The network-untangling problem: from interactions to activity timelines

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.