
Random kout subgraph leaves only O(n/k) intercomponent edges
Each vertex of an arbitrary simple graph on n vertices chooses k random ...
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Connectivity in Random Annulus Graphs and the Geometric Block Model
Random geometric graphs are the simplest, and perhaps the earliest possi...
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Deleting edges to restrict the size of an epidemic in temporal networks
A variety of potentially diseasespreading contact networks can be natur...
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Sharp Thresholds for a SIR Model on OneDimensional SmallWorld Networks
We study epidemic spreading according to a SusceptibleInfectiousRecove...
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Prague dimension of random graphs
The Prague dimension of graphs was introduced by Nesetril, Pultr and Rod...
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Plane and Planarity Thresholds for Random Geometric Graphs
A random geometric graph, G(n,r), is formed by choosing n points indepen...
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Temporal Cliques admit Sparse Spanners
Let G=(G,λ) be a labeled graph on n vertices with λ:E_G→N a locally inj...
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Sharp Thresholds in Random Simple Temporal Graphs
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 ErdősRényi 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 2n4 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 edgeordered graphs. In particular, our analyses can be extended to strengthen several known results in these fields.
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