All Questions
73 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
979
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
14
votes
2
answers
988
views
Properties of Some Random Graphs
Working in a problem the following family of graphs appears naturally. Consider the set $A_{n}=\{1,2,3,\ldots,n\}$ and let $\mathcal{C_{n}}$ be the set of all permutations of $A_{n}$ of order $n$ (...
- 3,626
14
votes
2
answers
387
views
What are some useful invariants for distinguishing between random graph models?
Quite a few probabilistic algorithms for generating random graphs exist in the literature, such as:
The Erdős-Rényi model
The Stochastic Block model
The Watts-Strogatz model
The Barabasi-Albert model
...
- 29.2k
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
3
answers
743
views
Rainbow matchings (in random graphs)
Suppose we have an $(n,n)$-bipartite graph with edges colored with $k$ colors. Is anything known about the existence of rainbow matchings (i.e. a matching that uses each color exactly once, for $k=n$) ...
- 3,323
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 ...
- 6,937
10
votes
3
answers
4k
views
Random bipartite graphs
Consider the following situation: I have a set $A$ of $n$ vertices and a set $B$ of $N = n^2$vertices. I consider the bipartite graph $(A, B)$ and put at random $M = n^{1 + \varepsilon}$ edges (or I ...
- 2,449
10
votes
2
answers
2k
views
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 ...
- 103
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
9
votes
1
answer
1k
views
Correlation-Function for Random Graph Ising Model
For non-Ising'ers: Given a graph, we study the probability-distribution on the set of colorings ("Spin-up" and "-down") generated by a given correlation ("force to equality") between adjacient nodes (...
- 1,846
6
votes
2
answers
725
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
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
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 ...
- 61
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
301
views
Probability that a random multigraph is simple
Question.
Consider a given sequence of $n$ integers $d_1$, $d_2$, $\cdots$, $d_n$ with $\sum_i d_i$ even and $d_i\le n$ for all $i$. One may sample a random multi-graph having this degree sequence ...
- 1,444
6
votes
0
answers
149
views
Does squaring a directed random graph more than double its out-degree?
As far as I know, it is an unsolved question
whether or not this is true:
If $G$ is a directed an oriented graph, $G^2$ always has some node whose outdegree is at least
double that of its ...
- 151k
5
votes
3
answers
314
views
Tracking automorphism groups of graph processes
Start with an edgeless graph on $n$ labeled vertices, and note that the automorphism group is $\Sigma_n$, the symmetric group on $n$ elements. Now imagine that we randomly start throwing in all of the ...
- 15.5k
5
votes
3
answers
840
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.
...
- 1,444
5
votes
1
answer
222
views
Switching oriented paths in a graph
Consider an oriented graph (e.g. a finite part of the standard grid with some random orientations).
Each minute the following operation takes place: we choose uniformly randomly an ordered pair $(A,B)...
- 5,063
5
votes
1
answer
705
views
Expectation of ratio of functions of i.i.d. Bernoullis: a concentration question
Consider the following $n \times n$ symmetric matrix of i.i.d. Bernoulli random variables, $X_{ij}$. For $i=1,...,n$ and $i<j\le n$. Let $X_{ij} \sim \text{Bernoulli}(p)$ when $i \ne j$ ($p$ fixed)...
- 123
4
votes
2
answers
882
views
The probability distribution for vertex degree in a unit disc graph generated from random points on a plane
Imagine I cover an arbitrarily large plane with randomly placed points at some density $\rho$ s.t. the number of points in any randomly sampled area $A$ (of arbitrary shape and size) is $\approx A*\...
4
votes
1
answer
245
views
Probability of a vertex being a "degree-celebrity" in a random graph
If $G(n,p)$ is a random graph of the Erdös-Rényi model,
what is the probability that $\mathrm{deg}(v)\gt\mathrm{deg}(u)\ \forall u\in\mathrm{adj}(v)$
Please feel free to relate answers to other ...
- 13.2k
4
votes
2
answers
4k
views
Expected global clustering coefficient for Erdős–Rényi graph
What is the expected global clustering coefficient $\mathbb{E}[C_{GC}]$ for the Erdős–Rényi random graph (ER-graph) $\mathcal{G}(n,p)$ (expectation is over the ensemble of all ER-graphs) as $n \...
- 151
4
votes
1
answer
217
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 ...
- 1,561
4
votes
1
answer
567
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 ...
user130903
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 $p$...
- 147
4
votes
1
answer
365
views
Expected number of leaf nodes in some theoretical graph models
If a leaf node of a graph refers to a node having the degree of 1, how can one compute the expected number of leaf nodes of:
(A) a random graph (e.g., Erdos-Renyi graph),
(B) a small-world graph (e....
- 355
4
votes
1
answer
2k
views
Probability of two vertices being connected in a random graph
Consider a random directed graph with $N$ vertices where each vertex $v$ has exactly one link to some vertex (maybe to itself) $u$ with known probability $a_{vu}$. What is the probability of ...
- 219
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
4
votes
0
answers
183
views
Concentration inequality for function of independent Bernoulli r.v.'s (related to random graph)
Consider a random undirected graph on a set of $n$ nodes, say $\{1,2,\ldots,n\}$, such that the probability of edge between nodes $i$ and $j$ is $p_{ij}$ (we may assume $p_{ij}=o(1)$ for all $i,j$, i....
- 163
4
votes
0
answers
617
views
Expected number of components with multiple cycles in a subgraph of a square lattice
Short version
Is there an understanding of the emergence and subsequent disappearance of components with zero, one, or more cycles in a random subgraph of a square or cubic lattice, as the edge-...
- 1,444
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
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
2
answers
1k
views
Non-backtracking random walk in regular (finite) graphs
I know that many things are known when dealing with random walks on a finite (or even infinite) graph: mixing time, returns to origin, etc. All is based in the use of the Markovian property of such a ...
- 1,561
3
votes
1
answer
337
views
Eigenvalues of random graphs
At time $t=0$, let $G_n(V,E)$ be a graph with $n$ vertices and $m < n$ edges. Then there exists a unique symmetric adjacency matrix $A_n$ associated with $G_n(V,E)$, defined as follows: $a_{ij} = 1$...
- 101
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
761
views
Removing edges from Erdős–Rényi graph to make two nodes disconnected
Consider a Erdős–Rényi graph on $n$ nodes, say $1,2,\ldots,n$, ($n\geq 3$) such that the probability of edge between any two nodes is $c/n$. I wish to know if there is a result that says
"There ...
- 163
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. ...
- 435
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
1
answer
276
views
Zero-one law in binomial random graph model $G(n,p)$
Consider the binomial random graph model $G(n,p)$ with $0<p<1$. We say that $G(n,p)$ satisfies the Zero-One law if for every first order property $Q$ one has $\lim\limits_{n \rightarrow \infty} ...
- 39
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
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_\...
- 6,353
3
votes
0
answers
151
views
Sequential generation of any random graph
The high-level question is: can we generate any random graph with size $d$ using a Markov chain?
For example, let $X^{(0)} = (1,0,\ldots,0) \in R^d$ be the initial state, and $X^{(t+1)} = f^{(t)}(X^{...
- 1,127
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
110
views
Difference between two largest degrees
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 $x$.) Let $S$ be the ...
- 1,209
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
286
views
Behaviour of global clustering for common random graph models
In order to develop some intuition for some of the commonly used random graph models, I've been looking at the global clustering coefficient as a means of comparing them. In particular, for the ...
- 649
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
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 ...
- 1,407