O(nrk1r2γDk), where r1 = min(DΔ/(q-2k+2), n), r2 = min(DΔ2/(q-2k+2), nD), and γk < 2 is a constant close to 2.

Quotes

"Finding cohesive subgraphs in a large graph has many important applications, such as community detection and biological network analysis." "As a relaxed clique model, k-plex was first introduced in [21], which is a graph where every vertex is adjacent to all but at most k vertices."

Key Insights Distilled From

Efficient Enumeration of Large Maximal k-Plexes

by Qihao Cheng,... at arxiv.org 05-07-2024

https://arxiv.org/pdf/2402.13008.pdf

Deeper Inquiries

How can the proposed techniques be extended to handle dynamic graphs where the graph structure changes over time

To extend the proposed techniques to handle dynamic graphs where the graph structure changes over time, we can implement incremental updates to the algorithm. Instead of recomputing the entire k-plex enumeration from scratch whenever the graph changes, we can track the changes in the graph structure and apply incremental updates to the existing k-plex enumeration. This can involve updating the candidate sets, reevaluating the upper bounds, and adjusting the pruning techniques based on the changes in the graph. By efficiently incorporating these incremental updates, the algorithm can adapt to dynamic graph changes without the need for complete recomputation.

What are the potential applications of the efficient k-plex enumeration algorithm beyond community detection and biological network analysis

The efficient k-plex enumeration algorithm has potential applications beyond community detection and biological network analysis. Some of the additional applications include: Social Network Analysis: Identifying cohesive subgroups in social networks can help in understanding user behavior, influence patterns, and community dynamics. Fraud Detection: Detecting fraudulent activities in financial transactions or online platforms by identifying groups of colluding entities. Telecommunications: Analyzing call detail records to identify groups of users with strong connections for targeted marketing or network optimization. Supply Chain Management: Identifying clusters of interconnected entities in the supply chain network to optimize logistics and improve efficiency. Traffic Flow Optimization: Analyzing traffic flow data to identify clusters of congested areas and optimize traffic management strategies. By applying the k-plex enumeration algorithm in these diverse domains, valuable insights can be gained to improve decision-making processes and optimize system performance.

How can the algorithm be further optimized to handle extremely large graphs that do not fit in the main memory

To optimize the algorithm for handling extremely large graphs that do not fit in the main memory, several strategies can be implemented: External Memory Algorithms: Implementing disk-based data structures and algorithms to process data in chunks from disk storage, reducing the memory footprint. Parallel Processing: Utilizing distributed computing frameworks like Apache Spark or Hadoop to distribute the computation across multiple nodes, enabling processing of large graphs in a distributed manner. Graph Partitioning: Dividing the large graph into smaller subgraphs and processing them independently, then merging the results to obtain the final output. Sampling Techniques: Employing sampling methods to work with a subset of the graph data, providing approximate results for analysis while reducing memory requirements. Streaming Algorithms: Implementing streaming algorithms that process data in a continuous stream, enabling real-time analysis of large graph data without the need to store the entire graph in memory. By combining these optimization techniques, the algorithm can efficiently handle extremely large graphs that exceed the main memory capacity, enabling scalable and effective analysis of complex network structures.