All Questions
44 questions
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 ...
- 7,857
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 ...
- 24.9k
21
votes
2
answers
548
views
Do these polynomials have alternating coefficients?
In answering another MathOverflow question, I stumbled across the sequence of polynomials $Q_n(p)$ defined by the recurrence
$$Q_n(p) = 1-\sum_{k=2}^{n-1} \binom{n-2}{k-2}(1-p)^{k(n-k)}Q_k(p).$$
Thus:
...
- 44.4k
13
votes
2
answers
383
views
Comparing two measures on trees on $n$ vertices
A standard measure on trees on $n$ vertices is the Uniform Spanning Tree (UST) on the complete graph. This is the measure where every tree has equal probability, $1 / n^{n-2}$ by Cayley's formula.
...
- 7,857
12
votes
3
answers
1k
views
A Modern Proof of Erdos and Renyi's 1959 Random Graph Paper?
In their paper, Erdos and Renyi consider a random graph with a fixed number of edges, as opposed to the more modern approach of adding each edge independently with probability $p$. From what I ...
- 470
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 ...
- 6,917
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 ...
- 383
8
votes
0
answers
240
views
Stepanov phase transition in random graph
Consider the classical random graph model G(n,p), with p=c/n, as proposed by Erd\"os and R\'enyi.
At this scaling, the most prominent feature is arguably the abrupt change of the topology that the ...
- 468
7
votes
4
answers
1k
views
Recent impressive combinatorial developments in probability theory
In the preface to the second edition of Daniel Stroock's book "Probability Theory: An Analytic View", there is this striking claim (on p. xv)
... I suspect that, for at least a decade, the most ...
- 1,302
7
votes
2
answers
417
views
Dynamics of a random "quadratic" directed graph
Let G be a directed graph on N vertices chosen at random, conditional on the requirement that the out-degree of each vertex is 1 and the in-degree of each vertex is either 0 or 2. The "periodic" ...
- 19.2k
7
votes
1
answer
156
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 ...
- 236
7
votes
1
answer
257
views
Collecting proofs of the birth of the giant component
I want to collect different proofs of Erdös-Rényi result on the double jump of the largest connected component on $G(n,p)$ (or in $G(n,M)$.
I know the original proof of Erdös-Rényi, the proof that ...
- 1,561
6
votes
2
answers
723
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 ...
- 557
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 ...
- 7,545
6
votes
0
answers
105
views
Long loops in critical random graphs
A simple calculation seems to show that the expected number $X_k$ of loops of length $k$ in a critical Erdös-Renyi random graph $G(n,n^{-1})$ is approximately given by
$$ \mathbb{E} X_k=\frac1{2k}{e^...
5
votes
1
answer
1k
views
Self Avoiding Walk Enumerations
Let $c(n)$ be the number of Self avoiding walks (SAW) of length $n$ on an infinite lattice $L$. Are there any known non-geometric interpretations of $c(n)$?. For example, is there a number theoretic ...
- 4,952
4
votes
1
answer
275
views
Probability of a subset of Bernoulli's being all 1's
Suppose we have $n$ iid Bernoulli's $X_1,\ldots,X_n$ with mean $p$, and a family $\mathcal{F}$ of subsets of $[n]$. The question is how to lower bound the probability that there is a set in the family ...
- 408
4
votes
0
answers
220
views
Navigation in a graph
The problem
Let $G=(V,E)$ be a graph. $k = O\left(\log(|V|)\right)$ distinct vertices are picked randomly from $V$. We call the set of chosen $k$ vertices $T$.
Assumptions about the graph: You may ...
- 323
3
votes
1
answer
822
views
Open Problems in Random Graphs [closed]
I am a PhD student in mathematics. I'm interested in probabilistic methods in combinatorics and especially random graphs. I am looking for an open problem in this area for my PhD proposal. I know that ...
- 57
3
votes
1
answer
108
views
Expected size of matchings in a cubic graph
Let $G$ be a random cubic graph on $n$ vertices. Let $M$ be the set of (not necessarily maximum) matchings of $G$. What is the expected size (i.e. number of edges) of an element of $M$?
In other ...
- 9,501
3
votes
1
answer
206
views
Component properties in Euclidean graphs with distance threshold
In the context of Euclidean graphs with vertices randomly embedded in either a 2D plane (for instance square with length $L$) or in 3D (similarly, cube of side $L$), where an edge between two given ...
- 649
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 ...
- 1,190
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 $...
- 1,190
2
votes
2
answers
256
views
Model for random graphs where clique number remains bounded
In the Erdös-Rényi model for random graphs,the clique number is seen to go to infinity as the number of vertices grows. Is anyone aware of models for random graphs with bounded clique number?
