Sharp Thresholds in Random Simple Temporal Graphs

Casteigts, Arnaud; Raskin, Mikhail; Renken, Malte; Zamaraev, Viktor

doi:10.1109/focs52979.2021.00040

Cited by 12 publications

(23 citation statements)

References 35 publications

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“…While the former result is an easy consequence of a theorem proved in [6], the latter requires a finer look at the early phases of densification. Before explaining the main lines of our approach, observe that reachability in the RSTG model only depends on the relative value of presence times of adjacent edges.…”

Section: Contributionsmentioning

confidence: 98%

“…temporally connected, i.e., for any (ordered) pair of distinct vertices , ∈ , there exists a temporal path from to in the RSTG. In order to prove this statement, we generalize the foremost tree technique from [6] (both the algorithm and its analysis), by growing a foremost forest from a set of sources rather than a foremost tree from a single source. The set of sources correspond to the nodes reachable from in 1 .…”

Section: Contributionsmentioning

confidence: 99%

“…The model of random simple temporal graphs (RSTGs) was introduced in [6] as a natural temporal generalization of the classical Erdős-Rényi model , of random graphs. An -vertex RSTG with the parameter ∈ [0, 1] is obtained by first drawing a static random Erdős-Rényi , and then defining a temporal order on its edges by ordering them according to a uniformly random permutation.…”

Section: Random Temporal Graph Modelsmentioning

confidence: 99%

“…In an attempt to quantify the different levels of temporal reachability that a graph may possess, some of the present authors recently considered a simple model of random temporal graphs inspired by the Erdős-Rényi model for static graphs [6]. In this model, a graph is drawn according to two parameters and , where is the number of vertices and is the probability that an edge exists between any two vertices (independently).…”

Section: Introductionmentioning

confidence: 99%

“…In the spirit of classical results on Erdős-Rényi graphs, a natural direction is to investigate the values of at which specific events occur asymptotically almost surely (a.a.s.) [6]. The fact that temporal reachability is neither transitive nor symmetrical turns out to cause a number of distinct thresholds for various levels of reachability that would occur simultaneously in standard graphs.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Giant Components in Random Temporal Graphs

Becker¹,

Casteigts²,

Crescenzi³

et al. 2022

Preprint

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A temporal graph is a graph whose edges appear only at certain points in time. In these graphs, reachability among the nodes relies on paths that traverse edges in chronological order (temporal paths). Unlike standard paths, these paths are not always composable, thus the reachability relation is intransitive and connected components do not form equivalence classes.We investigate the properties of reachability and connected components in random temporal graphs, using a simple model that consists of permuting uniformly at random the edges of an Erdös-Rényi graph and interpreting the position in this permutation as presence times. This model was introduced in [Casteigts et al., FOCS 2021], where thresholds for various reachability properties were identified; for example, sharp threshold for temporal connectivity (all-to-all reachability) is = 3 log / .We generalize several techniques from the above paper in order to characterize the emergence of a giant connected component, which answers an open question from that paper. The growth of a giant component turns out to be quite different from the static case, where a component of size 2/3 emerges at 0 = 1/ and subsequently absorbs a constant fraction of all vertices, with this fraction gradually approaching 1. In contrast, in temporal graphs, we show that the size of a giant connected component transitions abruptly from ( ) nodes to − ( ) nodes at = log / .

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Section: Contributionsmentioning

confidence: 98%

Section: Contributionsmentioning

confidence: 99%

Section: Random Temporal Graph Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Giant Components in Random Temporal Graphs

Becker¹,

Casteigts²,

Crescenzi³

et al. 2022

Preprint

Self Cite

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Invited Paper: Simple, Strict, Proper, Happy: A Study of Reachability in Temporal Graphs

Casteigts

Corsini

Sarkar

2022

Lecture Notes in Computer Science

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A temporal graph is a graph whose edges only appear at certain points in time. Reachability in these graphs is defined in terms of paths that traverse the edges in chronological order (temporal paths). This form of reachability is neither symmetric nor transitive, the latter having important consequences on the computational complexity of even basic questions, such as computing temporal connected components. In this paper, we introduce several parameters that capture how far a temporal graph G is from being transitive, namely, vertex-deletion distance to transitivity and arcmodification distance to transitivity, both being applied to the reachability graph of G. We illustrate the impact of these parameters on the temporal connected component problem, obtaining several tractability results in terms of fixed-parameter tractability and polynomial kernels. Significantly, these results are obtained without restrictions of the underlying graph, the snapshots, or the lifetime of the input graph. As such, our results isolate the impact of non-transitivity and confirm the key role that it plays in the hardness of temporal graph problems.

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