# Connectivity Thresholds for Bounded Size Rules

```@article{Einarsson2014ConnectivityTF,
title={Connectivity Thresholds for Bounded Size Rules},
author={H. Einarsson and J. Lengler and K. Panagiotou and Frank Mousset and A. Steger},
journal={arXiv: Probability},
year={2014}
}```
In an Achlioptas process, starting with a graph that has n vertices and no edge, in each round \$d \geq 1\$ edges are drawn uniformly at random, and using some rule exactly one of them is chosen and added to the evolving graph. For the class of Achlioptas processes we investigate how much impact the rule has on one of the most basic properties of a graph: connectivity. Our main results are twofold. First, we study the prominent class of bounded size rules, which select the edge to add according… Expand
1 Citations
Randomized Algorithms and Probabilistic Methods: Advanced Topics
• Mathematics
• 2015
For this, we use the canonical paths method. Thus, we should start by defining paths between any two elements A,B ∈ Ω. It turns out that the exact definition of the paths is not really all thatExpand

#### References

SHOWING 1-10 OF 30 REFERENCES
Avoiding small subgraphs in Achlioptas processes
• Mathematics
• 2009
For a fixed integer r, consider the following random process. At each round, one is presented with r random edges from the edge set of the complete graph on n vertices, and is asked to choose one ofExpand
The augmented multiplicative coalescent and critical dynamic random graph models
• Mathematics
• 2012
Random graph models with limited choice have been studied extensively with the goal of understanding the mechanism of the emergence of the giant component. One of the standard models are theExpand
Pursuing the Giant in Random Graph Processes
We study the evolution of random graph processes that are based on the paradigm of the power of multiple choices. The processes we consider begin with an empty graph on n vertices. In each subsequentExpand
Small subgraphs in random graphs and the power of multiple choices
• Mathematics, Computer Science
• J. Comb. Theory, Ser. B
• 2011
This work proposes an edge selection strategy that a.a.s. with probability 1-o(1) as n->~ (asymptotically almost surely) avoids creating a copy of F for as long as N=o(N"0), and proves that any online strategy will not create such a copy once [email protected](N" 0). Expand
The evolution of subcritical Achlioptas processes
• Mathematics, Computer Science
• Random Struct. Algorithms
• 2015
This paper shows that certain key statistics are tightly concentrated at least until the susceptibility (the expected size of the component containing a randomly chosen vertex) diverges, and believes that for a large class of rules the critical time where the susceptibility `blows up' coincides with the percolation threshold. Expand
On the connectivity threshold of Achlioptas processes
• Mathematics
• 2014
In this paper we study the connectivity threshold of Achlioptas processes. It is well known that the classical Erdős-Renyi random graph with n vertices becomes connected whp (with high probability,Expand
Hamiltonicity thresholds in Achlioptas processes
• Mathematics, Computer Science
• Random Struct. Algorithms
• 2010
This paper analyzes the appearance of a Hamilton cycle in the following random process, and shows that this problem has three regimes, depending on the value of K, including the intermediate regime where K=\Theta(\log n), and the threshold has order n. Expand
Birth control for giants
• Mathematics, Computer Science
• Comb.
• 2007
Computer aided solutions to the possible differential equations for susceptibility allow us to establish lower and upper bounds on the extent to which the authors can either delay or accelerate the birth of the giant component. Expand
The Birth of the Giant Component
• Mathematics, Computer Science
• Random Struct. Algorithms
• 1993
A “uniform” model of random graphs, which allows self-loops and multiple edges, is shown to lead to formulas that are substanitially simpler than the analogous formulas for the classical random graphs of Erdos and Renyi. Expand
Explosive Percolation in Erdős–Rényi-Like Random Graph Processes
• Computer Science, Mathematics
• Combinatorics, Probability and Computing
• 2012
This work proves discontinuous phase transitions for three random graph processes: all three start with the empty graph on n vertices and, depending on the process, they connect in every step to one vertex chosen randomly from all vertices, or two components chosenrandomly from the set of all components. Expand