All Questions
Tagged with pr.probability graph-theory
290 questions
7
votes
1
answer
191
views
Is there a Degenerate Dependency Local Lemma?
The Lovasz Local Lemma has several generalizations, with names usually starting with L, such as Lopsided or Lefthanded.
Here I ask whether another possible generalization (for which I could not yet ...
- 341
3
votes
2
answers
440
views
Graph game minimum vertex degree
Consider the following graph game, given a graph $G=(V,E)$ on $n$ vertices with minimum degree $ \gg \log(n)$. Players are BR and MA (BR moves first):
BR claims an unclaimed edge from $E$, adds it to ...
- 208
5
votes
0
answers
136
views
What's the variance in the Six Degrees model?
Recall the six degrees of Kevin Bacon game. You can even play the game at The Oracle of Bacon, and their search works via Breadth First Search.
I interpret the punchline as saying that if I start ...
CommunityBot
- 1
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
2
votes
1
answer
115
views
mean length of the non-crossing graphs on n points
My original question is rather vague so I'll start with a precise example and then indicate possible generalisations.
Given a n-tuple $x=(x_1,\dots,x_n)$ in, say, a square with side-length $1$ in the ...
CommunityBot
- 1
2
votes
2
answers
220
views
Removing subtrees
Let $T$ be a complete infinite rooted binary tree. Is it possible to remove (infinitely many) subtrees of $T$ and get a subgraph $G$ such that:
$G$ has no complete subtrees (the graph below any ...
- 370
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 ...
- 30.1k
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
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 ...
- 267
1
vote
1
answer
330
views
Probability of connected graph on torus
Let $G = (V, E)$ be a graph on n vertices constructed in the following way:
Each vertex $v \in V$ is positioned uniform randomly in $[0, 1] × [0, 1]$.
Connect two vertices $u, v \in V$ if $d(v,u) ≤ ...
- 680
2
votes
0
answers
66
views
Fixing (non)-independency of a the subfamilies of finitely many events.
I'm would be interesting in any construction of a probability space with n events (n is given), where for every subset of these events, it is also given whether or not, the family is mutually ...
- 24.8k
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
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)...
CommunityBot
- 1
2
votes
0
answers
83
views
Asymptotic results in unbalanced left $d$-regular expander graphs
Let $U = [n]$ and $V = [m]$ be sets of nodes with $n > m$ and $E = U\times V$ be a set of edges. Let $\mathcal{N}(S)$ be the set of neighbors of a subset $S$ from $U$ or $V$.
Call a graph $G = (U, ...
- 409
7
votes
2
answers
335
views
Wait time to grid network disconnection with failing edges
Let $G_n$ be an $n \times n$ planar toroidal grid graph, with each node
connected to its four neighbors, with the top row connected to the bottom,
and the right column connected to the left.
Suppose ...
- 6,101
4
votes
2
answers
2k
views
Do Random Walks on the Hexagonal Lattice have a limit?
For every positive integer $n$, consider a regular hexagon $\mathrm{H}_n$ such that
the distance of each vertex from the center is $\frac{1}{\sqrt{n}}$. That in turn
induces a tiling of $\mathbb{R}^...
- 3,069
4
votes
0
answers
184
views
Does the concept of connective constant make sense for any tiling of the plane?
First let me define what is the "connective constant" of a two dimensional lattice.
Let $c_{n}$ denote the number of $n$ step self-avoiding walks starting from a fixed origin point in the lattice. ...
- 3,245
21
votes
6
answers
3k
views
"The" random tree
One time I heard a talk about "the" random tree. This tree has one vertex for each natural number, and the edges are constructed probabilistically. Connect vertex $2$ to vertex $1$. Connect vertex $3$ ...
- 437
6
votes
0
answers
172
views
Uniformly sampling from the set of all simplicial maps
Let $K$ and $L$ be finite simplicial complexes that remain fixed throughout.
