Throughput Characterizations of Wireless Networks via Stochastic Geometry and Random Graph Theory

2021 ◽  
Author(s):  
Jeffrey William Wildman
Keyword(s):  
Wireless Networks ◽  
Graph Theory ◽  
Random Graph ◽  
Stochastic Geometry ◽  
Random Graph Theory

Getting a Directed Hamilton Cycle Two Times Faster

2012 ◽  
Vol 21 (5) ◽  
pp. 773-801 ◽  
Author(s):  
CHOONGBUM LEE ◽  
BENNY SUDAKOV ◽  
DAN VILENCHIK
Keyword(s):  
Graph Theory ◽  
Random Graph ◽  
Directed Graph ◽  
Classical Result ◽  
Hamilton Cycle ◽  
Empty Graph ◽  
Underlying Graph ◽  
Random Direction ◽  
Random Graph Theory ◽  
On Line

Consider the random graph process where we start with an empty graph on n vertices and, at time t, are given an edge et chosen uniformly at random among the edges which have not appeared so far. A classical result in random graph theory asserts that w.h.p. the graph becomes Hamiltonian at time (1/2+o(1))n log n. On the contrary, if all the edges were directed randomly, then the graph would have a directed Hamilton cycle w.h.p. only at time (1+o(1))n log n. In this paper we further study the directed case, and ask whether it is essential to have twice as many edges compared to the undirected case. More precisely, we ask if, at time t, instead of a random direction one is allowed to choose the orientation of et, then whether or not it is possible to make the resulting directed graph Hamiltonian at time earlier than n log n. The main result of our paper answers this question in the strongest possible way, by asserting that one can orient the edges on-line so that w.h.p. the resulting graph has a directed Hamilton cycle exactly at the time at which the underlying graph is Hamiltonian.

Counterexamples of the 0-1 Law for Fragments of Existential Second-Order Logic: an Overview

2000 ◽  
Vol 6 (1) ◽  
pp. 67-82 ◽  
Author(s):  
Jean-Marie Le Bars
Keyword(s):  
Graph Theory ◽  
Random Graph ◽  
Second Order ◽  
Order Logic ◽  
First Order ◽  
Random Graph Theory ◽  
Asymptotic Probability ◽  
Second Order Logic

AbstractWe propose an original use of techniques from random graph theory to find a Monadic(Minimal Scott without equality) sentence without an asymptotic probability. Our result implies that the 0-1 law fails for the logics(FO2) and](Minimal Gödel without equality). Therefore we complete the classification of first-order prefix classes with or without equality, according to the existence of the 0-1 law for the correspondingfragment. In addition, our counterexample can be viewed as a single explanation of the failure of the 0-1 law of all the fragments of existential second-order logic for which the failure is already known.

scholarly journals From trees to graphs: collapsing continuous-time branching processes

2018 ◽  
Vol 55 (3) ◽  
pp. 900-919
Author(s):  
A. Garavaglia ◽  
R. van der Hofstad
Keyword(s):  
Graph Theory ◽  
Random Graph ◽  
Real World ◽  
Continuous Time ◽  
Degree Distribution ◽  
Preferential Attachment ◽  
Branching Processes ◽  
Random Graph Theory ◽  
First Time ◽  
Continuous Time Branching Processes

Abstract Continuous-time branching processes (CTBPs) are powerful tools in random graph theory, but are not appropriate to describe real-world networks since they produce trees rather than (multi)graphs. In this paper we analyze collapsed branching processes (CBPs), obtained by a collapsing procedure on CTBPs, in order to define multigraphs where vertices have fixed out-degree m≥2. A key example consists of preferential attachment models (PAMs), as well as generalized PAMs where vertices are chosen according to their degree and age. We identify the degree distribution of CBPs, showing that it is closely related to the limiting distribution of the CTBP before collapsing. In particular, this is the first time that CTBPs are used to investigate the degree distribution of PAMs beyond the tree setting.

USING COMBINATORICA/MATHEMATICA FOR STUDENT PROJECTS IN RANDOM GRAPH THEORY

PRIMUS ◽  
2006 ◽  
Vol 16 (4) ◽  
pp. 314-319
Author(s):  
Thomas J. Pfaff ◽  
Michele Zaret
Keyword(s):  
Graph Theory ◽  
Random Graph ◽  
Random Graph Theory ◽  
Student Projects
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