scholarly journals Coloring Geographical Threshold Graphs

2010 ◽  
Vol Vol. 12 no. 3 (Graph and Algorithms) ◽  
Author(s):  
Milan Bradonjic ◽  
Tobias Mueller ◽  
Allon G. Percus
Keyword(s):  
Asymptotic Behavior ◽  
Euclidean Space ◽  
Chromatic Number ◽  
Clique Number ◽  
Threshold Function ◽  
The Internet ◽  
Geometric Graphs ◽  
Random Geometric Graphs ◽  
International Audience ◽  
Threshold Graph

Graphs and Algorithms International audience We propose a coloring algorithm for sparse random graphs generated by the geographical threshold graph (GTG) model, a generalization of random geometric graphs (RGG). In a GTG, nodes are distributed in a Euclidean space, and edges are assigned according to a threshold function involving the distance between nodes as well as randomly chosen node weights. The motivation for analyzing this model is that many real networks (e. g., wireless networks, the Internet, etc.) need to be studied by using a ''richer'' stochastic model (which in this case includes both a distance between nodes and weights on the nodes). Here, we analyze the GTG coloring algorithm together with the graph's clique number, showing formally that in spite of the differences in structure between GTG and RGG, the asymptotic behavior of the chromatic number is identical: chi = ln n/ln ln n(1 +o(1)). Finally, we consider the leading corrections to this expression, again using the coloring algorithm and clique number to provide bounds on the chromatic number. We show that the gap between the lower and upper bound is within C ln n/(ln ln n)(2), and specify the constant C.

scholarly journals Colouring random geometric graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Colin J. H. McDiarmid ◽  
Tobias Müller
Keyword(s):  
Chromatic Number ◽  
Analogous Result ◽  
Clique Number ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Random Geometric Graph ◽  
Random Geometric Graphs ◽  
Different Types ◽  
International Audience ◽  
Uniform Case

International audience A random geometric graph $G_n$ is obtained as follows. We take $X_1, X_2, \ldots, X_n ∈\mathbb{R}^d$ at random (i.i.d. according to some probability distribution ν on $\mathbb{R}^d$). For $i ≠j$ we join $X_i$ and $X_j$ by an edge if $║X_i - X_j ║< r(n)$. We study the properties of the chromatic number $χ _n$ and clique number $ω _n$ of this graph as n becomes large, where we assume that $r(n) →0$. We allow any choice $ν$ that has a bounded density function and $║. ║$ may be any norm on $ℝ^d$. Depending on the choice of $r(n)$, qualitatively different types of behaviour can be observed. We distinguish three main cases, in terms of the key quantity $n r^d$ (which is a measure of the average degree). If $r(n)$ is such that $\frac{nr^d}{ln n} →0$ as $n →∞$ then $\frac{χ _n}{ ω _n} →1$ almost surely. If n $\frac{r^d }{\ln n} →∞$ then $\frac{χ _n }{ ω _n} →1 / δ$ almost surely, where $δ$ is the (translational) packing density of the unit ball $B := \{ x ∈ℝ^d: ║x║< 1 \}$ (i.e. $δ$ is the proportion of $d$-space that can be filled with disjoint translates of $B$). If $\frac{n r^d }{\ln n} →t ∈(0,∞)$ then $\frac{χ _n }{ ω _n}$ tends almost surely to a constant that can be bounded in terms of $δ$ and $t$. These results extend earlier work of McDiarmid and Penrose. The proofs in fact yield separate expressions for $χ _n$ and $ω _n$. We are also able to prove a conjecture by Penrose. This states that when $\frac{n r^d }{ln n} →0$ then the clique number becomes focussed on two adjacent integers, meaning that there exists a sequence $k(n)$ such that $\mathbb{P}( ω _n ∈\{k(n), k(n)+1\}) →1$ as $n →∞$. The analogous result holds for the chromatic number (and for the maximum degree, as was already shown by Penrose in the uniform case).

scholarly journals High-Dimensional Random Geometric Graphs and their Clique Number

2011 ◽  
Vol 16 (0) ◽  
pp. 2481-2508 ◽  
Author(s):  
Luc Devroye ◽  
András György ◽  
Gábor Lugosi ◽  
Frederic Udina
Keyword(s):  
Clique Number ◽  
High Dimensional ◽  
Geometric Graphs ◽  
Random Geometric Graphs

scholarly journals Analysis of an algorithm catching elephants on the Internet

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Yousra Chabchoub ◽  
Christine Fricker ◽  
Frédéric Meunier ◽  
Danielle Tibi
Keyword(s):  
Asymptotic Behavior ◽  
Markov Chain ◽  
Stochastic Model ◽  
Limit Theorems ◽  
Bloom Filter ◽  
Simplified Model ◽  
The Internet ◽  
Huge Amount ◽  
International Audience ◽  
On Line

