scholarly journals Connectivity threshold for random subgraphs of the Hamming graph

2016 ◽  
Vol 21 (0) ◽  
Author(s):  
Lorenzo Federico ◽  
Remco van der Hofstad ◽  
Tim Hulshof
Keyword(s):  
Hamming Graph ◽  
Random Subgraphs ◽  
Connectivity Threshold

scholarly journals Random subgraphs of the 2D Hamming graph: the supercritical phase

2009 ◽  
Vol 147 (1-2) ◽  
pp. 1-41 ◽  
Author(s):  
Remco van der Hofstad ◽  
Malwina J. Luczak
Keyword(s):  
Hamming Graph ◽  
Supercritical Phase ◽  
Random Subgraphs

scholarly journals Connectivity threshold of Bluetooth graphs

2012 ◽  
Vol 44 (1) ◽  
pp. 45-66 ◽  
Author(s):  
Nicolas Broutin ◽  
Luc Devroye ◽  
Nicolas Fraiman ◽  
Gábor Lugosi
Keyword(s):  
Connectivity Threshold

Radius and diameter of random subgraphs of the hypercube

1993 ◽  
Vol 4 (2) ◽  
pp. 215-229 ◽  
Author(s):  
A. V. Kostochka ◽  
A. A. Sapozhenko ◽  
K. Weber
Keyword(s):  
Random Subgraphs

scholarly journals Topology of random -dimensional cubical complexes

2021 ◽  
Vol 9 ◽  
Author(s):  
Matthew Kahle ◽  
Elliot Paquette ◽  
Érika Roldán
Keyword(s):  
Fundamental Group ◽  
Random Graphs ◽  
High Probability ◽  
Cubical Complex ◽  
Sharp Threshold ◽  
Cubical Complexes ◽  
Simple Connectivity ◽  
Dimensional Cube ◽  
Dimensional Face ◽  
Connectivity Threshold

Abstract We study a natural model of a random $2$ -dimensional cubical complex which is a subcomplex of an n-dimensional cube, and where every possible square $2$ -face is included independently with probability p. Our main result exhibits a sharp threshold $p=1/2$ for homology vanishing as $n \to \infty $ . This is a $2$ -dimensional analogue of the Burtin and Erdoős–Spencer theorems characterising the connectivity threshold for random graphs on the $1$ -skeleton of the n-dimensional cube. Our main result can also be seen as a cubical counterpart to the Linial–Meshulam theorem for random $2$ -dimensional simplicial complexes. However, the models exhibit strikingly different behaviours. We show that if $p> 1 - \sqrt {1/2} \approx 0.2929$ , then with high probability the fundamental group is a free group with one generator for every maximal $1$ -dimensional face. As a corollary, homology vanishing and simple connectivity have the same threshold, even in the strong ‘hitting time’ sense. This is in contrast with the simplicial case, where the thresholds are far apart. The proof depends on an iterative algorithm for contracting cycles – we show that with high probability, the algorithm rapidly and dramatically simplifies the fundamental group, converging after only a few steps.

scholarly journals Topological Data Analysis reveals robust alterations in the whole-brain and frontal lobe functional connectomes in Attention-Deficit/Hyperactivity Disorder

2019 ◽  
Author(s):  
Zeus Gracia-Tabuenca ◽  
Juan Carlos Díaz-Patiño ◽  
Isaac Arelio ◽  
Sarael Alcauter
Keyword(s):  
Attention Deficit Hyperactivity Disorder ◽  
Data Analysis ◽  
Frontal Lobe ◽  
Attention Deficit ◽  
Brain Network ◽  
Topological Data Analysis ◽  
Hyperactivity Disorder ◽  
The Brain ◽  
Connectivity Threshold ◽  
Topological Data

AbstractThe functional organization of the brain network (connectome) has been widely studied as a graph; however, methodological issues may affect the results, such as the brain parcellation scheme or the selection of a proper threshold value. Instead of exploring the brain in terms of a static connectivity threshold, this work explores its algebraic topology as a function of the filtration value (i.e., the connectivity threshold), a process termed the Rips filtration in Topological Data Analysis. Specifically, we characterized the transition from all nodes being isolated to being connected into a single component as a function of the filtration value, in a public dataset of children with attention-deficit/hyperactivity disorder (ADHD) and typically developing children. Results were highly congruent when using four different brain segmentations (atlases), and exhibited significant differences for the brain topology of children with ADHD, both at the whole brain network and at the functional sub-network levels, particularly involving the frontal lobe and the default mode network. Therefore, this approach may contribute to identify the neurophysio-pathology of ADHD, reducing the bias of connectomics-related methods.HighlightsTopological Data Analysis was implemented in functional connectomes.Betti curves were assessed based on the area under the curve, slope and kurtosis.The explored variables were robust along four different brain atlases.ADHD showed lower areas, suggesting decreased functional segregation.Frontal and default mode networks showed the greatest differences between groups.Graphical Abstract

scholarly journals On the connectivity threshold of Achlioptas processes

2014 ◽  
Vol 5 (3) ◽  
pp. 291-304
Author(s):  
Mihyun Kang ◽  
Konstantinos Panagiotou
Keyword(s):  
Connectivity Threshold

scholarly journals Norton Algebras of the Hamming Graphs via Linear Characters

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Jia Huang
Keyword(s):  
Group Theory ◽  
Automorphism Group ◽  
Nonassociative Algebra ◽  
Bilinear Forms ◽  
Distance Regular Graph ◽  
Finite Abelian Groups ◽  
Hamming Graph ◽  
Large Subgroup ◽  
Special Cases ◽  
Hamming Graphs

The Norton product is defined on each eigenspace of a distance regular graph by the orthogonal projection of the entry-wise product. The resulting algebra, known as the Norton algebra, is a commutative nonassociative algebra that is useful in group theory due to its interesting automorphism group. We provide a formula for the Norton product on each eigenspace of a Hamming graph using linear characters. We construct a large subgroup of automorphisms of the Norton algebra of a Hamming graph and completely describe the automorphism group in some cases. We also show that the Norton product on each eigenspace of a Hamming graph is as nonassociative as possible, except for some special cases in which it is either associative or equally as nonassociative as the so-called double minus operation previously studied by the author, Mickey, and Xu. Our results restrict to the hypercubes and extend to the halved and/or folded cubes, the bilinear forms graphs, and more generally, all Cayley graphs of finite abelian groups.

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