scholarly journals Subgraph counts for dense random graphs with specified degrees

2020 ◽  
pp. 1-38
Author(s):  
Catherine Greenhill ◽  
Mikhail Isaev ◽  
Brendan D. McKay
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Spanning Trees ◽  
Degree Sequence ◽  
Complex Function ◽  
Random Permutation ◽  
Expected Number ◽  
Asymptotic Expressions ◽  
Number Of Spanning Trees

Abstract We prove two estimates for the expectation of the exponential of a complex function of a random permutation or subset. Using this theory, we find asymptotic expressions for the expected number of copies and induced copies of a given graph in a uniformly random graph with degree sequence(d 1 , …, d n ) as n→ ∞. We also determine the expected number of spanning trees in this model. The range of degrees covered includes d j = λn + O(n1/2+ε) for some λ bounded away from 0 and 1.

scholarly journals On the Number of Spanning Trees in Random Regular Graphs

10.37236/3752 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Catherine Greenhill ◽  
Matthew Kwan ◽  
David Wind
Keyword(s):  
Asymptotic Distribution ◽  
Regular Graph ◽  
Spanning Trees ◽  
Regular Graphs ◽  
Cubic Graph ◽  
Expected Number ◽  
Fixed Integer ◽  
Numerical Evidence ◽  
Number Of Spanning Trees

Let $d\geq 3$ be a fixed integer.   We give an asympotic formula for the expected number of spanning trees in a uniformly random $d$-regular graph with $n$ vertices. (The asymptotics are as $n\to\infty$, restricted to even $n$ if $d$ is odd.) We also obtain the asymptotic distribution of the number of spanning trees in a uniformly random cubic graph, and conjecture that the corresponding result holds for arbitrary (fixed) $d$. Numerical evidence is presented which supports our conjecture.

scholarly journals Connectivity for Random Graphs from a Weighted Bridge-Addable Class

10.37236/2596 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Colin McDiarmid
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Planar Graphs ◽  
Additional Assumption ◽  
General Distribution ◽  
Expected Number ◽  
Recent Interest ◽  
Main Conjecture

There has been much recent interest in random graphs sampled uniformly from the $n$-vertex graphs in a suitable structured class, such as the class of all planar graphs. Here we consider a general bridge-addable class $\cal A$ of graphs -- if a graph is in $\cal A$ and $u$ and $v$ are vertices in different components   then the graph obtained by adding an edge (bridge) between $u$ and $v$ must also be in $\cal A$. Various bounds are known concerning the probability of a random graph from such a   class being connected or having many components, sometimes under the additional assumption that bridges can be deleted as well as added. Here we improve or amplify or generalise these bounds (though we do not resolve the main conjecture). For example, we see that the expected number of vertices left when we remove a largest component is less than 2. The generalisation is to consider `weighted' random graphs, sampled from a suitable more general distribution, where the focus is on the bridges.

scholarly journals A Note on the Warmth of Random Graphs with Given Expected Degrees

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Statistical Mechanics ◽  
Chromatic Number ◽  
Random Graph ◽  
Lower Bounds ◽  
Random Graphs ◽  
Upper Bound ◽  
Degree Sequence ◽  
Graph Model ◽  
Upper And Lower Bounds ◽  
Random Graph Model

We consider the random graph modelG(w)for a given expected degree sequencew=(w1,w2,…,wn). Warmth, introduced by Brightwell and Winkler in the context of combinatorial statistical mechanics, is a graph parameter related to lower bounds of chromatic number. We present new upper and lower bounds on warmth ofG(w). In particular, the minimum expected degree turns out to be an upper bound of warmth when it tends to infinity and the maximum expected degreem=O(nα)with0<α<1/2.

scholarly journals Upper bounds on the non-3-colourability threshold of random graphs