International audience The paper deals with the problem of catching the elephants in the Internet traffic. The aim is to investigate an algorithm proposed by Azzana based on a multistage Bloom filter, with a refreshment mechanism (called $\textit{shift}$ in the present paper), able to treat on-line a huge amount of flows with high traffic variations. An analysis of a simplified model estimates the number of false positives. Limit theorems for the Markov chain that describes the algorithm for large filters are rigorously obtained. The asymptotic behavior of the stochastic model is here deterministic. The limit has a nice formulation in terms of a $M/G/1/C$ queue, which is analytically tractable and which allows to tune the algorithm optimally.

scholarly journals Plane and planarity thresholds for random geometric graphs

2019 ◽  
Vol 12 (01) ◽  
pp. 2050005
Author(s):  
Ahmad Biniaz ◽  
Evangelos Kranakis ◽  
Anil Maheshwari ◽  
Michiel Smid
Keyword(s):  
Euclidean Distance ◽  
Independent Set ◽  
Geometric Graph ◽  
Threshold Function ◽  
Geometric Graphs ◽  
Connected Subgraph ◽  
Random Geometric Graph ◽  
Distance Threshold ◽  
Random Geometric Graphs ◽  
Straight Line

A random geometric graph, [Formula: see text], is formed by choosing [Formula: see text] points independently and uniformly at random in a unit square; two points are connected by a straight-line edge if they are at Euclidean distance at most [Formula: see text]. For a given constant [Formula: see text], we show that [Formula: see text] is a distance threshold function for [Formula: see text] to have a connected subgraph on [Formula: see text] points. Based on this, we show that [Formula: see text] is a distance threshold for [Formula: see text] to be plane, and [Formula: see text] is a distance threshold to be planar. We also investigate distance thresholds for [Formula: see text] to have a non-crossing edge, a clique of a given size, and an independent set of a given size.

scholarly journals On the chromatic number of random geometric graphs

COMBINATORICA ◽  
2011 ◽  
Vol 31 (4) ◽  
pp. 423-488 ◽  
Author(s):  
Colin Mcdiarmid ◽  
Tobias Müller
Keyword(s):  
Chromatic Number ◽  
Geometric Graphs ◽  
Random Geometric Graphs

scholarly journals Improper colouring of (random) unit disk graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Ross J. Kang ◽  
Tobias Müller ◽  
Jean-Sébastien Sereni
Keyword(s):  
Unit Disk ◽  
Chromatic Number ◽  
Maximum Degree ◽  
Clique Number ◽  
Unit Disk Graphs ◽  
Colour Class ◽  
International Audience

International audience For any graph $G$, the $k$-improper chromatic number $χ ^k(G)$ is the smallest number of colours used in a colouring of $G$ such that each colour class induces a subgraph of maximum degree $k$. We investigate the ratio of the $k$-improper chromatic number to the clique number for unit disk graphs and random unit disk graphs to extend results of [McRe99, McD03] (where they considered only proper colouring).

scholarly journals An upper bound for the chromatic number of line graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Andrew D. King ◽  
Bruce A. Reed ◽  
Adrian R. Vetta
Keyword(s):  
Chromatic Number ◽  
Polynomial Time ◽  
Upper Bound ◽  
Maximum Degree ◽  
Line Graph ◽  
Polynomial Time Algorithm ◽  
Clique Number ◽  
Time Algorithm ◽  
Line Graphs ◽  
International Audience

International audience It was conjectured by Reed [reed98conjecture] that for any graph $G$, the graph's chromatic number $χ (G)$ is bounded above by $\lceil Δ (G) +1 + ω (G) / 2\rceil$ , where $Δ (G)$ and $ω (G)$ are the maximum degree and clique number of $G$, respectively. In this paper we prove that this bound holds if $G$ is the line graph of a multigraph. The proof yields a polynomial time algorithm that takes a line graph $G$ and produces a colouring that achieves our bound.

scholarly journals Normalized Sombor Indices as Complexity Measures of Random Networks

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 976
Author(s):  
R. Aguilar-Sánchez ◽  
J. Méndez-Bermúdez ◽  
José Rodríguez ◽  
José Sigarreta
Keyword(s):  
Random Matrix ◽  
Adjacency Matrix ◽  
Matrix Theory ◽  
Computational Study ◽  
Random Networks ◽  
Geometric Graphs ◽  
Theory Approach ◽  
Complexity Measures ◽  
Random Geometric Graphs ◽  
Highly Correlated

We perform a detailed computational study of the recently introduced Sombor indices on random networks. Specifically, we apply Sombor indices on three models of random networks: Erdös-Rényi networks, random geometric graphs, and bipartite random networks. Within a statistical random matrix theory approach, we show that the average values of Sombor indices, normalized to the order of the network, scale with the average degree. Moreover, we discuss the application of average Sombor indices as complexity measures of random networks and, as a consequence, we show that selected normalized Sombor indices are highly correlated with the Shannon entropy of the eigenvectors of the adjacency matrix.

Sign in / Sign up

Export Citation Format

Share Document