DeepAI AI Chat
Log In Sign Up

Sharp Thresholds in Random Simple Temporal Graphs

11/07/2020
by   Arnaud Casteigts, et al.
0

A graph whose edges only appear at certain points in time is called a temporal graph (among other names). Such a graph is temporally connected if each ordered pair of vertices is connected by a path which traverses edges in chronological order (i.e. a temporal path). In this paper, we consider a simple model of random temporal graph, obtained by assigning to every edge of an Erdős-Rényi random graph G_n,p a uniformly random presence time in the real interval [0, 1]. It turns out that this model exhibits a surprisingly regular sequence of thresholds related to temporal reachability. In particular, we show that at p=log n/n any fixed pair of vertices can a.a.s. reach each other, at 2 log n/n at least one vertex (and in fact, any fixed node) can a.a.s. reach all others, and at 3 log n/n all the vertices can a.a.s. reach each other (i.e. the graph is temporally connected). All these thresholds are sharp. In addition, at p=4 log n/n the graph contains a spanning subgraph of minimum possible size that preserves temporal connectivity, i.e. it admits a temporal spanner with 2n-4 edges. Another contribution of this paper is to connect our model and the above results with existing ones in several other topics, including gossip theory (rumor spreading), population protocols (with sequential random scheduler), and edge-ordered graphs. In particular, our analyses can be extended to strengthen several known results in these fields.

READ FULL TEXT

page 1

page 2

page 3

page 4

05/30/2022

Giant Components in Random Temporal Graphs

A temporal graph is a graph whose edges appear only at certain points in...
06/22/2022

Sparse Temporal Spanners with Low Stretch

A temporal graph is an undirected graph G=(V,E) along with a function th...
09/24/2019

Random k-out subgraph leaves only O(n/k) inter-component edges

Each vertex of an arbitrary simple graph on n vertices chooses k random ...
02/11/2022

The complexity of computing optimum labelings for temporal connectivity

A graph is temporally connected if there exists a strict temporal path, ...
05/17/2018

Deleting edges to restrict the size of an epidemic in temporal networks

A variety of potentially disease-spreading contact networks can be natur...
11/18/2020

Prague dimension of random graphs

The Prague dimension of graphs was introduced by Nesetril, Pultr and Rod...
11/16/2021

On The Complexity of Maximizing Temporal Reachability via Trip Temporalisation

We consider the problem of assigning appearing times to the edges of a d...