scholarly journals A Remark on the Tournament Game

10.37236/5142 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Dennis Clemens ◽  
Mirjana Mikalački
Keyword(s):  
Random Graph ◽  
Threshold Probability

We study the Maker-Breaker tournament game played on the edge set of a given graph $G$. Two players, Maker and Breaker, claim unclaimed edges of $G$ in turns, while Maker additionally assigns orientations to the edges that she claims. If by the end of the game Maker claims all the edges of a pre-defined goal tournament, she wins the game. Given a tournament $T_k$ on $k$ vertices, we determine the threshold bias for the $(1:b)$ $T_k$-tournament game on $K_n$. We also look at the $(1:1)$ $T_k$-tournament game played on the edge set of a random graph ${\mathcal{G}_{n,p}}$ and determine the threshold probability for Maker's win. We compare these games with the clique game and discuss whether a random graph intuition is satisfied. 

scholarly journals Connector-Breaker Games on Random Boards

10.37236/9381 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Dennis Clemens ◽  
Laurin Kirsch ◽  
Yannick Mogge
Keyword(s):  
Random Graph ◽  
Complete Graph ◽  
Threshold Probability

By now, the Maker-Breaker connectivity game on a complete graph $K_n$ or on a random graph $G\sim G_{n,p}$ is well studied. Recently, London and Pluhár suggested a variant in which Maker always needs to choose her edges in such a way that her graph stays connected. By their results it follows that for this connected version of the game, the threshold bias on $K_n$ and the threshold probability on $G\sim G_{n,p}$ for winning the game drastically differ from the corresponding values for the usual Maker-Breaker version, assuming Maker's bias to be 1. However, they observed that the threshold biases of both versions played on $K_n$ are still of the same order if instead Maker is allowed to claim two edges in every round. Naturally, this made London and Pluhár ask whether a similar phenomenon can be observed when a $(2:2)$ game is played on $G_{n,p}$. We prove that this is not the case, and determine the threshold probability for winning this game to be of size $n^{-2/3+o(1)}$.

scholarly journals The Threshold Probability for Long Cycles

2016 ◽  
Vol 26 (2) ◽  
pp. 208-247 ◽  
Author(s):  
ROMAN GLEBOV ◽  
HUMBERTO NAVES ◽  
BENNY SUDAKOV
Keyword(s):  
Random Graph ◽  
Complete Graph ◽  
Minimum Degree ◽  
Hamilton Cycle ◽  
Threshold Probability ◽  
Spanning Subgraph ◽  
Classic Result ◽  
Long Cycles

For a given graph G of minimum degree at least k, let Gp denote the random spanning subgraph of G obtained by retaining each edge independently with probability p = p(k). We prove that if p ⩾ (logk + loglogk + ωk(1))/k, where ωk(1) is any function tending to infinity with k, then Gp asymptotically almost surely contains a cycle of length at least k + 1. When we take G to be the complete graph on k + 1 vertices, our theorem coincides with the classic result on the threshold probability for the existence of a Hamilton cycle in the binomial random graph.

Random graphs

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Large Scale ◽  
Random Network ◽  
Graph Model ◽  
Clustering Coefficient ◽  
Giant Component ◽  
Random Graph Model ◽  
Definition Of ◽  
Average Size

An introduction to the mathematics of the Poisson random graph, the simplest model of a random network. The chapter starts with a definition of the model, followed by derivations of basic properties like the mean degree, degree distribution, and clustering coefficient. This is followed with a detailed derivation of the large-scale structural properties of random graphs, including the position of the phase transition at which a giant component appears, the size of the giant component, the average size of the small components, and the expected diameter of the network. The chapter ends with a discussion of some of the shortcomings of the random graph model.

