All Questions
Tagged with random-graphs pr.probability
137 questions
4
votes
1
answer
388
views
Proofs of inequalities used by Erdos-Renyi in their Random Graphs Paper 1
Please refer to this, it is Erdos-Renyi 1959 paper 1 on Random Graphs. I am currently working on this, but I am stuck on the fifth page, where they use two estimates. More specifically, here's the ...
CommunityBot
- 1
2
votes
0
answers
51
views
Subgraphs of random graphs with a given degree sequence
Let $\mathbf{d}=(d_1,\dots, d_n)$ be a given degree sequence with $3\leq d_i\leq \Delta$ for every $i$, where $\Delta$ is constant. Let $G(n,\mathbf{d})$ denote the random graph uniformly distributed ...
- 143
6
votes
2
answers
717
views
Threshold function for a graph not being planar
A graph property $\mathcal{P}$ is monotone increasing if $G\in \mathcal{P}$ implies $G+e \in \mathcal{P}$, i.e., adding an edge to a graph does not destroy the property.
It is well-known that every ...
- 13.6k
3
votes
0
answers
81
views
Can we remove the restriction on a parameter in Talagrand concentration inequality?
Recently I am trying to use Talagrand concentration inequality to do something on graphs. I find a version from the book of Molloy and Reed ''Graph Colouring and Probabilistics Method''. I attached a ...
- 4,219
3
votes
0
answers
87
views
Is the probability distribution of a graphon given as a graph limit computable?
Let $G_i$ be a sequence of finite graphs that is Cauchy in the space of graphons. That is, for every $\epsilon \in \mathbb Q_+$ there is a $N \in \mathbb N$ such that $$\forall n, m > N. \delta_\...
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7
votes
2
answers
235
views
Evolution of the empirical mean of a list as we remove elements proportional to their value
Consider a list of $N$ integers $k_1,k_2,\dots k_N$, drawn independently from some distribution $P(k)$ with $k_i \geq 1$. We denote its mean with $\langle k\rangle=\sum_{k=1}kP(k)$. The first two ...
- 10.3k
2
votes
2
answers
286
views
Finding an easy example applying the general Lovász local lemma
Is there any easy application for the general local lemma as follows? If someone knows, please tell me the references or just post an example here. Thanks.
General Lovász local lemma: Consider a set $...
- 10.4k
7
votes
1
answer
155
views
Nearest neighbors on random complete graph
Consider the complete graph on $2n$ vertices, where the ${2n \choose 2}$ edges have distinct lengths in uniform random
order. So each vertex $v$ has a nearest neighbor $N(v)$, across the shortest ...
- 61.9k
2
votes
2
answers
165
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Example of random walk in a random environment (RWRE) saying things on the environment
I was wondering if anyone is aware of works/articles/examples where random walks in a random environment (RWRE) are actually used for obtaining information on the random environment.
To clarify a bit, ...
- 1,897
1
vote
0
answers
68
views
A one-sided/monotone version of min/max-stable distributions -- does this have a name?
In a couple of papers I am working on (in random graph theory) I have encountered the following property of certain probability distributions, which I will describe shortly, and I am wondering if this ...
- 5,429
2
votes
1
answer
248
views
Connected components in random regular graphs
Suppose we take a random regular graph $G_{2n, r}$, where $n$ is large. Let us also assume that $r$ is fixed, (not dependent on $n$). Let's say that half of the vertices of the graph are colored black ...
- 3,937
1
vote
0
answers
46
views
Diameter of component graph of uniform spanning forests on the amenable transitive graph with super polynomial growth
According to the paper Benjamini, Kesten, Peres, and Schramm - Geometry of the uniform spanning forest: transitions in dimensions 4, 8, 12 (Annals, 2004), the diameter of the component graph of the ...
- 63.9k
2
votes
0
answers
148
views
Union of two copies of uniform spanning forest on $\mathbb{Z}^3$ is transient? [closed]
Let $G$ be the (random) graph which is the union of two independent copies of the uniform spanning forest on $\mathbb{Z}^3$.
