## Abstract

In this paper we study a problem of determining when entities are active based on their interactions with each other. More formally, we consider a set of entities V and a sequence of time-stamped edges E among the entities. Each edge (u,v,t)∈E denotes an interaction between entities u and v that takes place at time t. We view this input as a temporal network. We then assume a simple activity model in which each entity is active during a short time interval. An interaction (u, v, t) can be explained if at least one of u or v are active at time t. Our goal is to reconstruct the activity intervals, for all entities in the network, so as to explain the observed interactions. This problem, which we refer to as the network-untangling problem, can be applied to discover timelines of events from complex interactions among entities.

We provide two formulations for the network-untangling problem: (i) minimizing the total interval length over all entities, and (ii) minimizing the maximum interval length. We show that the sum problem is NP-hard, while, surprisingly, the max problem can be solved optimally in linear time, using a mapping to 2-SAT. For the sum problem we provide efficient and effective algorithms based on realistic assumptions. Furthermore, we complement our study with an evaluation on synthetic and real-world datasets, which demonstrates the validity of our concepts and the good performance of our algorithms.

We provide two formulations for the network-untangling problem: (i) minimizing the total interval length over all entities, and (ii) minimizing the maximum interval length. We show that the sum problem is NP-hard, while, surprisingly, the max problem can be solved optimally in linear time, using a mapping to 2-SAT. For the sum problem we provide efficient and effective algorithms based on realistic assumptions. Furthermore, we complement our study with an evaluation on synthetic and real-world datasets, which demonstrates the validity of our concepts and the good performance of our algorithms.

Original language | English |
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Title of host publication | Machine Learning and Knowledge Discovery in Databases |

Subtitle of host publication | European Conference, ECML PKDD 2017, Skopje, Macedonia, September 18–22, 2017, Proceedings, Part I |

Editors | Michelangelo Ceci, Jaakko Hollmén, Ljupčo Todorovski, Celine Vens, Sašo Džeroski |

Place of Publication | Cham |

Pages | 701-716 |

Number of pages | 16 |

Volume | 1 |

ISBN (Electronic) | 978-3-319-71249-9 |

DOIs | |

Publication status | Published - 2017 |

MoE publication type | A3 Part of a book or another research book |

Event | European Conference on Principles and Practice of Knowledge Discovery in Databases - Croke Park Conference Centre, Skopje, Macedonia, The Former Yugoslav Republic of Duration: 18 Sep 2017 → 22 Sep 2017 Conference number: 10 |

### Publication series

Name | Lecture Notes in Computer Science |
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Publisher | Springer |

Volume | 10534 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | European Conference on Principles and Practice of Knowledge Discovery in Databases |
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Abbreviated title | ECML PKDD |

Country/Territory | Macedonia, The Former Yugoslav Republic of |

City | Skopje |

Period | 18/09/2017 → 22/09/2017 |

## Keywords

- Temporal networks
- Complex networks
- Timeline reconstruction
- Vertex cover
- Linear programming
- 2-SAT