Panchromatic colorings of random hypergraphs

2021 ◽  
Vol 31 (1) ◽  
pp. 19-41
Author(s):  
D. A. Kravtsov ◽  
N. E. Krokhmal ◽  
D. A. Shabanov
Keyword(s):  
Binomial Model ◽  
Threshold Probability ◽  
The Difference ◽  
Random Hypergraphs

Abstract We study the threshold probability for the existence of a panchromatic coloring with r colors for a random k-homogeneous hypergraph in the binomial model H(n, k, p), that is, a coloring such that each edge of the hypergraph contains the vertices of all r colors. It is shown that this threshold probability for fixed r < k and increasing n corresponds to the sparse case, i. e. the case p = c n / ( n k ) $p = cn/{n \choose k}$ for positive fixed c. Estimates of the form c 1(r, k) < c < c 2(r, k) for the parameter c are found such that the difference c 2(r, k) − c 1(r, k) converges exponentially fast to zero if r is fixed and k tends to infinity.

scholarly journals A Note on the Majority Dynamics in Inhomogeneous Random Graphs

2021 ◽  
Vol 76 (3) ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Random Graph ◽  
Initial State ◽  
Time Step ◽  
Dynamic Monopoly ◽  
Inhomogeneous Random Graphs ◽  
Assignment Probability ◽  
Edge Probability ◽  
Consensus Reaching Process ◽  
Consensus Reaching ◽  
Inhomogeneous Random Graph

AbstractIn this note, we study discrete time majority dynamics over an inhomogeneous random graph G obtained by including each edge e in the complete graph $$K_n$$ K n independently with probability $$p_n(e)$$ p n ( e ) . Each vertex is independently assigned an initial state $$+1$$ + 1 (with probability $$p_+$$ p + ) or $$-1$$ - 1 (with probability $$1-p_+$$ 1 - p + ), updated at each time step following the majority of its neighbors’ states. Under some regularity and density conditions of the edge probability sequence, if $$p_+$$ p + is smaller than a threshold, then G will display a unanimous state $$-1$$ - 1 asymptotically almost surely, meaning that the probability of reaching consensus tends to one as $$n\rightarrow \infty $$ n → ∞ . The consensus reaching process has a clear difference in terms of the initial state assignment probability: In a dense random graph $$p_+$$ p + can be near a half, while in a sparse random graph $$p_+$$ p + has to be vanishing. The size of a dynamic monopoly in G is also discussed.

scholarly journals PHI density prospectively improves prostate cancer detection

2021 ◽  
Author(s):  
Carsten Stephan ◽  
Klaus Jung ◽  
Michael Lein ◽  
Hannah Rochow ◽  
Frank Friedersdorff ◽  
...  
Keyword(s):  
Prostate Cancer ◽  
Specific Antigen ◽  
Roc Curves ◽  
Grey Zone ◽  
Net Benefit ◽  
Threshold Probability ◽  
Free Psa ◽  
Prostate Health Index ◽  
Magnetic Resonance Imaging Mri ◽  
Prostate Health

Abstract Purpose To evaluate the Prostate Health Index (PHI) density (PHID) in direct comparison with PHI in a prospective large cohort. Methods PHID values were calculated from prostate-specific antigen (PSA), free PSA and [− 2]proPSA and prostate volume. The 1057 patients included 552 men with prostate cancer (PCa) and 505 with no evidence of malignancy (NEM). In detail, 562 patients were biopsied at the Charité Hospital Berlin and 495 patients at the Sana Hospital Offenbach. All patients received systematic or magnetic resonance imaging (MRI)/ultrasound fusion-guided biopsies. The diagnostic accuracy was evaluated by receiver operating characteristic (ROC) curves comparing areas under the ROC-curves (AUC). The decision curve analysis (DCA) was performed with the MATLAB Neural Network Toolbox. Results PHID provided a significant larger AUC than PHI (0.835 vs. 0.801; p = 0.0013) in our prospective cohort of 1057 men from 2 centers. The DCA had a maximum net benefit of ~ 5% for PHID vs. PHI between 35 and 65% threshold probability. In those 698 men within the WHO-calibrated PSA grey-zone up to 8 ng/ml, PHID was also significantly better than PHI (AUC 0.819 vs. 0.789; p = 0.0219). But PHID was not different from PHI in the detection of significant PCa. Conclusions Based on ROC analysis and DCA, PHID had an advantage in comparison with PHI alone to detect any PCa but PHI and PHID performed equal in detecting significant PCa.

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