The network-untangling problem: from interactions to activity timelines
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The network-untangling problem: from interactions to activity timelines Polina Rozenshtein1,2
· Nikolaj Tatti3,4 · Aristides Gionis2,5
Received: 26 October 2018 / Accepted: 16 September 2020 © The Author(s), under exclusive licence to Springer Science+Business Media LLC, part of Springer Nature 2020
Abstract In this paper we study a problem of determining when entities are active based on their interactions with each other. 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 at time t. We assume an activity model where each entity is active during at most k time intervals. 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, the network-untangling problem, can be applied to discover event timelines from complex entity interactions. We provide two formulations of the network-untangling problem: (i) minimizing the total interval length over all entities (sum version), and (ii) minimizing the maximum interval length (max version). We study separately the two problems for k = 1 and k > 1 activity intervals per entity. For the case k = 1, we show that the sum problem is NP-hard, while the max problem can be solved optimally in linear time. For the sum problem we provide efficient algorithms motivated by realistic assumptions. For the case of k > 1, we show that both formulations are inapproximable. However, we propose efficient algorithms based on alternative optimization. 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. Keywords Temporal networks · Complex networks · Timeline reconstruction · Vertex cover · Linear programming · 2- sat
1 Introduction New data abstractions emerging from modern applications require new definitions for data-summarization and synthesis tasks. In particular, for many data that are typically
Responsible editor: Tina Eliassi-Rad, Johannes Fürnkranz. Extended author information available on the last page of the article
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modeled as networks, temporal information is nowadays readily available, leading to temporal networks (Holme and Saramäki 2012; Michail 2016). In a temporal network G = (V , E), edges describe interactions over a set of entities V . For each edge (u, v, t) ∈ E, the time of interaction t, between entities u, v ∈ V is also available. In this paper we introduce a new problem formulation for summarizing temporal networks. Network summarization is a well-established problem with applications to data compression, visualization, interactive analysis, and noise elimination. However, temporal network summarization is a rather novel and challenging topic, because of the large variety of temporal network summaries being proposed. An extensive survey f
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