Infinitary logics and very sparse random graphs

1997 ◽  
Vol 62 (2) ◽  
pp. 609-623 ◽  
Author(s):  
James F. Lynch
Keyword(s):  
Growth Rate ◽  
Random Graph ◽  
Random Graphs ◽  
First Order ◽  
Order Language ◽  
Edge Probability ◽  
Asymptotic Probability

AbstractLet be the infinitary language obtained from the first-order language of graphs by closure under conjunctions and disjunctions of arbitrary sets of formulas, provided only finitely many distinct variables occur among the formulas. Let p(n) be the edge probability of the random graph on n vertices. It is shown that if p(n) ≪ n−1 satisfies certain simple conditions on its growth rate, then for every , the probability that σ holds for the random graph on n vertices converges. In fact, if p(n) = n−α, α > 1, then the probability is either smaller than for some d > 0, or it is asymptotic to cn−d for some c > 0, d ≥ 0. Results on the difficulty of computing the asymptotic probability are given.

scholarly journals Packing Tree Factors in Random and Pseudo-Random Graphs

10.37236/3285 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Deepak Bal ◽  
Alan Frieze ◽  
Michael Krivelevich ◽  
Po-Shen Loh
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
High Probability ◽  
Edge Disjoint ◽  
Edge Probability ◽  
Vertex Disjoint ◽  
Fixed Tree

For a fixed graph $H$ with $t$ vertices, an $H$-factor of a graph $G$ with $n$ vertices, where $t$ divides $n$, is a collection of vertex disjoint (not necessarily induced) copies of $H$ in $G$ covering all vertices of $G$. We prove that for a fixed tree $T$ on $t$ vertices and $\epsilon>0$, the random graph $G_{n,p}$, with $n$ a multiple of $t$, with high probability contains a family of edge-disjoint $T$-factors covering all but an $\epsilon$-fraction of its edges, as long as $\epsilon^4 n p \gg \log^2 n$. Assuming stronger divisibility conditions, the edge probability can be taken down to $p>\frac{C\log n}{n}$. A similar packing result is proved also for pseudo-random graphs, defined in terms of their degrees and co-degrees.

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.

On the spectra of first-order language properties for random graphs

2015 ◽  
Vol 92 (1) ◽  
pp. 503-506 ◽  
Author(s):  
J. H. Spencer ◽  
M. E. Zhukovskii
Keyword(s):  
Random Graphs ◽  
First Order ◽  
Order Language

scholarly journals Closing Gaps in Problems related to Hamilton Cycles in Random Graphs and Hypergraphs

10.37236/5025 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Asaf Ferber
Keyword(s):  
Random Graph ◽  
Directed Graph ◽  
Random Graphs ◽  
High Probability ◽  
Hamilton Cycle ◽  
Uniform Hypergraph ◽  
Hamilton Cycles ◽  
Edge Probability

We show how to adjust a very nice coupling argument due to McDiarmid in order to prove/reprove in a novel way results concerning Hamilton cycles in various models of random graph and hypergraphs. In particular, we firstly show that for $k\geq 3$, if $pn^{k-1}/\log n$ tends to infinity, then a random $k$-uniform hypergraph on $n$ vertices, with edge probability $p$, with high probability (w.h.p.) contains a loose Hamilton cycle, provided that $(k-1)|n$. This generalizes results of Frieze, Dudek and Frieze, and reproves a result of Dudek, Frieze, Loh and Speiss. Secondly, we show that there exists $K>0$ such for every $p\geq (K\log n)/n$ the following holds: Let $G_{n,p}$ be a random graph on $n$ vertices with edge probability $p$, and suppose that its edges are being colored with $n$ colors uniformly at random. Then, w.h.p. the resulting graph contains a Hamilton cycle with for which all the colors appear (a rainbow Hamilton cycle). Bal and Frieze proved the latter statement for graphs on an even number of vertices, where for odd $n$ their $p$ was $\omega((\log n)/n)$. Lastly, we show that for $p=(1+o(1))(\log n)/n$, if we randomly color the edge set of a random directed graph $D_{n,p}$ with $(1+o(1))n$ colors, then w.h.p. one can find a rainbow Hamilton cycle where all the edges are directed in the same way.

scholarly journals The t-Improper Chromatic Number of Random Graphs

2009 ◽  
Vol 19 (1) ◽  
pp. 87-98 ◽  
Author(s):  
ROSS J. KANG ◽  
COLIN McDIARMID
Keyword(s):  
Asymptotic Behaviour ◽  
Chromatic Number ◽  
Random Graph ◽  
Random Graphs ◽  
Maximum Degree ◽  
Colour Class ◽  
Edge Probability

We consider the t-improper chromatic number of the Erdős–Rényi random graph Gn,p. The t-improper chromatic number χt(G) is the smallest number of colours needed in a colouring of the vertices in which each colour class induces a subgraph of maximum degree at most t. If t = 0, then this is the usual notion of proper colouring. When the edge probability p is constant, we provide a detailed description of the asymptotic behaviour of χt(Gn,p) over the range of choices for the growth of t = t(n).