How does one efficiently sample (according to the uniform distribution) elements from the finite set of simplicial maps ...
- 15.5k
1
vote
1
answer
313
views
Expected length of the shortest polygonal chain connecting N random points in the unit square
N points are selected uniformly at random in the unit square. Let L(N) be the expected length of the shortest (possibly self-intersecting) polygonal chain connecting all the points. It can be proved ...
CommunityBot
- 1
12
votes
1
answer
419
views
Coloring $K_n$ via edge-weight sums
This is a question inspired by
and tangential to "A Question on 1, 2 ,3 Conjecture"—and certainly
much easier!
Suppose one assigns a random edge weight among $\{1,2,3,\ldots,k\}$ to each
edge ...
CommunityBot
- 1
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 ...
CommunityBot
- 1
4
votes
3
answers
4k
views
Stationary distribution for bipartite graph
I was wondering if there is any stationary distribution for bipartite graph? Can we apply random walks on bipartite graph? since we know the stationary distribution can be found from Markov chain, but ...
- 1,037
8
votes
3
answers
1k
views
Singularity of sparse random matrices
The following topic came up in conversation with my office-mate Lionel: Let $p$ be a fixed prime, $c$ a fixed positive real parameter and $n$ a large number. Consider a random $(0,1)$ matrix with ...
- 747
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....
12
votes
3
answers
782
views
Connectedness of random distance graph on integers
This is not my field, a friend needs the answer for the following question. Suppose we have a decreasing probability function, $p: N \rightarrow [0,1]$ such that $sum_n p(n) = \infty$. Take the graph ...
- 747
13
votes
1
answer
2k
views
Counting subtrees of a random tree ("random Catalan numbers")
Given a rooted tree $T$ and an integer $k \geq 1$, let $N_k(T)$ be the number
of subtrees of $T$ containing the root and having exactly $k$ nodes (take $N_k(T)=0$ if $T$ has less than $k$ nodes).
...
- 6,210
1
vote
1
answer
295
views
Equivalent Markov Random Fields
Hi,
Is it possible to have topologically different Markov Random Fields (few different edges) and yet yielding the same inference results ?
Thanks!
- 200
0
votes
1
answer
4k
views
Calculate the probability of winning for a selected tic-tac-toe player
I am not a mathematician, I am a programmer. Sorry, if formulation of the problem is inexact.
I want to calculate the probability of winning for a selected tic-tac-toe player.
I have a directed graph ...
- 73.2k
3
votes
0
answers
474
views
What is the expected Cheeger constant of a random graph?
Recall that the Cheeger constant (AKA isoperimetric constant) of a graph $G$ is the infimum of $\frac{\partial S}{vol S}$ over all subsets $S$ of $G$ with volume no larger than $vol(G)/2$. I would ...
- 29.2k
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 ...
- 655
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
2
answers
1k
views
Probability distribution over cluster size in Erdős–Rényi random graph.
My question is about the probability distribution over the possible size of the containing cluster of a randomly chosen node in an Erdős–Rényi random graph.
Let G(n,p) be an Erdős–Rényi random graph (...
- 248
6
votes
1
answer
644
views
Random path in a graph
Consider a finite graph $G$. I would like to define a random path between two vertices $s$ and $t$ of the graph $G$ by looking at a measure $\mu$ on all spanning trees. Then the probability of a given ...
- 4,432
6
votes
2
answers
720
views
Local concentration of measure on Erdos-Rényi graph
Let $G_n=(V_n,E_n)$ be an Erdos-Rényi random graph, precisely the vertex set is $V_n=(1,\dots,n)$ and the edge set is $E_n=(ij\in\mathcal{P}_2(V_n)\ |\ \epsilon_{ij}=1)$ where $(\epsilon_{ij})_{ij}$ ...
- 7,514
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 ...
