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Second-order random walk-based proximity measures in graph analysis: formulations and algorithms

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Abstract

Measuring the proximity between different nodes is a fundamental problem in graph analysis. Random walk-based proximity measures have been shown to be effective and widely used. Most existing random walk measures are based on the first-order Markov model, i.e., they assume that the next step of the random surfer only depends on the current node. However, this assumption neither holds in many real-life applications nor captures the clustering structure in the graph. To address the limitation of the existing first-order measures, in this paper, we study the second-order random walk measures, which take the previously visited node into consideration. While the existing first-order measures are built on node-to-node transition probabilities, in the second-order random walk, we need to consider the edge-to-edge transition probabilities. Using incidence matrices, we develop simple and elegant matrix representations for the second-order proximity measures. A desirable property of the developed measures is that they degenerate to their original first-order forms when the effect of the previous step is zero. We further develop Monte Carlo methods to efficiently compute the second-order measures and provide theoretical performance guarantees. Experimental results show that in a variety of applications, the second-order measures can dramatically improve the performance compared to their first-order counterparts.

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Notes

  1. The entire graph is publicly available at http://webdatacommons.org.

  2. The data are publicly available at http://dblp.uni-trier.de/xml/.

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Acknowledgements

This work was partially supported by the National Science Foundation Grants IIS-11623-74, CAREER, and the NIH Grant R01GM115833.

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Authors and Affiliations

  1. Department of Computer Science, Georgia State University, Atlanta, GA, USA

    Yubao Wu, Zhipeng Cai, Xueting Liao & Fengpan Zhao

  2. College Information Sciences and Technology, The Pennsylvania State University, State College, PA, USA

    Xiang Zhang & Yuchen Bian

  3. Department of Computer Science, Kent State University, Kent, OH, USA

    Xiang Lian

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  1. Yubao Wu
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  2. Xiang Zhang
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  3. Yuchen Bian
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  4. Zhipeng Cai
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Correspondence to Yubao Wu.

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Wu, Y., Zhang, X., Bian, Y. et al. Second-order random walk-based proximity measures in graph analysis: formulations and algorithms. The VLDB Journal 27, 127–152 (2018). https://doi.org/10.1007/s00778-017-0490-5

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