All Questions
Tagged with pr.probability graph-theory
290 questions
1
vote
0
answers
45
views
What is the number of iterations needed for the message passing algorithm to converge when applied to an acyclic factor graph?
I understand that the message passing algorithm (Belief Propagation algorithm), when applied to a factor graph consists in an exchange in messages between the factor nodes and the variable nodes, ...
- 39
1
vote
0
answers
61
views
What is the minimal $m$ for which the independence graph is $n$-universal?
Suppose, an $m$ sided die is rolled. Let's define the independence graph $I_m$ as a graph with the set of all possible events as vertices, and edges between two events iff they are independent.
...
- 2,618
1
vote
0
answers
117
views
Entropy of endpoints of a random walk in a dense graph
Let $p\in[0,1]$ be a constant and let $G$ be a graph with $n$ vertices and $\approx p\binom{n}{2}$ edges. If you'd like, consider $p=1/2$.
Let $X$ be a random vertex of $G$ chosen proportional to ...
- 761
1
vote
0
answers
86
views
A Random Graph Process
I'd like to understand the following random graph process. I'm not sure if it's difficult or straightforward, so apologies if this is below the level of mathoverflow, but I've gotten no response on ...
- 345
1
vote
0
answers
43
views
a question about probabilities on spaces of digraphs
Let $G$ be a directed graph with fixed nodes $s$ and $t$. Assume that each edge $e$ in the graph comes with a number $n(e)\in[0,1]$.
We consider probability spaces $S$ whose points are directed ...
- 61
1
vote
0
answers
109
views
Number of $H$-free graphs
Sorry if this is basic for MO. But the people at SE couldn't help me.
I'd like to get an estimate on the number of (labeled) $H$-free graphs on $n$ vertices, i.e. graphs in which no set of $|V(H)|$ ...
- 11
1
vote
0
answers
87
views
How to estimate the size of balanced biclique in random bipartite graph?
We have a random bipartite graph $G=(V,U,E)$ and $|V|=|U|=n$, in which any vertex pair $<v,u>$ ($v\in V$,$u\in U$) exists an edge with probability $p$. A balanced bipartite complete graph is a ...
- 11
1
vote
0
answers
159
views
Probabilistic proof for expander existence [closed]
I am new to probabilistic proofs and trying to understand them better. Apparently, a common probabilistic proof focuses on the existence of expanders (eg. vertex expanders).
I've been using the search ...
- 111
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
1
vote
0
answers
46
views
Is there an effective algorithm for finding "minimal discovery times" for large graphs?
Consider a large, probably sparse graph with Markovian random walkers on it.
Define the discovery time as the expected time to first reach a vertex by
random walk from a uniform start. Are there ...
- 355
0
votes
1
answer
1k
views
Probability of an edge appearing in a spanning tree
Hi guys, let's say I have a connected, undirected graph with many nodes. I am interested in finding the probability that an edge appears in any spanning tree of the graph. I could apply some of the ...
0
votes
1
answer
463
views
Expected number of connected components as $V(G)$ grows large
Let $E^c_n$ be the expected number of connected components of a simple undirected graph on the vertex set $\{1,\ldots,n\}$. (Every possible edge in $\big\{\{a, b\}: a, b\in \{1,\ldots,n\} \land a \neq ...
- 48.9k
0
votes
1
answer
307
views
How to estimate the fraction of graphs with small clique among the graphs with certain edges
Among all $n$-vertex graphs with $M$ edges and constant $k$, how to estimate the fraction of graphs of clique less than $k$? Thanks.
- 3
0
votes
1
answer
128
views
Non-linear diffusion on networks
The diffusion equation with constant diffusion $D$ can be represented as:
\begin{equation}
\frac{\partial \phi(r, t)}{\partial t}=D \Delta \phi(r, t)
\end{equation}
where
$\Delta$ is the Laplace ...
- 117
0
votes
1
answer
560
views
Random walk on the hypercube
Consider the hypercube $Q_4$. I would like to know how to compute the number of steps of a random walk in this graph such that the probability to be at a vertex is a given number $x$. I think I just ...
- 119
0
votes
1
answer
75
views
The probability of generating a ring graph by following the Erdos-Renyi model G(N,p) [closed]
The Erdos-Renyi random graph model G(N,p) describes a way to generate a network with N nodes, the probability that there is a link between any two nodes is p. I am wondering about the probability of ...
- 13
0
votes
1
answer
77
views
Fourth moment of a random-variable with block-tridiagonal structure
Let x be a random variable in $\mathbb{R}^d$, $J$ a block tridiagonal $d\times d$ matrix, and probability of $x$ is defined as follows
$$p(x)\propto \exp(-x'Jx)$$
For a fixed $d\times d$ matrix $v$ ...
