### Abstract

The Sylvester equation offers a powerful and unifying primitive for a variety of important graph mining tasks, including network alignment, graph kernel, node similarity, subgraph matching, etc. A major bottleneck of Sylvester equation lies in its high computational complexity. Despite tremendous effort, state-of-the-art methods still require a complexity that is at least quadratic in the number of nodes of graphs, even with approximations. In this paper, we propose a family of Krylov subspace based algorithms (FASTEN) to speed up and scale up the computation of Sylvester equation for graph mining. The key idea of the proposed methods is to project the original equivalent linear system onto a Kronecker Krylov subspace. We further exploit (1) the implicit representation of the solution matrix as well as the associated computation, and (2) the decomposition of the original Sylvester equation into a set of inter-correlated Sylvester equations of smaller size. The proposed algorithms bear two distinctive features. First, they provide the exact solutions without any approximation error. Second, they significantly reduce the time and space complexity for solving Sylvester equation, with two of the proposed algorithms having a linear complexity in both time and space. Experimental evaluations on a diverse set of real networks, demonstrate that our methods (1) are up to 10, 000× faster against Conjugate Gradient method, the best known competitor that outputs the exact solution, and (2) scale up to million-node graphs.

Original language | English (US) |
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Title of host publication | KDD 2018 - Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining |

Publisher | Association for Computing Machinery |

Pages | 1339-1347 |

Number of pages | 9 |

ISBN (Print) | 9781450355520 |

DOIs | |

State | Published - Jul 19 2018 |

Event | 24th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD 2018 - London, United Kingdom Duration: Aug 19 2018 → Aug 23 2018 |

### Other

Other | 24th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD 2018 |
---|---|

Country | United Kingdom |

City | London |

Period | 8/19/18 → 8/23/18 |

### Fingerprint

### ASJC Scopus subject areas

- Software
- Information Systems

### Cite this

*KDD 2018 - Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining*(pp. 1339-1347). Association for Computing Machinery. https://doi.org/10.1145/3219819.3220002

**FASTEN : Fast sylvester equation solver for graph mining.** / Du, Boxin; Tong, Hanghang.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*KDD 2018 - Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining.*Association for Computing Machinery, pp. 1339-1347, 24th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD 2018, London, United Kingdom, 8/19/18. https://doi.org/10.1145/3219819.3220002

}

TY - GEN

T1 - FASTEN

T2 - Fast sylvester equation solver for graph mining

AU - Du, Boxin

AU - Tong, Hanghang

PY - 2018/7/19

Y1 - 2018/7/19

N2 - The Sylvester equation offers a powerful and unifying primitive for a variety of important graph mining tasks, including network alignment, graph kernel, node similarity, subgraph matching, etc. A major bottleneck of Sylvester equation lies in its high computational complexity. Despite tremendous effort, state-of-the-art methods still require a complexity that is at least quadratic in the number of nodes of graphs, even with approximations. In this paper, we propose a family of Krylov subspace based algorithms (FASTEN) to speed up and scale up the computation of Sylvester equation for graph mining. The key idea of the proposed methods is to project the original equivalent linear system onto a Kronecker Krylov subspace. We further exploit (1) the implicit representation of the solution matrix as well as the associated computation, and (2) the decomposition of the original Sylvester equation into a set of inter-correlated Sylvester equations of smaller size. The proposed algorithms bear two distinctive features. First, they provide the exact solutions without any approximation error. Second, they significantly reduce the time and space complexity for solving Sylvester equation, with two of the proposed algorithms having a linear complexity in both time and space. Experimental evaluations on a diverse set of real networks, demonstrate that our methods (1) are up to 10, 000× faster against Conjugate Gradient method, the best known competitor that outputs the exact solution, and (2) scale up to million-node graphs.

AB - The Sylvester equation offers a powerful and unifying primitive for a variety of important graph mining tasks, including network alignment, graph kernel, node similarity, subgraph matching, etc. A major bottleneck of Sylvester equation lies in its high computational complexity. Despite tremendous effort, state-of-the-art methods still require a complexity that is at least quadratic in the number of nodes of graphs, even with approximations. In this paper, we propose a family of Krylov subspace based algorithms (FASTEN) to speed up and scale up the computation of Sylvester equation for graph mining. The key idea of the proposed methods is to project the original equivalent linear system onto a Kronecker Krylov subspace. We further exploit (1) the implicit representation of the solution matrix as well as the associated computation, and (2) the decomposition of the original Sylvester equation into a set of inter-correlated Sylvester equations of smaller size. The proposed algorithms bear two distinctive features. First, they provide the exact solutions without any approximation error. Second, they significantly reduce the time and space complexity for solving Sylvester equation, with two of the proposed algorithms having a linear complexity in both time and space. Experimental evaluations on a diverse set of real networks, demonstrate that our methods (1) are up to 10, 000× faster against Conjugate Gradient method, the best known competitor that outputs the exact solution, and (2) scale up to million-node graphs.

UR - http://www.scopus.com/inward/record.url?scp=85051469136&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85051469136&partnerID=8YFLogxK

U2 - 10.1145/3219819.3220002

DO - 10.1145/3219819.3220002

M3 - Conference contribution

SN - 9781450355520

SP - 1339

EP - 1347

BT - KDD 2018 - Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining

PB - Association for Computing Machinery

ER -