Concurrency and reachability in treelike temporal networks

Abstract

Network properties govern the rate and extent of various spreading processes, from simple contagions to complex cascades. Recently, the analysis of spreading processes has been extended from static networks to temporal networks, where nodes and links appear and disappear. We focus on the effects of "accessibility", whether there is a temporally consistent path from one node to another, and "reachability", the density of the corresponding "accessibility graph" representation of the temporal network. The level of reachability thus inherently limits the possible extent of any spreading process on the temporal network. We study reachability in terms of the overall levels of temporal concurrency between edges and the structural cohesion of the network agglomerating over all edges. We use simulation results and develop heterogeneous mean field model predictions for random networks to better quantify how the properties of the underlying temporal network regulate reachability.

Concurrency measures in the era of temporal network epidemiology: a review

“…By definition, the reachability ignores infectivity and therefore does not directly translate into observed epidemics. However, since high average reachability is positively correlated with the final outbreak size, the results in [26,27,30,76] are consistent with the common claim that the concurrency positively contributes to the severity of outbreaks.…”

“…Assuming that the event time and its duration are randomly generated and independent for different edges, different authors defined the temporal coherency as the probability that the time windows of activation on two edges overlap [ 26 , 30 ]. For empirical data of temporal networks, the same quantity can be measured as the fraction of edge pairs that have overlapping active time, where the edge may be optionally assumed to be active all the time between its first and last event times [ 27 ].…”

“…The reachability is defined for a temporal network as the expected fraction of node pairs i and j with a time-respecting path from i to j starting at a random time between the beginning and end of the data [80][81][82] (also see [83] for a mathematical analysis). The reachability of the temporal network generally increases as the temporal coherency increases [26,27,30,76]. Concurrency implies that two edges sharing a node simultaneously exist.…”

“…These early articles rarely state how one could hypothetically reduce the full information of a temporal network of sexual interactions to a network of partnerships-should one casual, non-recurring interaction count as a partnership? Furthermore, papers state a multitude of quantitative definitions of concurrency [23][24][25][26][27][28][29][30][31][32], and several studies point out the difficulty of studying an issue without an agreed definition [27,[33][34][35]. Some of the existing confusion and controversy probably stems from the vagueness of the partnership concept.…”

Directed Percolation in Random Temporal Network Models with Heterogeneities

“…These phenomena are governed by the convoluted effects of intrinsic cognitive processes and the environment, which in turn induce temporal correlations in the sequence of human actions and interactions. As opposed to systems governed by homogeneous and independent processes, these correlations and the induced temporal dynamics may have significant effects on various dynamical processes evolving on temporal networks such as spreading [31,32], reachability [33,34], diffusion [9,35], and opinion formation [36].…”