Question: Is (the simple random walk on) $G$ transient almost surely?
- 24.2k
10
votes
2
answers
2k
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Probability of Generating a Connected Graph
$N$ points are generated randomly within a unit square, with a uniform distribution.
What is the probability that the points form a connected graph, given that two points are connected if the distance ...
- 2,821
1
vote
1
answer
543
views
Vertex degree on random graphs
Let $p = d/n$ with $d$ constant. How do I prove that, with high probability, $G_{n,p}$ contains a vertex of degree at least $(\log n)^{1/2}$,
where $G_{n,p}$ is a graph with $n$ vertices and the ...
CommunityBot
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4
votes
1
answer
216
views
Quasi-random vs pseudo-random graphs
My question is somehow concerning terminology on extremal graph theory.
Is there any difference concerning the notion of quasi-random graph and the notion of pseudo-random graph? My feeling is that ...
- 242
1
vote
0
answers
96
views
Limit of alternating sum of factorial moments which diverge
Consider the non-negative, integer valued random variable $X$, and its $i^{\text{th}}$ factorial moment $E_{i}[X]$. Then we have that
$$
P(X=0) = \sum _{i=0}^{\infty} \frac{(-1)^i E_{r}[X]}{ i!}
$$
...
- 640
1
vote
0
answers
82
views
Concavity of expected size of a maximum matching (in a bipartite graph) w.r.t. edge probability
Given a n*n bipartite graph where each edge (between any two nodes on the opposite side) is formed i.i.d. with probability $p$, can we show a concavity result on the expected size of a maximum ...
- 53
1
vote
1
answer
313
views
What is the exact definition of a sharp transition?
In "Sharp threshold phenomena in statistical physics", H. Duminil-Copin, Japanese J. of Math. 14, 2019, a sharp transition of a boolean function is defined as follows:
A sequence of ...
- 640
0
votes
0
answers
69
views
Likelihood ratio of non-trivial cycles in an inhomogeneous random square lattice graph embedded on a toroidal surface
Consider a square lattice (random) graph $G$ embedded on a toroidal surface. Each edge $(i, j)$ of the graph has an associated likelihood probability $p_{ij}$. The probabilities $p_{ij}$ come from a ...
- 269
0
votes
0
answers
181
views
What to do when the second moment method does not provide a sufficient bound for $P(X=0)$
We have that for a real valued random variable $X$,
$$
P(X=0) \leq \frac{\text{Var}(X)}{\left(\mathbb{E}(X)\right)^2}
$$
known as Chebyshev's inequality. Consider a random variable $X \in \{0,1,2,\...
- 640
6
votes
0
answers
164
views
Hamilton cycles in random graphs with just enough connectivity
What is the asymptotic probability that $G$ has a Hamilton cycle if $G$ is a random $n$ vertex $\frac{4}{3}n$ edge graph, with minimum degree 2 and without degree 2 vertices at distance 1 or 2 to each ...
- 7,525
1
vote
0
answers
167
views
Width of the critical window in a random graph
In an Erdős–Rényi random graph $G(n,p)$, the giant component emerges with thresholding function $p(n) = c/n$, where $c>1$.
When $c=1$, and $\lambda \in \mathbb{R}$, we can write or "...
- 640
1
vote
1
answer
152
views
Discrepancy of random bipartite graphs (2)
This question is a modification of the one asked here, which turned out to ask for something too strong to be true.
Given $k>0$ and a positive integer $n$, let $X, Y$ be two vertex sets of size $n$ ...
- 3,446
3
votes
1
answer
192
views
Discrepancy of random bipartite graphs
This is a crosspost from MathStackExchange (original question).
Fix $k>0$ and let $X, Y$ be two vertex sets of size $n$ a positive integer (we're interested in the limit $n\to \infty$).
Define a ...
