Interference-free Walks in Time: Temporally Disjoint Paths

We investigate the computational complexity of finding temporally disjoint paths or walks in temporal graphs. There, the edge set changes over discrete time steps and a temporal path (resp. walk) uses edges that appear at monotonically increasing time steps. Two paths (or walks) are temporally disjoint if they never use the same vertex at the same time; otherwise, they interfere. This reflects applications in robotics, traffic routing, or finding safe pathways in dynamically changing networks. On the one extreme, we show that on general graphs the problem is computationally hard. The "walk version" is W[1]-hard when parameterized by the number of routes. However, it is polynomial-time solvable for any constant number of walks. The "path version" remains NP-hard even if we want to find only two temporally disjoint paths. On the other extreme, restricting the input temporal graph to have a path as underlying graph, quite counterintuitively, we find NP-hardness in general but also identify natural tractable cases.

Correlating Covid-19 mortality and infection levels

Covid-19 deaths and positive cases show a remarkable heterogeneity across countries which cannot be easily explained on the basis of similarities or differences in the quality of healthcare, access to healthcare, testing facilities, or preventive measures such as lockdowns. Here we show that there is a distinct correlation between the mortality level and the infection level across countries, which can explain the mortality levels for a wide spectrum of countries. This implies that the number of deaths per 100 infected individuals is approximately the same across diverse countries and can be estimated from the slope of the mortality level-infection level plot. The correlation presented here can potentially be combined with estimates of infection spread to forecast future mortality levels and therefore future needs in terms of healthcare and other resources. Tracking of an individual location’s temporal path on this plot can potentially serve as a visual assessment of the nature of the epidemic. Methods presented here are not specific to the current epidemic. This is a preliminary report and uses data from a single source at a single time-point to demonstrate the capability of such an analysis.

Hypothesis: protein and RNA attributes are continuously optimized over time

Background Little is known why proteins and RNAs exhibit half-lives varying over several magnitudes. Despite many efforts, a conclusive link between half-lives and gene function could not be established suggesting that other determinants may influence these molecular attributes. Results Here, I find that with increasing gene age there is a gradual and significant increase of protein and RNA half-lives, protein structure, and other molecular attributes that tend to affect protein abundance. These observations are accommodated in a hypothesis which posits that new genes at ‘birth’ are not optimized and thus their products exhibit low half-lives and less structure but continuous mutagenesis eventually improves these attributes. Thus, the protein and RNA products of the oldest genes obtained their high degrees of stability and structure only after billions of years while the products of younger genes had less time to be optimized and are therefore less stable and structured. Because more stable proteins with lower turnover require less transcription to maintain the same level of abundance, reduced transcription-associated mutagenesis (TAM) would fixate the changes by increasing gene conservation. Conclusions Consequently, the currently observed diversity of molecular attributes is a snapshot of gene products being at different stages along their temporal path of optimization.

Approximating the Temporal Neighbourhood Function of Large Temporal Graphs

Temporal networks are graphs in which edges have temporal labels, specifying their starting times and their traversal times. Several notions of distances between two nodes in a temporal network can be analyzed, by referring, for example, to the earliest arrival time or to the latest starting time of a temporal path connecting the two nodes. In this paper, we mostly refer to the notion of temporal reachability by using the earliest arrival time. In particular, we first show how the sketch approach, which has already been used in the case of classical graphs, can be applied to the case of temporal networks in order to approximately compute the sizes of the temporal cones of a temporal network. By making use of this approach, we subsequently show how we can approximate the temporal neighborhood function (that is, the number of pairs of nodes reachable from one another in a given time interval) of large temporal networks in a few seconds. Finally, we apply our algorithm in order to analyze and compare the behavior of 25 public transportation temporal networks. Our results can be easily adapted to the case in which we want to refer to the notion of distance based on the latest starting time.