All Questions
Tagged with random-graphs reference-request
32 questions
2
votes
2
answers
165
views
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, ...
- 59
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
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
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( ...
- 767
1
vote
0
answers
88
views
In percolation on a lattice, how is "infected" status correlated for points in a region around the origin?
Consider independent bond percolation on $\mathbb{Z}^2$, with $p>p_c$ so that the process is supercritical. For any site $x$ let $Y_x$ be the indicator of $x$ belonging to the infinite open cluster....
- 1,068
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
0
answers
72
views
Another betweenness centrality measure: neighbourhood centrality
Among the many centrality measures that I have heard of, I miss the following (but maybe I'm just blind).
Consider a graph $G$ with $k$ connected components $G_i$ of size $|G_i|$. The number of node ...
- 9,720
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
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
2
votes
1
answer
124
views
Relation between random graph models $G^{(B)}_{n,m}$ and $G_{n,m}$
In Frieze, Alan; Karoński, Michał, Introduction to random graphs, in Section 1.3 Pseudo-Graphs, there is a model of random multi-graphs, which is denoted as $\mathbb{G}^{(B)}_{n,m}$.
Def. A random ...
- 655
1
vote
1
answer
192
views
Survey/references on random geometric $K$-NN – $K$-nearest-neighbour graphs?
[Edit:] Some related info on number of connected components of NN-graphs can be found here: https://cstheory.stackexchange.com/a/47037/2408
Sample $N$ points in $\mathbb{R}^d$ from some distribution $...
- 24.8k
3
votes
1
answer
230
views
Unique maximum degree in tournament
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 $p(n)$ denote ...
- 1,209
1
vote
1
answer
104
views
Limit of normalized sum of Dirac measures at first $\lfloor p/2\rfloor$ eigenvalues of the sample covariance matrix, with Marcenko-Pastur assumptions?
Let $\lfloor{*}\rfloor$ denotes the nearest integer $\le *$. I'm asking myself the question what's the limit of the part of the empirical spectral distribution corresponding to the first $\lfloor{p/2}...
- 1,465
8
votes
1
answer
447
views
Graphons and Graphs
The situation is as follows: assume we have a sequence of simple weighted graphs $(G_n)_{n\in\Bbb{N}}$. For the terminology that follows I refer to Limits of dense graph sequences by László Lovász and ...
- 81
2
votes
1
answer
154
views
Electrode assignment problem in resistive networks
Main question
In the context of resistor networks, and particularly purely from a graph theory point of view, is there a consistent way of assigning the two electrode nodes in order to compare the ...
- 649
6
votes
0
answers
116
views
The properties of almost all directed graphs
A mathematician on the forum previously requested a reference on human brains modelled as directed graphs. This makes sense as neurons are mostly unidirectional and I have been thinking about similar ...
- 3,871
3
votes
1
answer
596
views
Asymptotic formula for the number of connected graphs
It can be shown that the set of graphs with $N$ vertices $G_N$ has cardinality:
\begin{equation}
\lvert G_N \rvert = 2^{N \choose 2} \tag{1}
\end{equation}
Recently, I wondered how much bigger $\...
- 3,871
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
94
views
What is the expected distance between the sides of a random subgraph of the grid?
Let $G$ be the $n \times n$ grid, in which each vertex is connected to the vertices above it, below it and on either side. Let $G_p$ be the random subgraph of $G$ obtained by keeping each edge with ...
- 1,633
6
votes
1
answer
304
views
Citations graphs: what is known?
There has been much research related to web graphs and social graphs.
They can be thought of as a kind of random graphs, but the point is that
they are different from the well-known Erdős–Rényi model.
...
- 24.8k
2
votes
1
answer
97
views
References on the structure of bond percolation on the (finite) 2D-grid in the sub-critical regime (e.g p=1/10)
Would appreciate references to the most up-to-date results for the structure of bond percolation on the (finite) 2D-grid in the sub-critical regime (e.g, $p=1/10$).
Thank you.
- 131
0
votes
0
answers
80
views
Not exactly directed percolation
Is the following problem known/well-studies? I'm looking for references or a name that I can look up.
I start with $N$ cell, each one divides into two cells, each one of the new cells either dies ...
- 371
2
votes
1
answer
261
views
Minimum dominating sets in tournaments
It is known that in any tournament with $n$ vertices, there is a dominating set of size no more than $\lceil \log_2 n\rceil$. (See Fact 2.5 here.)
What about when the tournament is chosen ...
- 1,209
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^...
1
vote
1
answer
167
views
Vertex reachability in random graph
Given $n$ vertices, one of which is $z$. Consider a uniform random tournament: Between any two vertices $x,y$, with probability $0.5$ draw an edge from $x$ to $y$; otherwise draw an edge from $y$ to $...
- 1,209
4
votes
2
answers
254
views
first order languages over graphs (and other discrete models)
A lot of research has been devoted to the study of first order language of graphs (FO). Formulas in this language are constructed using variables $x, y,\dots$ ranging over the vertices of a graph, the ...
- 1,561
2
votes
1
answer
302
views
Finding loops and double edges ASAP in configuration model random graph
A common approach (at least in theory) to generating a random $n$ vertex graph uniformly subject to having a given (feasible) degree sequence $(d_i)_{i = 1}^n$ is to use the configuration model, i.e. ...
- 253
0
votes
0
answers
107
views
Maximum Independent set of sparse graphs with few triangles
Notations used
$\alpha(G) = $ Max sized independent set of graph $G$.
$n(G) = $ Number of vertex in graph $G$.
Theorem (by Ajtai et al.): For a triangle-free graph $G$ and max degree being $\Delta$,...
- 189
1
vote
0
answers
372
views
counting k-cliques not also (k+1) on random graphs
consider the set of graphs with $n$ vertices and exactly half of all $\binom n 2$ possible edges.
looking for a formula that counts the number of these graphs that have a $k$-clique but not a $(k+...
- 529
4
votes
2
answers
675
views
between "giant-component" and "fully connected"
This is a request for reference. Where can I find discussion of the Erdős–Rényi random graph in the regime between "giant-component" and "fully connected"?
e.g. a detailed picture for say, $p_n=\frac{(...
- 103
29
votes
3
answers
2k
views
Growing random trees on a lattice $\rightarrow$ Voronoi diagrams
Imagine growing trees from $k$ seeds on a square $n \times n$ region
of $\mathbb{Z}^2$.
At each step, a unit-length edge $e$ between two points of
$\mathbb{Z}^2$ is added.
The edge $e$ is chosen ...
- 151k
3
votes
0
answers
156
views
Large Deviation Bounds for Number of Forests (or Tutte polynomial) in G(n,p)
Does anyone know of results/references related to large deviation bounds on the number of subforests (or the Tutte polynomial) in G(n,p) (Erdos-Renyi random graphs)?
- 53