- 3,446
9
votes
1
answer
1k
views
Vertex connectivity of random graphs?
Consider simple, undirected Erdős–Rényi graphs $G(n,p)$, where $n$ is the number of vertices and $p$ is the probability for each pair of vertices to form an edge. Many properties of these graphs are ...
- 1,354
10
votes
1
answer
492
views
(Asymmetric) matrix power series in closed form: $\sum_{i=0}^{\infty} A^i \left(A^i\right)^{\top}={?}$
Let $A\in \mathbb{S}^{N\times N}$ be a symmetric, real and stable matrix, i.e., $\rho(A)<1$, where $\rho(A)$ stands for the spectral radius of $A$. Then, $$\sum\limits_{i=0}^{\infty} A^{2i}=\left( ...
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2
votes
2
answers
183
views
Which infinite random graphs with percolation threshold $p_c=0$ are transient?
I am interested in long-range percolation models with heavy-tailed degree distributions such as DHH13, GLM21, Y6. The simplest example is scale-free percolation in which vertices are elements $\mathbb{...
- 775
5
votes
3
answers
836
views
Probability of an edge in a random graph
Consider a vertex set $V$ and a degree sequence $(d_v)_{v\in V}$. I want to know the probability that an edge exists between two given vertices $u$ and $v$ in a random graph with this degree sequence.
...
- 184
1
vote
1
answer
602
views
Hammersley-Clifford theorem
The paper spatial interaction and the statistical analysis of lattice systems by Besag (1974) presents an alternative proof for the Hammersley-Clifford theorem.
In order to prove the HM theorem, Besag ...
- 128k
2
votes
1
answer
165
views
Is a random $(r+1,r)$-biregular bipartite graph $r$-edge connected w.h.p?
A uniformly random $r$-regular bipartite graph on $n$ vertices is known to be $r$-edge connected. That is, with high probability as $n$ grows large, the minimum size of a cut in a random $r$-regular ...
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1
vote
1
answer
131
views
how to compute the probability that a random graph has two components? [closed]
This question is an example in the book Introduction to Probability Models 11th edition (Sheldon M.Ross). 3.6.2 A random graph:
A graph has $V$ nodes and a set $A$ of pairs of nodes in $V$ called arcs....
- 14.2k
57
votes
4
answers
15k
views
Connectivity of the Erdős–Rényi random graph
It is well-known that if $\omega=\omega(n)$ is any function such that $\omega \to \infty$ as $n \to \infty$, and if $p \ge (\log{n}+\omega) / n$ then the Erdős–Rényi random graph $G(n,p)$ is ...
- 4,713
1
vote
0
answers
78
views
Canonical representation of the a probability distribution for Hammersley Clifford Theorem
I'm reading the following paper
http://www2.stat.duke.edu/~scs/Courses/Stat376/Papers/GibbsFieldEst/BesagJRSSB1974.pdf
On page 7 they give the result that
$$Q(\textbf{x}) = \sum_{1 \leq i \leq n} ...
- 903
1
vote
1
answer
124
views
Empirical degree distribution of random $n$ vertices labeled rooted tree converges to Poisson distribution
I am reading Louigi's lecture note on random trees and graphs here. I get stuck on part (b), Exercise 1.2.3 on page 19, which says the following:
Let $T_n$ be uniformly drawn from $\mathcal{T}_n$, ...
- 715
6
votes
1
answer
521
views
Graphs resembling the math genealogy graph must have concentration in a small number of families?
I was talking with a non-mathematician the other week at a workshop about the fact that many mathematicians, like myself, are indexed in the math genealogy database. We talked a little about how many ...
- 3,209
1
vote
0
answers
97
views
Longest close common subsequence but for continuous random variables
We have two copied sequences of correlated continuous positive random variables that are independent of each other $(X_{n})\perp(Y_{n})$ and equal in distribution $X_{n}\stackrel{dis}{=}Y_{n}$ for ...
