scholarly journals Balanced Online Ramsey Games in Random Graphs

10.37236/100 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Anupam Prakash ◽  
Reto Spöhel ◽  
Henning Thomas
Keyword(s):  
Large Class ◽  
Random Graphs ◽  
Arbitrary Number ◽  
Vertex Coloring ◽  
Empty Graph ◽  
Threshold Functions ◽  
Current Graph ◽  
Similar Threshold ◽  
Player Game ◽  
Monochromatic Copy

Consider the following one-player game. Starting with the empty graph on $n$ vertices, in every step $r$ new edges are drawn uniformly at random and inserted into the current graph. These edges have to be colored immediately with $r$ available colors, subject to the restriction that each color is used for exactly one of these edges. The player's goal is to avoid creating a monochromatic copy of some fixed graph $F$ for as long as possible. We prove explicit threshold functions for the duration of this game for an arbitrary number of colors $r$ and a large class of graphs $F$. This extends earlier work for the case $r=2$ by Marciniszyn, Mitsche, and Stojaković. We also prove a similar threshold result for the vertex-coloring analogue of this game.

scholarly journals Upper Bounds for Online Ramsey Games in Random Graphs

2009 ◽  
Vol 18 (1-2) ◽  
pp. 259-270 ◽  
Author(s):  
MARTIN MARCINISZYN ◽  
RETO SPÖHEL ◽  
ANGELIKA STEGER
Keyword(s):  
Large Class ◽  
Lower Bounds ◽  
Random Graphs ◽  
Upper Bound ◽  
Upper Bounds ◽  
Arbitrary Size ◽  
Threshold Functions ◽  
Current Graph ◽  
Player Game ◽  
Monochromatic Copy

Consider the following one-player game. Starting with the empty graph onnvertices, in every step a new edge is drawn uniformly at random and inserted into the current graph. This edge has to be coloured immediately with one ofravailable colours. The player's goal is to avoid creating a monochromatic copy of some fixed graphFfor as long as possible. We prove an upper bound on the typical duration of this game ifFis from a large class of graphs including cliques and cycles of arbitrary size. Together with lower bounds published elsewhere, explicit threshold functions follow.

scholarly journals Online Ramsey Games in Random Graphs

2009 ◽  
Vol 18 (1-2) ◽  
pp. 271-300 ◽  
Author(s):  
MARTIN MARCINISZYN ◽  
RETO SPÖHEL ◽  
ANGELIKA STEGER
Keyword(s):  
Lower Bound ◽  
Random Graphs ◽  
Empty Graph ◽  
Current Graph ◽  
Player Game ◽  
Monochromatic Copy

Consider the following one-player game. Starting with the empty graph onnvertices, in every step a new edge is drawn uniformly at random and inserted into the current graph. This edge has to be coloured immediately with one ofravailable colours. The player's goal is to avoid creating a monochromatic copy of some fixed graphFfor as long as possible. We prove a lower bound ofnβ(F,r)on the typical duration of this game, where β(F,r) is a function that is strictly increasing inrand satisfies limr→∞β(F,r) = 2 − 1/m2(F), wheren2−1/m2(F)is the threshold of the corresponding offline colouring problem.

scholarly journals On the Path-Avoidance Vertex-Coloring Game

10.37236/650 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Torsten Mütze ◽  
Reto Spöhel
Keyword(s):  
Lower Bound ◽  
Large Class ◽  
Random Graph ◽  
Bipartite Graphs ◽  
Vertex Coloring ◽  
Arbitrary Size ◽  
Greedy Strategy ◽  
Player Game ◽  
Ramsey Type ◽  
Complete Bipartite Graphs

For any graph $F$ and any integer $r\geq 2$, the online vertex-Ramsey density of $F$ and $r$, denoted $m^*(F,r)$, is a parameter defined via a deterministic two-player Ramsey-type game (Painter vs. Builder). This parameter was introduced in a recent paper [arXiv:1103.5849], where it was shown that the online vertex-Ramsey density determines the threshold of a similar probabilistic one-player game (Painter vs. the binomial random graph $G_{n,p}$). For a large class of graphs $F$, including cliques, cycles, complete bipartite graphs, hypercubes, wheels, and stars of arbitrary size, a simple greedy strategy is optimal for Painter and closed formulas for $m^*(F,r)$ are known. In this work we show that for the case where $F=P_\ell$ is a long path, the picture is very different. It is not hard to see that $m^*(P_\ell,r)= 1-1/k^*(P_\ell,r)$ for an appropriately defined integer $k^*(P_\ell,r)$, and that the greedy strategy gives a lower bound of $k^*(P_\ell,r)\geq \ell^r$. We construct and analyze Painter strategies that improve on this greedy lower bound by a factor polynomial in $\ell$, and we show that no superpolynomial improvement is possible.