CommunityBot
- 1
4
votes
0
answers
128
views
Metrized categories
Motivation: Let $\Gamma = (V,E)$ be a directed graph. To each edge $e \in E$, choose a value $\kappa^e \in \mathbb R$, representing the cost of transporting one unit of "stuff" through the edge. Let $\...
- 8,512
5
votes
1
answer
272
views
Random graphs nonisomorphic to unit distance graphs
I've encountered an interesting problem but can solve it only partially:
Prove that random graph $G\sim G\left(n,\frac cn\right)$, $c=const$, almost surely is isomorphic to some unit distance graph ...
- 14.4k
2
votes
0
answers
108
views
Shortest loop containing 0 in continuum percolation
I am interersted in continuum percolation with intensity $\lambda>0$. Formally, let $X$ be a Poisson point process in $\mathbb{R}^d$ with intensity $\lambda$ and $G$ the graph obtained by ...
- 1,352
5
votes
1
answer
774
views
Probabilities of a random walk exiting a set
Let $F$ be a finite connected set in a graph (soon to be the Cayley graph of a group) and $\mathrm{Ex}_x^F$ be the function on the vertices in $F^c$ which are neighbour to vertices in $F$ defined as ...
- 1,791
4
votes
0
answers
256
views
Graph distance of close points within the minimum spanning tree
My question is the following: Given $N$ uniform IID points $X=(X_1,...,X_N)$ on the unit cube of $\mathbb{R}^d$, take $X_1$ and another point, say $X_{(1)}$, "close" to $X_1$ (i.e. connected to $X_1$ ...
CommunityBot
- 1
0
votes
1
answer
816
views
Two different definitions of Erdos-Rényi random graph
There are two or more ways to define an Erdos-Rényi random graph. Let consider the following two:
1) $G_n=(V_n,E_n)$ with vertex set $V_n=(1,\dots,n)$ and edge set $E_n=(ij\in\mathcal{P}_2(V_n)\ |\ \...
- 293
2
votes
1
answer
141
views
Spanning subgaph with trivial Poisson boundaries
Assume $\Gamma$ is the Cayley graph of an amenable$^{*}$ group and that the simple random walk has non-trivial Poisson boundary$^{**}$. Is there a spanning connected subgraph $\Gamma'$ of some $k$-...
- 17k
12
votes
3
answers
552
views
Estimate on currents in Cayley graphs
Take a Cayley graph $\Gamma$ (thought of as an electrical network with all edges having equal resistance) and break one edge $e$ and put a battery there. (Assume the graph has only one end* so that ...
- 4,432
2
votes
2
answers
1k
views
expected number of cycles in a "random" bipartite directed graph
Consider a "random" bipartite directed graph where (1) on each side, the set of vertices has cardinality n and (2) for each vertex i, we add one (and only one) directed edge i->j at random (drawn ...
- 37.7k
3
votes
1
answer
443
views
What is the expected value for this
If there are $8$ random points in the plane whose horizontal coordinate
and vertical coordinate are uniformly distributed on the open interval
$\left(0,1\right)$, what is the expected largest size of ...
- 1,411
-2
votes
1
answer
283
views
How to work with infinite random graph(s) ?
Hi,
In the case where we are dealing with an infinite random graph (RG with infinite nodes).
How do we model/work with notions like degrees, degree distribution ? How are they defined ?
Thanks!
- 14k
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
2
votes
2
answers
304
views
Uniformly random planar map
Is there a way to sample a planar map uniformly at random? I am aware of the Cori-Vauquelin-Schaeffer bijection that can be used to sample and study uniformly random quadrangulations. There are other ...
- 1,411
3
votes
1
answer
376
views
The degrees in a random subgraph
Fix some positive integers $N$ and $d_k$, $k=1,2,\dots$ with $N=\sum_{k=1}^\infty d_k$.
Suppose you have a graph $G$ taken randomly uniformly among the set of all (unoriented) graphs with $N$ ...
- 28k