Efficient Computation of Optimal Temporal Walks Under Waiting-Time Constraints

Abstract

Node connectivity plays a central role in temporal network analysis. We provide a comprehensive study of various concepts of walks in temporal graphs, that is, graphs with fixed vertex sets but edge sets changing over time. Importantly, the temporal aspect results in a rich set of optimization criteria for "shortest" walks. Extending and significantly broadening state-ofthe-art work of Wu et al. [IEEE TKDE 2016], we provide a quasi-linear-time algorithm for shortest walk computation that is capable to deal with various optimization criteria and any linear combination of these. A central distinguishing factor to Wu et al.'s work is that our model allows to, motivated by real-world applications, respect waiting-time constraints for vertices, that is, the minimum and maximum waiting time allowed in intermediate vertices of a walk. Moreover, other than Wu et al. our algorithm does not request a strictly increasing time evolvement of the walk and can optimize a richer set of optimization criteria. Our experimental studies indicate that our richer modeling can be achieved without significantly worsening the running time. * Supported by the DFG, projects DAMM (NI 369/13) and FPTinP (NI 369/16).

Betweenness centrality for temporal multiplexes

“…A valid path in this temporal multiplex is a sequence of edges such that if is incident to node on layer and disappears (‘arrives’) at time t , then leaves from node j on any layer and appears (‘departs’) at time , where is a minimum connecting time (as also introduced in 15 ).…”

“…A number of previous works has focused on paths, distances and betweenness in single-layer temporal networks. The networks considered typically have instantaneous links 17 – 20 , but in some cases a non-zero link travel time has been considered 14 , 15 , 21 . In 21 , the travel time is assumed to be the same for all links and equal to the discretisation interval.…”

“…Ref. 15 also considers constraints on the minimum and maximum connecting time in a node (a maximum time could apply e.g. to disease spreading, to describe a finite infectious period).…”

“…Wu et al 14 considers an additional type, the ones departing latest. Himmel et al 15 consider 8 types of shortest paths, including all the previous ones and adding the paths that minimise the time spent travelling (duration minus total connecting time), the ones that minimise the waiting time at intermediate nodes (total connecting time), the cheapest (if a cost is associated to each link), the most likely (if a probability is associated to each link). An attempt to reconcile different definitions has been done by Tsalouchidou et al 19 , defining the length of a path as a combination of the number of links n and the duration , with a relative weight : .…”

“…minimum connecting time between two flights). These characteristics have rarely been considered in previous works on temporal networks (though see 13 – 15 ).…”