4 research outputs found
Acquaintance time of random graphs near connectivity threshold
Benjamini, Shinkar, and Tsur stated the following conjecture on the
acquaintance time: asymptotically almost surely for a random graph , provided that is connected. Recently,
Kinnersley, Mitsche, and the second author made a major step towards this
conjecture by showing that asymptotically almost surely , provided that has a Hamiltonian cycle. In this paper, we finish the
task by showing that the conjecture holds in the strongest possible sense, that
is, it holds right at the time the random graph process creates a connected
graph. Moreover, we generalize and investigate the problem for random
hypergraphs
The acquaintance time of (percolated) random geometric graphs
In this paper, we study the acquaintance time \AC(G) defined for a
connected graph . We focus on \G(n,r,p), a random subgraph of a random
geometric graph in which vertices are chosen uniformly at random and
independently from , and two vertices are adjacent with probability
if the Euclidean distance between them is at most . We present
asymptotic results for the acquaintance time of \G(n,r,p) for a wide range of
and . In particular, we show that with high probability
\AC(G) = \Theta(r^{-2}) for G \in \G(n,r,1), the "ordinary" random
geometric graph, provided that (that is, above
the connectivity threshold). For the percolated random geometric graph G \in
\G(n,r,p), we show that with high probability \AC(G) = \Theta(r^{-2} p^{-1}
\ln n), provided that p n r^2 \geq n^{1/2+\eps} and p < 1-\eps for some
\eps>0
Aspects of random graphs
The present report aims at giving a survey of my work since the end of my PhD thesis "Spectral Methods for Reconstruction Problems". Since then I focussed on the analysis of properties of different models of random graphs as well as their connection to real-world networks. This report's goal is to capture these problems in a common framework. The very last chapter of this thesis about results in bootstrap percolation is different in the sense that the given graph is deterministic and only the decision of being active for each vertex is probabilistic; since the proof techniques resemble very much results on random graphs, we decided to include them as well. We start with an overview of the five random graph models, and with the description of bootstrap percolation corresponding to the last chapter. Some properties of these models are then analyzed in the different parts of this thesis