Equilibrium points for three games based on the Poisson process

1993 ◽  
Vol 30 (03) ◽  
pp. 627-638
Author(s):  
M. T. Dixon
Keyword(s):  
Poisson Process ◽  
Equilibrium Point ◽  
Uniform Distribution ◽  
Arbitrary Number ◽  
Transition Matrix ◽  
Equilibrium Points ◽  
Expected Value ◽  
Time Limit ◽  
The Other ◽  
Player Game

An arbitrary number of competitors are presented with independent Poisson streams of offers consisting of independent and identically distributed random variables having the uniform distribution on [0, 1]. The players each wish to accept a single offer before a known time limit is reached and each aim to maximize the expected value of their offer. Rejected offers may not be recalled, but they are passed on to the other players according to a known transition matrix. This paper finds equilibrium points for two such games, and demonstrates a two-player game with an equilibrium point under which the player with the faster stream of offers has a lower expected reward than his opponent.

Equilibrium points for three games based on the Poisson process

1993 ◽  
Vol 30 (3) ◽  
pp. 627-638 ◽  
Author(s):  
M. T. Dixon
Keyword(s):  
Poisson Process ◽  
Equilibrium Point ◽  
Uniform Distribution ◽  
Arbitrary Number ◽  
Transition Matrix ◽  
Equilibrium Points ◽  
Expected Value ◽  
Time Limit ◽  
The Other ◽  
Player Game

An arbitrary number of competitors are presented with independent Poisson streams of offers consisting of independent and identically distributed random variables having the uniform distribution on [0, 1]. The players each wish to accept a single offer before a known time limit is reached and each aim to maximize the expected value of their offer. Rejected offers may not be recalled, but they are passed on to the other players according to a known transition matrix. This paper finds equilibrium points for two such games, and demonstrates a two-player game with an equilibrium point under which the player with the faster stream of offers has a lower expected reward than his opponent.

scholarly journals Convergence of Achlioptas Processes via Differential Equations with Unique Solutions

2015 ◽  
Vol 25 (1) ◽  
pp. 154-171 ◽  
Author(s):  
OLIVER RIORDAN ◽  
LUTZ WARNKE
Keyword(s):  
Differential Equations ◽  
Stochastic Processes ◽  
Large Class ◽  
Unique Solution ◽  
Random Graph ◽  
Giant Component ◽  
General Idea ◽  
Product Rule ◽  
System Of Differential Equations ◽  
Empty Graph

In Achlioptas processes, starting from an empty graph, in each step two potential edges are chosen uniformly at random, and using some rule one of them is selected and added to the evolving graph. The evolution of the rescaled size of the largest component in such variations of the Erdős–Rényi random graph process has recently received considerable attention, in particular for Bollobás's ‘product rule’. In this paper we establish the following result for rules such as the product rule: the limit of the rescaled size of the ‘giant’ component exists and is continuous provided that a certain system of differential equations has a unique solution. In fact, our result applies to a very large class of Achlioptas-like processes.Our proof relies on a general idea which relates the evolution of stochastic processes to an associated system of differential equations. Provided that the latter has a unique solution, our approach shows that certain discrete quantities converge (after appropriate rescaling) to this solution.

scholarly journals Emergent Structures in Large Networks

2013 ◽  
Vol 50 (03) ◽  
pp. 883-888 ◽  
Author(s):  
David Aristoff ◽  
Charles Radin
Keyword(s):  
Phase Transition ◽  
Large Class ◽  
Random Graph ◽  
Random Graphs ◽  
Essential Feature ◽  
Exponential Random Graph Models ◽  
Graph Models ◽  
Random Graph Models ◽  
Graph Limits ◽  
Exponential Random Graph

We consider a large class of exponential random graph models and prove the existence of a region of parameter space corresponding to the emergent multipartite structure, separated by a phase transition from a region of disordered graphs. An essential feature is the formalism of graph limits as developed by Lovász et al. for dense random graphs.

scholarly journals Network dismantling

2016 ◽  
Vol 113 (44) ◽  
pp. 12368-12373 ◽  
Author(s):  
Alfredo Braunstein ◽  
Luca Dall’Asta ◽  
Guilhem Semerjian ◽  
Lenka Zdeborová
Keyword(s):  
Statistical Mechanics ◽  
Large Class ◽  
Random Graph ◽  
Random Graphs ◽  
Degree Distribution ◽  
Connected Components ◽  
Minimal Size ◽  
Epidemic Spreading ◽  
Minimal Set ◽  
Heavy Tailed

We study the network dismantling problem, which consists of determining a minimal set of vertices in which removal leaves the network broken into connected components of subextensive size. For a large class of random graphs, this problem is tightly connected to the decycling problem (the removal of vertices, leaving the graph acyclic). Exploiting this connection and recent works on epidemic spreading, we present precise predictions for the minimal size of a dismantling set in a large random graph with a prescribed (light-tailed) degree distribution. Building on the statistical mechanics perspective, we propose a three-stage Min-Sum algorithm for efficiently dismantling networks, including heavy-tailed ones for which the dismantling and decycling problems are not equivalent. We also provide additional insights into the dismantling problem, concluding that it is an intrinsically collective problem and that optimal dismantling sets cannot be viewed as a collection of individually well-performing nodes.

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