Detect overlapping and hierarchical community structure in networks (0810.3093v3)

Abstract: Clustering and community structure is crucial for many network systems and the related dynamic processes. It has been shown that communities are usually overlapping and hierarchical. However, previous methods investigate these two properties of community structure separately. This paper proposes an algorithm (EAGLE) to detect both the overlapping and hierarchical properties of complex community structure together. This algorithm deals with the set of maximal cliques and adopts an agglomerative framework. The quality function of modularity is extended to evaluate the goodness of a cover. The examples of application to real world networks give excellent results.

• The paper introduces the EAGLE algorithm as a novel method to simultaneously detect overlapping and hierarchical communities using maximal cliques.
• It employs an agglomerative process with an extended modularity function to iteratively merge communities based on similarity.
• Applications to word association and scientific collaboration networks demonstrate that EAGLE outperforms traditional methods in revealing complex community structures.

Detecting Overlapping and Hierarchical Community Structure in Networks: An Analysis of EAGLE Algorithm

The paper, Detect Overlapping and Hierarchical Community Structure in Networks by Huawei Shen, Xueqi Cheng, Kai Cai, and Mao-Bin Hu, addresses the complex issue of community detection in networks, proposing the EAGLE algorithm as a novel solution to simultaneously uncover both overlapping and hierarchical community structures.

Introduction

The significance of detecting community structure within networks is well-known, influencing the understanding of various network systems, including the internet, biological, and social networks. Traditional methods for community detection generally fall into two categories: partition-based methods, which assign each node to exactly one community, and overlapping methods, where nodes can belong to multiple communities. What distinguishes EAGLE is its ability to integrate both hierarchies and overlaps into the community detection process.

Algorithmic Details

EAGLE (agglomerative framework for clique-percolation) leverages the concept of maximal cliques to initiate the detection process. This algorithm differs from conventional agglomerative approaches by treating maximal cliques, rather than individual vertices, as the initial communities.

The algorithm proceeds as follows:

1. Initialization:
   • Maximal Cliques: Identify all maximal cliques using the Bron-Kerbosch algorithm. Subordinate maximal cliques, i.e., those which are subsets of larger cliques, are discarded by setting a threshold k for clique size.
2. Agglomerative Process:
   • Similarity Calculation: Compute community similarity based on a modified modularity function.
   • Community Merging: Iteratively merge the most similar pairs of communities, recalculating similarity at each stage until a single community remains.
3. Hierarchical Detection:
   • The dendrogram is analyzed to determine the optimal cut, yielding the most significant community partition (best cover) by maximizing an extended modularity EQ.

Computational Complexity

The initial calculation of similarities between pairs of communities involves $O(n^2)$ operations. Each step of merging, requiring recalculations, takes $O(n \times s)$ operations, where $n$ is the number of vertices and $s$ is the number of maximal cliques. The algorithm acknowledges the non-polynomial complexity of finding maximal cliques, but mitigates this by the typical sparseness of real-world networks.

Application and Results

The EAGLE algorithm is applied to two networks: a word association network and a scientific collaboration network.

1. Word Association Network:
   • Primary Communities: Initial application reveals 17 significant overlapping communities. Further granularity shows sub-communities with semantic relevance, suggesting EAGLE's capability to discern meaningfully coherent subgroups within larger communities.
2. Scientific Collaboration Network:
   • Primary Communities: EAGLE detects 1754 communities with a high modularity score $EQ \approx 0.85$. The hierarchical structure corresponds well to subfields within the scientific community, correlating with authors' regional affiliations or research specialties.
   • Comparison with Established Methods: The Newman fast algorithm and k-clique algorithm, while useful in their domains, fall short of EAGLE in detecting overlapping structures and providing hierarchical insights. Unlike partition-based methods, EAGLE reveals overlapping nodes and comprehensive context, such as the prominent role of certain researchers across multiple fields.

Implications and Future Work

The practical implications of the EAGLE algorithm are significant for real-world networks where community structures are not strictly partitioned but inherently hierarchical and overlapping. This dual recognition is critical for fields like sociology (e.g., understanding social groups), biology (e.g., protein interaction networks), and technology (e.g., distributed systems).

Future work will focus on extending EAGLE to handle weighted and directed networks and improving its computational efficiency. Considering the high computational cost, optimizing the algorithm for large-scale networks is imperative. Furthermore, enhancing EAGLE's parallelizability could significantly impact its widespread application.

Conclusion

EAGLE stands as a robust method for detecting the nuanced structure of communities within networks, acknowledging the inherent overlaps and hierarchies present in complex systems. It provides a comprehensive approach, offering both theoretical advances in community detection algorithms and practical tools for real-world network analysis. The detailed insights into community structures open avenues for deeper research into the dynamics of complex networks.