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

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

Connectivity threshold for random chordal graphs

1991 ◽  
Vol 7 (2) ◽  
pp. 177-181 ◽  
Author(s):  
F. R. McMorris ◽  
Edward R. Scheinerman
Keyword(s):  
Chordal Graphs ◽  
Connectivity Threshold

scholarly journals Acquaintance Time of Random Graphs Near Connectivity Threshold

2016 ◽  
Vol 30 (1) ◽  
pp. 555-568 ◽  
Author(s):  
Andrzej Dudek ◽  
Paweł Prałat
Keyword(s):  
Random Graphs ◽  
Connectivity Threshold

On the connectivity threshold for general uniform metric spaces

2010 ◽  
Vol 110 (10) ◽  
pp. 356-359 ◽  
Author(s):  
Gady Kozma ◽  
Zvi Lotker ◽  
Gideon Stupp
Keyword(s):  
Metric Spaces ◽  
Connectivity Threshold

A Sharp Threshold for Network Reliability

2002 ◽  
Vol 11 (5) ◽  
pp. 465-474 ◽  
Author(s):  
MICHAEL KRIVELEVICH ◽  
BENNY SUDAKOV ◽  
VAN H. VU
Keyword(s):  
Classical Result ◽  
Network Reliability ◽  
Average Degree ◽  
Edge Connectivity ◽  
Sharp Threshold ◽  
Random Subgraph ◽  
Connectivity Threshold

Given a graph G on n vertices with average degree d, form a random subgraph Gp by choosing each edge of G independently with probability p. Strengthening a classical result of Margulis we prove that, if the edge connectivity k(G) satisfies k(G) [Gt ] d/log n, then the connectivity threshold in Gp is sharp. This result is asymptotically tight.

Sign in / Sign up

Export Citation Format

Share Document