- 2,630
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 ...
- 129
2
votes
1
answer
607
views
Component size distribution in small Erdos-Renyi networks
I'm looking at $\mathcal{G}(n,p)$ (I'll call these Erdos-Renyi networks) where $n$ is, say, at most 10.
I would like to know the probability a random node is in a component of size $m$.
It's ...
- 121
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 ...
- 153
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?
- 41
2
votes
0
answers
83
views
Zero-One law for Hamiltonian path subgraphs of Hamming Distance Graphs?
$(\alpha,\beta,d)$-Hamming Distance Graph $G_d(\alpha,\beta)$ for $\alpha,\beta\in(0,1]$ is a graph on $2^d$ vertices $v_0,\dots,v_{2^d-1}$ with edges $(v_i,v_j)\in\mathcal E(G_d)$ iff $0<\sum_{t=1}...
- 13.9k
2
votes
0
answers
93
views
Erdös-Renyi Model with prescribed subgraph
In the Erdös-Rényi model for random graphs there is a lot of results stating sharp phase transitions for the probability of a random graph to contain a fixed prescribed ...
- 2,630
1
vote
2
answers
2k
views
Proving a random bipartite graph contains a perfect matching
I have the following problem
consider a random bipartite with vertex classes $A$ and $B$ of size $|A|=|B|=\mathrm{log}^{2}(n)$ graph in which every possible edge is chosen independently with ...
- 903
1
vote
1
answer
638
views
Threshold for perfect Matchings in Bipartite graph
Consider the random bipartite graph with vertex classes of size $n$ and each edge being present independently with probability $p(n)$.
I know one way to prove the threshold of a perfect matching is ...
- 903
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 ...
1
vote
2
answers
116
views
How to use probability to find a matching in a family of graphs?
In a conference, I heard that we can use some probabilistic methods to find a matching in some kind of graphs. I would like to see some examples of such technics. Can someone provide some references ...
- 57
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
1
vote
1
answer
436
views
Size of minimum cut in random graph
Consider a uniform random tournament with $n$ vertices. (Between any two vertices $x,y$, with probability $0.5$ draw an edge from $x$ to $y$; otherwise draw an edge from $y$ to $x$.) The score of each ...
- 1,209
1
vote
1
answer
188
views
KPZ relation $\chi = 2 \xi -1$ in a random geometric graph
If I have $n$ points uniformly distributed on the surface of a torus, and form a graph by adding an edge between pairs whenever they are within a unit distance (induced by the Euclidean metric), I ...
- 640
1
vote
2
answers
2k
views
Expected matching in a bipartite graph
Consider a random bipartite graph constructed on vertex classes of size $n$ with each edge present independently with probability $p$. How could I go about calculating the size of the expected ...
- 903
1
vote
1
answer
420
views
Poisson approximation of random sub-graphs
I add the edges of $G(n)$ the complete graph on $n$ vertices one by one, at random and without replacement, and denote by $G(n,m)$ the resulting Erdos Renyi random graph process. At step $m$ in the ...
- 468
1
vote
0
answers
140
views
Count shortest path with different lengths in random graph
Let $G(n,p)$ be an Erdos-Renyi random graph on $n$ vertices with probability $p$, i.e. for each pair of vertices, they are connected directly by an undirected edge with probability $p$. Suppose we are ...
- 1,495
1
vote
0
answers
255
views
Multiple Bipartite graphs and matchings
I've been told recently that it's better i just for help regarding my 'specific' problem rather than lots of little questions around the same topic which appear somewhat unclear. I would first like to ...
- 903
1
vote
1
answer
353
views
Probability of each edge in K-clique [closed]
For $c \in R$ and $k \in N$, $k \geq 3$ let
$p_{k,c} := n^{\frac{−2}{k+1}}log^c(n)$.
I would like to prove that exists $c\in R$ such that every edge in the random graph $G(n,p_{k,c})$ lies in a copy ...
- 75
0
votes
1
answer
3k
views
How to compute the clustering coefficient of a random graph?
How is the clustering coefficient defined for random graphs? For example, a first definition could be calling clustering coefficient of a random graph the expected value of the clustering coefficient ...
- 23
0
votes
0
answers
216
views
Computation on Random Bipartite graphs
I'm looking at a random bipartite graph $K_{\omega(n)}*K_{\omega(n)}$ where $\mathrm{log}(n)\leq \omega(n) \leq n^{1/2}$, in which each of the $\omega(n)^{2}$ edges is placed randomly with probability ...
- 903