2002 ◽  
Vol Vol. 5 ◽  
Author(s):  
Nikolaos Fountoulakis ◽  
Colin McDiarmid
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Upper Bounds ◽  
Average Degree ◽  
Expected Number ◽  
International Audience ◽  
Full Analysis ◽  
A Minor ◽  
Minor Improvement

International audience We present a full analysis of the expected number of 'rigid' 3-colourings of a sparse random graph. This shows that, if the average degree is at least 4.99, then as n → ∞ the expected number of such colourings tends to 0 and so the probability that the graph is 3-colourable tends to 0. (This result is tight, in that with average degree 4.989 the expected number tends to ∞.) This bound appears independently in Kaporis \textitet al. [Kap]. We then give a minor improvement, showing that the probability that the graph is 3-colourable tends to 0 if the average degree is at least 4.989.

scholarly journals Power of $k$ Choices and Rainbow Spanning Trees in Random Graphs

10.37236/4642 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Deepak Bal ◽  
Patrick Bennett ◽  
Alan Frieze ◽  
Paweł Prałat
Keyword(s):  
Stochastic Process ◽  
Random Graph ◽  
Random Graphs ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Special Importance ◽  
Sharp Threshold

We consider the Erdős-Rényi random graph process, which is a stochastic process that starts with $n$ vertices and no edges, and at each step adds one new edge chosen uniformly at random from the set of missing edges. Let $\mathcal{G}(n,m)$ be a graph with $m$ edges obtained after $m$ steps of this process. Each edge $e_i$ ($i=1,2,\ldots, m$) of $\mathcal{G}(n,m)$ independently chooses precisely $k \in\mathbb{N}$ colours, uniformly at random, from a given set of $n-1$ colours (one may view $e_i$ as a multi-edge). We stop the process prematurely at time $M$ when the following two events hold: $\mathcal{G}(n,M)$ is connected and every colour occurs at least once ($M={n \choose 2}$ if some colour does not occur before all edges are present; however, this does not happen asymptotically almost surely). The question addressed in this paper is whether $\mathcal{G}(n,M)$ has a rainbow spanning tree (that is, multicoloured tree on $n$ vertices). Clearly, both properties are necessary for the desired tree to exist.In 1994, Frieze and McKay investigated the case $k=1$ and the answer to this question is "yes" (asymptotically almost surely). However, since the sharp threshold for connectivity is $\frac {n}{2} \log n$ and the sharp threshold for seeing all the colours is $\frac{n}{k} \log n$, the case $k=2$ is of special importance as in this case the two processes keep up with one another. In this paper, we show that asymptotically almost surely the answer is "yes" also for $k \ge 2$.

An improved MCMC algorithm for generating random graphs from constrained distributions

2016 ◽  
Vol 4 (1) ◽  
pp. 117-139 ◽  
Author(s):  
TREVOR TAO
Keyword(s):  
Random Graphs ◽  
Degree Sequence ◽  
Search Space ◽  
General Interest ◽  
Mcmc Algorithm ◽  
Expected Number ◽  
Typical Application ◽  
Mcmc Algorithms ◽  
The World ◽  
Repeated Sampling

AbstractWe consider the problem of generating uniformly random graphs from a constrained distribution. A graph is valid if it obeys certain constraints such as a given number of nodes, edges, k-stars or degree sequence, and each graph must occur with equal probability. A typical application is to confirm the correctness of a model by repeated sampling and comparing statistical properties against empirical data. Markov Chain Monte Carlo (MCMC) algorithms are often used, but have certain difficulties such as the inability to search the space of all possible valid graphs. We propose an improved algorithm which overcomes these difficulties. Although each individual iteration of the MCMC algorithm takes longer, we obtain better coverage of the search space in the same amount of time. This leads to better estimates of various quantities such as the expected number of transitive triads given the constraints. The algorithm should be of general interest with many possible applications, including the world wide web, biological, and social networks.