- 5,474
1
vote
0
answers
87
views
Contiguity of uniform random regular graphs and uniform random regular graphs which have a perfect matching
Let us consider $\cal{G}_{_{n,d}}$ as the uniform probability space of d-regular graphs
on the n vertices $\{1, \ldots, n \}$ (where $dn$ is even). We say that an event $H_{_{n}}$ occurs a.a.s. (...
- 63.9k
1
vote
1
answer
286
views
Bound on $i$th largest eigenvalue in a large Erdos-Renyi graphs
Typical magnitude of $i$th largest eigenvalue of an Erdos-Renyi random graph seems to decay at least exponentially with $i$. Is there an analytic expression for the constant in the exponent, or a nice ...
- 2,442
1
vote
1
answer
119
views
Does exponential degree distribution entail Log-normal distance distribution in large complex graphs?
We've been exploring the graph structure of a large genealogical data base (WikiTree) of which main connected component contains about 23 million nodes. The graph edges are defined by any direct ...
- 184
3
votes
1
answer
333
views
Probabilistic bound to the number of edge disjoint triangles in a random graph
Let $G$ be a random graph with $n$ vertices, and let $\delta(G)$ be the maximum number of triangles in $G$.
Question. How to prove the bound $$P(\delta(G)) \leq m - t \sqrt{f(m)}) \leq 2e^{-t^2 / 4}$...
- 66.3k
2
votes
0
answers
74
views
Cycle statistics of random endomorphism
Let $S$ be a set with $n$ elements and let $f:S\to S$ be a random function, chosen uniformly among the $n^n$ possibilities. Considering $f$ as a directed graph of constant outdegree $1$, i. e. with ...
- 3,017
4
votes
1
answer
566
views
Random graphs and Benjamini-Schramm convergence
I am looking for literature on the question whether a randomly chosen sequence of $k$-regular graphs converges in the Benjamini-Schramm sense to the universal covering with probability one.
There are ...
- 14.2k
-1
votes
2
answers
419
views
How to define probability over graphs?
How can one formally define a random graph variable?
If G is a random graph variable, then any finite graph is a realization of G. Formally a r.v maps the set of outcomes to a measurable space (may be ...
- 128k
2
votes
1
answer
426
views
Random subgraph properties
Consider a graph $G$ of $N$ vertices and $M$ edges, and assume $G$ has typical complex network properties: it is not necessarily connected, but it has a high clustering coefficient and a giant ...
CommunityBot
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23
votes
4
answers
978
views
What nodes of a graph should be vaccinated first?
Consider a graph, choose some "p: 0<p<1" (probability to infect the neighbor node).
Choose some random number "K" of nodes which are "infected" initially.
So we ...
- 1,054
3
votes
1
answer
166
views
Reference request - random regular graphs vs random graphs w/ degree sequence
There are some properties that are easily studied for random d-regular graphs, but that are very hard to extend to random graphs with a given degree sequence (e.g. whether a graph is w.h.p. ...
- 14.2k
11
votes
1
answer
370
views
Graph with path of length $\geq n$ along grid diagonals - a known result in graph theory?
Is the following lemma a well known result in graph theory?
I am studying a basic existence result that appears to be simple yet powerful. I have not seen it stated as an important result in graph ...
- 290
6
votes
1
answer
421
views
Probability in Chromatic number upper bound of induced subgraph
Let $G=(V, E)$ be a graph with chromatic number $\chi(G)=1000 .$ Let $U \subset V$ be a random subset of $V$ chosen uniformly from among all $2^{|V|}$ subsets of $V$. Let $H=G[U]$ be the induced ...
- 4,589
4
votes
1
answer
1k
views
Critical probability for Erdos-Renyi digraphs to be strongly connected
Given $p \in [0,1]$, an Erdos-Renyi graph ${ER}(n,p)$ on $n$ vertices is constructed by defining, for each unordered couple of distinct vertices ${i,j}$ an edge between $i$ and $j$ with probability $p$...
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