Random graphs

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Large Scale ◽  
Random Network ◽  
Graph Model ◽  
Clustering Coefficient ◽  
Giant Component ◽  
Random Graph Model ◽  
Definition Of ◽  
Average Size

An introduction to the mathematics of the Poisson random graph, the simplest model of a random network. The chapter starts with a definition of the model, followed by derivations of basic properties like the mean degree, degree distribution, and clustering coefficient. This is followed with a detailed derivation of the large-scale structural properties of random graphs, including the position of the phase transition at which a giant component appears, the size of the giant component, the average size of the small components, and the expected diameter of the network. The chapter ends with a discussion of some of the shortcomings of the random graph model.

scholarly journals A Note on the Majority Dynamics in Inhomogeneous Random Graphs

2021 ◽  
Vol 76 (3) ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Random Graph ◽  
Initial State ◽  
Time Step ◽  
Dynamic Monopoly ◽  
Inhomogeneous Random Graphs ◽  
Assignment Probability ◽  
Edge Probability ◽  
Consensus Reaching Process ◽  
Consensus Reaching ◽  
Inhomogeneous Random Graph

AbstractIn this note, we study discrete time majority dynamics over an inhomogeneous random graph G obtained by including each edge e in the complete graph $$K_n$$ K n independently with probability $$p_n(e)$$ p n ( e ) . Each vertex is independently assigned an initial state $$+1$$ + 1 (with probability $$p_+$$ p + ) or $$-1$$ - 1 (with probability $$1-p_+$$ 1 - p + ), updated at each time step following the majority of its neighbors’ states. Under some regularity and density conditions of the edge probability sequence, if $$p_+$$ p + is smaller than a threshold, then G will display a unanimous state $$-1$$ - 1 asymptotically almost surely, meaning that the probability of reaching consensus tends to one as $$n\rightarrow \infty $$ n → ∞ . The consensus reaching process has a clear difference in terms of the initial state assignment probability: In a dense random graph $$p_+$$ p + can be near a half, while in a sparse random graph $$p_+$$ p + has to be vanishing. The size of a dynamic monopoly in G is also discussed.

Weakly atomic-compact relational structures

1971 ◽  
Vol 36 (1) ◽  
pp. 129-140 ◽  
Author(s):  
G. Fuhrken ◽  
W. Taylor
Keyword(s):  
Abelian Group ◽  
Model Theory ◽  
Compact Abelian Group ◽  
Finite Subset ◽  
Relational Structure ◽  
Similarity Type ◽  
First Order ◽  
Relational Structures ◽  
Order Language ◽  
Weakly Atomic

A relational structure is called weakly atomic-compact if and only if every set Σ of atomic formulas (taken from the first-order language of the similarity type of augmented by a possibly uncountable set of additional variables as “unknowns”) is satisfiable in whenever every finite subset of Σ is so satisfiable. This notion (as well as some related ones which will be mentioned in §4) was introduced by J. Mycielski as a generalization to model theory of I. Kaplansky's notion of an algebraically compact Abelian group (cf. [5], [7], [1], [8]).

Supercritical Fluid Deposition of Ruthenium Thin Films: A Kinetic Study

2007 ◽  
Vol 992 ◽  
Author(s):  
Christos F. Karanikas ◽  
James J. Watkins
Keyword(s):  
Thin Films ◽  
Growth Rate ◽  
Deposition Temperature ◽  
Zero Order ◽  
Measurable Effect ◽  
Deposition Rates ◽  
Reaction Pressure ◽  
Deposition Kinetics ◽  
First Order ◽  
Kinetics Of

AbstractThe kinetics of the deposition of ruthenium thin films from the hydrogen assisted reduction of bis(2,2,6,6-tetramethyl-3,5-heptanedionato)(1,5-cyclooctadiene)ruthenium(II), [Ru(tmhd)2cod], in supercritical carbon dioxide was studied in order to develop a rate expression for the growth rate as well as to determine a mechanism for the process. The deposition temperature was varied from 240°C to 280°C and the apparent activation energy was 45.3 kJ/mol. Deposition rates up to 30 nm/min were attained. The deposition rate dependence on precursor concentrations between 0 and 0.2 wt. % was studied at 260°C with excess hydrogen and revealed first order deposition kinetics with respect to precursor at concentrations lower then 0.06 wt. % and zero order dependence at concentrations above 0.06 wt. %. The effect of reaction pressure on the growth rate was studied at a constant reaction temperature of 260°C and pressures between 159 bar to 200 bar and found to have no measurable effect on the growth rate.

scholarly journals On the Isolated Vertices and Connectivity in Random Intersection Graphs

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Random Graphs ◽  
Intersection Graph ◽  
Intersection Graphs ◽  
Random Intersection Graph ◽  
Probability Of Connectivity ◽  
Asymptotic Probability

We study isolated vertices and connectivity in the random intersection graph . A Poisson convergence for the number of isolated vertices is determined at the threshold for absence of isolated vertices, which is equivalent to the threshold for connectivity. When and , we give the asymptotic probability of connectivity at the threshold for connectivity. Analogous results are well known in Erdős-Rényi random graphs.

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