On the Structure of Graphs of Markoff Triples

2020 ◽  
Vol 71 (2) ◽  
pp. 637-648
Author(s):  
Sergei V Konyagin ◽  
Sergey V Makarychev ◽  
Igor E Shparlinski ◽  
Ilya V Vyugin
Keyword(s):  
Giant Component ◽  
Connected Components ◽  
Functional Graph ◽  
Modulo P

Abstract We sharpen the bounds of J. Bourgain, A. Gamburd and P. Sarnak (2016) on the possible number of nodes outside the ‘giant component’ and on the size of individual connected components in the suitably defined functional graph of Markoff triples modulo $p$. This is a step towards the conjecture that there are no such nodes at all.

Grasping the connectivity of random functional graphs

2005 ◽  
Vol 42 (1) ◽  
pp. 1-19 ◽  
Author(s):  
David Romero ◽  
Federico Zertuche
Keyword(s):  
Directed Graph ◽  
Connected Components ◽  
Asymptotic Formulas ◽  
Expected Number ◽  
Functional Graph

A functional graph is a directed graph where every node has out-degree one (loops allowed). This paper deals with connectivity aspects of random functional graphs, like the expected number and size of connected components, cycles, and trajectories. Both exact and asymptotic formulas are provided.

scholarly journals Homological Percolation: The Formation of Giant k-Cycles

2020 ◽  
Author(s):  
Omer Bobrowski ◽  
Primoz Skraba
Keyword(s):  
Thermodynamic Limit ◽  
Exponential Decay ◽  
Algebraic Topology ◽  
Percolation Model ◽  
Critical Values ◽  
Giant Component ◽  
Connected Components ◽  
Continuum Percolation ◽  
Dimensional Torus ◽  
Higher Dimensional

Abstract In this paper we introduce and study a higher dimensional analogue of the giant component in continuum percolation. Using the language of algebraic topology, we define the notion of giant $k$-dimensional cycles (with $0$-cycles being connected components). Considering a continuum percolation model in the flat $d$-dimensional torus, we show that all the giant $k$-cycles ($1\le k \le d-1$) appear in the regime known as the thermodynamic limit. We also prove that the thresholds for the emergence of the giant $k$-cycles are increasing in $k$ and are tightly related to the critical values in continuum percolation. Finally, we provide bounds for the exponential decay of the probabilities of giant cycles appearing.

On the Number of NK-Kauffman Networks Mapped into a Functional Graph

2016 ◽  
Vol 25 (4) ◽  
pp. 329-345
Author(s):  
Federico Zertuche ◽  
Keyword(s):  
Functional Graph

The algorithm of the color signal recognition at landing an unmanned aerial vehicle on an aircraft carrier in autonomous mode

2020 ◽  
pp. 78
Keyword(s):  
Unmanned Aerial Vehicle ◽  
Treatment Method ◽  
Recognition Algorithm ◽  
Connected Components ◽  
Preliminary Treatment ◽  
Signal Recognition ◽  
Aircraft Carrier ◽  
Halftone Image ◽  
Glide Path ◽  
Aerial Vehicle

The movement along the glide path of an unmanned aerial vehicle during landing on an aircraft carrier is investigated. The implementation of this task is realized in the conditions of radio silence of the aircraft carrier. The algorithm for treatment information from an optical landing system installed on an aircraft carrier is developed. The algorithm of the color signal recognition assumes the usage of the image frame preliminary treatment method via a downsample function, that performs the decimation process, the HSV model, the Otsu’s method for calculating the binarization threshold for a halftone image, and the method of separating the connected Two-Pass components. Keywords unmanned aerial vehicle; aircraft carrier; approach; glide path; optical landing system; color signal recognition algorithm; decimation; connected components; halftone image binarization

The configuration model

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Degree Sequence ◽  
Graph Model ◽  
Network Models ◽  
Clustering Coefficient ◽  
Giant Component ◽  
Configuration Model ◽  
Bipartite Networks ◽  
Small Component ◽  
Random Graph Model ◽  
Average Size

A discussion of the most fundamental of network models, the configuration model, which is a random graph model of a network with a specified degree sequence. Following a definition of the model a number of basic properties are derived, including the probability of an edge, the expected number of multiedges, the excess degree distribution, the friendship paradox, and the clustering coefficient. This is followed by derivations of some more advanced properties including the condition for the existence of a giant component, the size of the giant component, the average size of a small component, and the expected diameter. Generating function methods for network models are also introduced and used to perform some more advanced calculations, such as the calculation of the distribution of the number of second neighbors of a node and the complete distribution of sizes of small components. The chapter ends with a brief discussion of extensions of the configuration model to directed networks, bipartite networks, networks with degree correlations, networks with high clustering, and networks with community structure, among other possibilities.

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.

scholarly journals Upperbounds on the probability of finding marked connected components using quantum walks

2021 ◽  
Vol 20 (1) ◽  
Author(s):  
Adam Glos ◽  
Nikolajs Nahimovs ◽  
Konstantin Balakirev ◽  
Kamil Khadiev
Keyword(s):  
Quantum Walks ◽  
Connected Components

The Cognitive Social Network in Dreams: Transitivity, Assortativity, and Giant Component Proportion Are Monotonic

2015 ◽  
Vol 40 (3) ◽  
pp. 671-696 ◽  
Author(s):  
Hye Joo Han ◽  
Richard Schweickert ◽  
Zhuangzhuang Xi ◽  
Charles Viau-Quesnel
Keyword(s):  
Social Network ◽  
Giant Component
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