Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userLocal cliques in ER-perturbed random geometric graphs
157
0
0.0
(
0
)
Added by
Minghao Tian
Publication date
2018
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
Random graphs are mathematical models that have applications in a wide range of domains. We study the following model where one adds ErdH{o}s--Renyi (ER) type perturbation to a random geometric graph. More precisely, assume $G_mathcal{X}^{*}$ is a random geometric graph sampled from a nice measure on a metric space $mathcal{X} = (X,d)$. The input observed graph $widehat{G}(p,q)$ is generated by removing each existing edge from $G_mathcal{X}^*$ with probability $p$, while inserting each non-existent edge to $G_mathcal{X}^{*}$ with probability $q$. We refer to such random $p$-deletion and $q$-insertion as ER-perturbation. Although these graphs are related to the objects in the continuum percolation theory, our understanding of them is still rather limited. In this paper we consider a localized version of the classical notion of clique number for the aforementioned ER-perturbed random geometric graphs: Specifically, we study the edge clique number for each edge in a graph, defined as the size of the largest clique(s) in the graph containing that edge. The clique number of the graph is simply the largest edge clique number. Interestingly, given a ER-perturbed random geometric graph, we show that the edge clique number presents two fundamentally different types of behaviors, depending on which type of randomness it is generated from. As an application of the above results, we show that by using a filtering process based on the edge clique number, we can recover the shortest-path metric of the random geometric graph $G_mathcal{X}^*$ within a multiplicative factor of $3$, from an ER-perturbed observed graph $widehat{G}(p,q)$, for a significantly wider range of insertion probability $q$ than in previous work.
rate research
Read More
Online routing in a planar embedded graph is central to a number of fields and has been studied extensively in the literature. For most planar graphs no $O(1)$-competitive online routing algorithm exists. A notable exception is the Delaunay triangulation for which Bose and Morin [Online routing in triangulations. SIAM Journal on Computing, 33(4):937-951, 2004] showed that there exists an online routing algorithm that is $O(1)$-competitive. However, a Delaunay triangulation can have $Omega(n)$ vertex degree and a total weight that is a linear factor greater than the weight of a minimum spanning tree. We show a simple construction, given a set $V$ of $n$ points in the Euclidean plane, of a planar geometric graph on $V$ that has small weight (within a constant factor of the weight of a minimum spanning tree on $V$), constant degree, and that admits a local routing strategy that is $O(1)$-competitive. Moreover, the technique used to bound the weight works generally for any planar geometric graph whilst preserving the admission of an $O(1)$-competitive routing strategy.
For two graphs $G$ and $H$, write $G stackrel{mathrm{rbw}}{longrightarrow} H$ if $G$ has the property that every {sl proper} colouring of its edges yields a {sl rainbow} copy of $H$. We study the thresholds for such so-called {sl anti-Ramsey} properties in randomly perturbed dense graphs, which are unions of the form $G cup mathbb{G}(n,p)$, where $G$ is an $n$-vertex graph with edge-density at least $d$, and $d$ is a constant that does not depend on $n$. Our results in this paper, combined with our results in a companion paper, determine the threshold for the property $G cup mathbb{G}(n,p) stackrel{mathrm{rbw}}{longrightarrow} K_s$ for every $s$. In this paper, we show that for $s geq 9$ the threshold is $n^{-1/m_2(K_{leftlceil s/2 rightrceil})}$; in fact, our $1$-statement is a supersaturation result. This turns out to (almost) be the threshold for $s=8$ as well, but for every $4 leq s leq 7$, the threshold is lower; see our companion paper for more details. In this paper, we also consider the property $G cup mathbb{G}(n,p) stackrel{mathrm{rbw}}{longrightarrow} C_{2ell - 1}$, and show that the threshold for this property is $n^{-2}$ for every $ell geq 2$; in particular, it does not depend on the length of the cycle $C_{2ell - 1}$. It is worth mentioning that for even cycles, or more generally for any fixed bipartite graph, no random edges are needed at all.
For two graphs $G$ and $H$, write $G stackrel{mathrm{rbw}}{longrightarrow} H$ if $G$ has the property that every emph{proper} colouring of its edges yields a emph{rainbow} copy of $H$. We study the thresholds for such so-called emph{anti-Ramsey} properties in randomly perturbed dense graphs, which are unions of the form $G cup mathbb{G}(n,p)$, where $G$ is an $n$-vertex graph with edge-density at least $d >0$, and $d$ is independent of $n$. In a companion article, we proved that the threshold for the property $G cup mathbb{G}(n,p) stackrel{mathrm{rbw}}{longrightarrow} K_ell$ is $n^{-1/m_2(K_{leftlceil ell/2 rightrceil})}$, whenever $ell geq 9$. For smaller $ell$, the thresholds behave more erratically, and for $4 le ell le 7$ they deviate downwards significantly from the aforementioned aesthetic form capturing the thresholds for emph{large} cliques. In particular, we show that the thresholds for $ell in {4, 5, 7}$ are $n^{-5/4}$, $n^{-1}$, and $n^{-7/15}$, respectively. For $ell in {6, 8}$ we determine the threshold up to a $(1 + o(1))$-factor in the exponent: they are $n^{-(2/3 + o(1))}$ and $n^{-(2/5 + o(1))}$, respectively. For $ell = 3$, the threshold is $n^{-2}$; this follows from a more general result about odd cycles in our companion paper.
We study some percolation problems on the complete graph over $mathbf N$. In particular, we give sharp sufficient conditions for the existence of (finite or infinite) cliques and paths in a random subgraph. No specific assumption on the probability, such as independency, is made. The main tools are a topological version of Ramsey theory, exchangeability theory and elementary ergodic theory.
We study biplane graphs drawn on a finite planar point set $S$ in general position. This is the family of geometric graphs whose vertex set is $S$ and can be decomposed into two plane graphs. We show that two maximal biplane graphs---in the sense that no edge can be added while staying biplane---may differ in the number of edges, and we provide an efficient algorithm for adding edges to a biplane graph to make it maximal. We also study extremal properties of maximal biplane graphs such as the maximum number of edges and the largest maximum connectivity over $n$-element point sets.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