Connectivity of finite anisotropic random graphs and directed graphs

1986 ◽  
Vol 99 (2) ◽  
pp. 315-330 ◽  
Author(s):  
Joel E. Cohen
Keyword(s):  
Communication Networks ◽  
Random Graphs ◽  
Spanning Trees ◽  
Directed Graphs ◽  
Nuclear Forces ◽  
Finite Set ◽  
Recursive Formulas ◽  
Finite Directed Graphs ◽  
And Control ◽  
Number Of Spanning Trees

AbstractFor graphs on a finite set of vertices with arbitrary probabilities of independently occurring edges, the reliability is defined as the probability that the graph is connected, and the redundancy as the expected number of spanning trees of the graph. Analogous measures of connectivity are defined for random finite directed graphs with arbitrary probabilities of independently occurring directed edges. Recursive formulas for computing the reliability are known. Determinantal formulas, based on matrix-tree theorems, for computing the redundancy are given here. Among random graphs with a given sum of edge probabilities, the more evenly the probabilities are distributed over potential edges, the larger the redundancy. This inequality, proved using the theory of majorization, in combination with examples shows unexpectedly that conflicts between reliability and redundancy can arise in the design of communication networks modelled by such random graphs. The significance of these calculations for the command and control of nuclear forces is sketched.

scholarly journals Subgraphs of Dense Random Graphs with Specified Degrees

2011 ◽  
Vol 20 (3) ◽  
pp. 413-433 ◽  
Author(s):  
BRENDAN D. McKAY
Keyword(s):  
Saddle Point ◽  
Random Graph ◽  
Random Graphs ◽  
Degree Sequence ◽  
Bipartite Graphs ◽  
Average Degree ◽  
Point Method ◽  
Saddle Point Method ◽  
Induced Subgraph ◽  
Basic Facts

Let d = (d1, d2, . . ., dn) be a vector of non-negative integers with even sum. We prove some basic facts about the structure of a random graph with degree sequence d, including the probability of a given subgraph or induced subgraph.Although there are many results of this kind, they are restricted to the sparse case with only a few exceptions. Our focus is instead on the case where the average degree is approximately a constant fraction of n.Our approach is the multidimensional saddle-point method. This extends the enumerative work of McKay and Wormald (1990) and is analogous to the theory developed for bipartite graphs by Greenhill and McKay (2009).

scholarly journals The Critical Phase for Random Graphs with a Given Degree Sequence

2008 ◽  
Vol 17 (1) ◽  
pp. 67-86 ◽  
Author(s):  
M. KANG ◽  
T. G. SEIERSTAD
Keyword(s):  
Phase Transition ◽  
Random Graph ◽  
Generating Function ◽  
Random Graphs ◽  
Branching Process ◽  
Degree Sequence ◽  
Singularity Analysis ◽  
Fixed Degree ◽  
Critical Phase ◽  
Before And After

We consider random graphs with a fixed degree sequence. Molloy and Reed [11, 12] studied how the size of the giant component changes according to degree conditions. They showed that there is a phase transition and investigated the order of components before and after the critical phase. In this paper we study more closely the order of components at the critical phase, using singularity analysis of a generating function for a branching process which models the random graph with a given degree sequence.

scholarly journals The Degree Sequence of Random Graphs from Subcritical Classes

2009 ◽  
Vol 18 (5) ◽  
pp. 647-681 ◽  
Author(s):  
NICLA BERNASCONI ◽  
KONSTANTINOS PANAGIOTOU ◽  
ANGELIKA STEGER
Keyword(s):  
Random Graphs ◽  
Degree Sequence ◽  
Expected Number ◽  
Graph Class ◽  
Expected Values ◽  
Clique Graphs ◽  
Exponentially Small

In this work we determine the expected number of vertices of degreek=k(n) in a graph withnvertices that is drawn uniformly at random from asubcritical graph class. Examples of such classes are outerplanar, series-parallel, cactus and clique graphs. Moreover, we provide exponentially small bounds for the probability that the quantities in question deviate from their expected values.

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