### 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.

Original language | English (US) |
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Pages (from-to) | 13-24 |

Number of pages | 12 |

Journal | Proceedings of the VLDB Endowment |

Volume | 10 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2016 |

Event | 43rd International Conference on Very Large Data Bases, VLDB 2017 - Munich, Germany Duration: Aug 28 2017 → Sep 1 2017 |

### All Science Journal Classification (ASJC) codes

- Computer Science (miscellaneous)
- Computer Science(all)

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## Cite this

*Proceedings of the VLDB Endowment*,

*10*(1), 13-24. https://doi.org/10.14778/3015270.3015272