- 2,252
0
votes
1
answer
181
views
Bound on queries to a tree with unusual probabilties -- follow-up
This question follows up on Bound on queries to a tree with unusual probabilities, where @fedja was able to disprove my conjecture under only constraints (1-4) below. I restate the relevant facts here ...
- 209
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
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 ...
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
0
votes
1
answer
292
views
Probability of preserving connectivity between pair of vertices in weighted graph
Let $G=(V,E)$ be an undirected graph and $p \colon E \mapsto (0,1]$ defines weights of its edges.
Let's fix two connected vertices $v_1, v_2 \in V$.
Random graph $G'=(V,E')$ is obtained from $G$ by ...
- 3
0
votes
0
answers
45
views
Functional inequalities on neighbourhood graphs
Consider an open domain $\Omega \in \mathbb{R}^d$, say the unit disk in $\mathbb{R}^2$ with $N$ points sampled i.i.d. on it. One of the simplest possible (unnormalised) discrete Laplacian of a ...
- 111
0
votes
0
answers
63
views
Arrangements of fixed $k$-polyplets in a $n\times n$ matrix
Recently, I asked a question about the number of arrangements of $k$ elements inside a $n\times n$ matrix with certain restrictions. The one I´m actually interested in for this question is in its 2. ...
- 47
0
votes
0
answers
55
views
Counting matrix paths for (n,m>2) matrices
Given a $n\times m$ matrix with $k$ elements inside it, I need to calculate the number of arrangements of those $k$ elements that form at least 1 path from the top to bottom matrix row composed of the ...
- 47
0
votes
0
answers
94
views
"Cut norm" of conditional expectation has supremum on products of sets in sub-$\sigma$-algebra, or not?
I am reading Lovasz's book "Large networks and graph limits", and encountered the exercise that the stepping operator for graphons is contractive under the cut norm:
$$||W_P||_\square\leq||W|...
- 715
0
votes
0
answers
133
views
is there an example in planar graph that using probabilistic methods
The probabilistic method is a technique for proving the
existence of an object with certain properties by showing that
a random object chosen from an appropriate probability
distribution has the ...
- 1,851
0
votes
0
answers
39
views
hypergraph product that preserve expansion properties
I am looking for a hypergraphs product of hypergraph H1,H2 that preserves some expansion properties of H1,H2.
The expansion property I am looking at is HD-random walk.
The product I am looking for is ...
- 340
0
votes
0
answers
72
views
A random variable standing for the size of connected component including a given node in a tree
Suppose we have a tree $T = (V,E)$, in which each nodes $v_i \in V$ has a probability $p_i$ to vanish. Let $v_0\in V$, we define random variable $\boldsymbol{X} = \boldsymbol{X}(T, v_0)$ stands for ...
- 1,551
0
votes
0
answers
34
views
What kind of prior on edge existence would form graphs that are unions of complete (sub)graphs?
Suppose a graph has $n$ vertices.
First question: is it possible to give a (nontrivial) prior probability on edge existence so that if a graph is created by querying the prior on the $\binom{n}{2}$ ...
- 118
0
votes
0
answers
168
views
A path optimisation problem
Consider a graph of $n$ nodes randomly located in $[0,1]^2$. Each node moves following a path randomly chosen from the set of all possible paths. Regard nodes as attackers. A policeman seeks an ...
- 367
0
votes
0
answers
165
views
Expected length of minimum spanning trees
For a simple, finite, connected and complete graph $K_n = (V(K_n), E(K_n))$ with vertex set $V(K_n)$ and edge set $E(K_n)$, we assign a non-negative independent and identical distributed random weight ...
- 1
0
votes
0
answers
320
views
Gromov-Hausdorff distance measure between minimum spanning trees
I am trying to compare minimum spanning trees through time. I have two questions:
1-Is it possible to measure the similarity between two minimum spanning trees with Gromov-Hausdorff distance measure ...
- 1
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
0
votes
0
answers
145
views
Finding expectation of size of a subgraph
I have been trying to implement a algorithm but got stuck in finding expectation of the size of the subgraph.
n - size of the network.
d - at most number of communities a node could participate ...
- 1
-1
votes
2
answers
421
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 ...
- 1
-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!
- 167
-2
votes
1
answer
181
views
Stationary distribution of a weighted directed acyclic graph
Is there any way to calculate the equilibrium (stationary) distribution for a weighted directed acyclic graph?
Some references emphasized adjacency matrix to be symmetric.
https://arxiv.org/abs/1012....
-3
votes
1
answer
144
views
Count arrangements with pairs of attacking kings [closed]
I have a $1\times n$ chessboard and $2$ pairs of kings in it. Both components of each pair of kings must be adjacent in the chessboard, that is, they must be attacking.
Now, I want to calculate the ...
- 47