[Paper Review] Efficient Truss Maintenance in Evolving Networks

Rui Zhou, Chengfei Liu+3 more|arXiv (Cornell University)|Feb 12, 2014

Manufacturing Process and Optimization8 references16 citations

TL;DR

This paper proposes efficient algorithms for maintaining maximal k-trusses in evolving networks subject to dynamic edge insertions and deletions. By leveraging edge-triangle relationships and incremental update techniques, the method achieves O(|El|) time complexity per update, where |El| is the number of affected edges, significantly outperforming batch recomputation for moderate update frequencies on real-world datasets.

ABSTRACT

Truss was proposed to study social network data represented by graphs. A k-truss of a graph is a cohesive subgraph, in which each edge is contained in at least k-2 triangles within the subgraph. While truss has been demonstrated as superior to model the close relationship in social networks and efficient algorithms for finding trusses have been extensively studied, very little attention has been paid to truss maintenance. However, most social networks are evolving networks. It may be infeasible to recompute trusses from scratch from time to time in order to find the up-to-date $k$-trusses in the evolving networks. In this paper, we discuss how to maintain trusses in a graph with dynamic updates. We first discuss a set of properties on maintaining trusses, then propose algorithms on maintaining trusses on edge deletions and insertions, finally, we discuss truss index maintenance. We test the proposed techniques on real datasets. The experiment results show the promise of our work.

Motivation & Objective

• Address the challenge of maintaining k-trusses in real-time evolving networks where frequent edge updates make batch recomputation infeasible.
• Develop incremental algorithms for edge insertions and deletions that efficiently propagate truss number updates only to affected edges.
• Introduce and maintain k-truss indexes to accelerate query response times for dynamic truss queries.
• Evaluate the performance of incremental maintenance against batch recomputation and index-based approaches on real-world social networks.
• Demonstrate that incremental maintenance is more efficient than batch processing for moderate update frequencies, especially in decentralized networks.

Proposed method

• Propose a set of theoretical properties governing how truss numbers change upon edge insertions and deletions.
• Design an incremental algorithm that propagates truss number updates only to edges in the affected neighborhood, minimizing unnecessary computation.
• Utilize triangle counting to determine edge participation in k-trusses, focusing on local changes rather than global recomputation.
• Introduce a k-truss index structure to store and efficiently query truss numbers, enabling fast lookups during dynamic updates.
• Maintain the k-truss index incrementally by updating only the affected portions after each edge change.
• Implement progressive and indexed update strategies that avoid full truss decomposition after each update, reducing time overhead.

Experimental results

Research questions

• RQ1 How can k-truss maintenance be efficiently performed in response to dynamic edge insertions and deletions in evolving networks?
• RQ2 What are the key properties governing the propagation of truss number changes after a single edge update?
• RQ3 How does incremental maintenance compare to batch recomputation in terms of performance for moderate update frequencies?
• RQ4 What is the impact of network structure (e.g., clustering coefficient, centrality) on the efficiency of incremental truss maintenance?
• RQ5 How can k-truss indexes be built and maintained efficiently to support fast query evaluation in dynamic environments?

Key findings

• The proposed incremental maintenance algorithms achieve O(|El|) time complexity per update, where |El| is the number of edges whose truss numbers are affected.
• For the Slashdot dataset (lower clustering, more decentralized), progressiveUpdate and indexedUpdate outperform batchUpdate up to 16,000 updates.
• For the Epinions dataset (higher clustering, more centralized), the incremental methods remain superior up to approximately 10,000 updates.
• The Enron email dataset, being smaller and more centralized, shows that incremental methods are effective only for fewer than 10,000 updates, after which batch processing becomes more efficient.
• Truss indexing significantly improves query performance over non-indexed approaches, especially under moderate update frequencies.
• The incremental approach is most effective in decentralized networks with lower clustering coefficients, where updates affect fewer nodes and edges.