## Abstract

The number of triangles is a fundamental metric for analyzing the structure and function of a network. In this paper, for the first time, we investigate the triangle minimization problem in a network under edge (node) attack, where the attacker aims to minimize the number of triangles in the network by removing $$k$$k edges (nodes). We show that the triangle minimization problem under edge (node) attack is a submodular function maximization problem, which can be solved efficiently. Specifically, we propose a degree-based edge (node) removal algorithm and a near-optimal greedy edge (node) removal algorithm for approximately solving the triangle minimization problem under edge (node) attack. In addition, we introduce two pruning strategies and an approximate marginal gain evaluation technique to further speed up the greedy edge (node) removal algorithm. We conduct extensive experiments over 12 real-world datasets to evaluate the proposed algorithms, and the results demonstrate the effectiveness, efficiency and scalability of our algorithms.

Original language | English |
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Pages (from-to) | 617-643 |

Number of pages | 27 |

Journal | Knowledge and Information Systems |

Volume | 45 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Dec 2015 |

Externally published | Yes |

## Keywords

- FM sketch
- Large networks
- Submodular function
- Triangle minimization