The random-graphs tag has no wiki summary.
4
votes
1answer
103 views
Statistics of strongly connected components in random directed graphs
I'm interested in the statistics of strongly connected components in random directed graphs. However, I'm unable to find any results on this, partly because I don't know the terminology to search for.
...
3
votes
0answers
34 views
Probability of a giant component existing in a $G(n,p)$ random graph with $p=\omega(\frac 1n)$
Consider a random $G(n,p)$ graph where $p=\omega(\frac 1n)$, and let $x$ denote the probability that the graph has a connected component of size linear in $n$.
It is well known that $x$ tends to $1$ ...
1
vote
0answers
80 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 ...
1
vote
2answers
175 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 ...
1
vote
0answers
147 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 ...
1
vote
2answers
157 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 ...
0
votes
0answers
54 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 ...
1
vote
0answers
61 views
Matchings in random bipartite graphs
I was wondering if anyone could point me in the direction of a text or paper which would help deal with the following problem
Suppose i am given a $K_{\mathrm{log}(n)} \times K_{\mathrm{log}(n)}$ ...
1
vote
1answer
141 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 ...
4
votes
0answers
186 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 ...
0
votes
1answer
176 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 ...
9
votes
5answers
318 views
Do there exist sparse graphs with large crossing number?
Does there exist a sequence of graphs $\{ G_n \}$ such that
$G_n$ has $n$ vertices,
the number of edges of $G_n$ is $O(n)$, and
the crossing number of $G_n$ is $\Omega(n)$?
In particular, do ...
4
votes
1answer
155 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 $i \ne j$ ($p$ ...
2
votes
0answers
32 views
“drift” of a random graph $G(n,p)$ with $p=\alpha\ln{n}/n$
Suppose $G\sim G(n,p)$ with $p=\alpha\ln{n}/n$ for some large constant $\alpha$. I wish to show a certain "drift" property of $G$, which can roughly be phrased as follows: if $u$ is "not that far" ...
3
votes
1answer
117 views
Independent Sets in random geometric graphs
I was wondering if a lot is known about independent sets in Random Geometric graphs? Most google searches don't bring up much. In addition I'm interested in any algorithms used for finding independent ...
2
votes
1answer
202 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. ...
2
votes
1answer
70 views
Hamiltonicity of random graphs with high girth
We say that $G\sim G_{n,f}$ (for $f=f(n)$) if $G$ is chosen uniformly at random from all graphs on $n$ vertices with girth $g(G)\ge f(n)$. Is there any threshold function $F(n)$ such that when $f\ll ...
9
votes
0answers
330 views
First passage percolation on a random geometric graph in the large connectivity limit
Let $V_\rho\subset\mathbb{R}^2$ be a point set in the plane obtained from a Poisson process of density $\rho$. The random geometric graph $G_\rho$ is obtained from $V_\rho$ by connecting points that ...
2
votes
1answer
92 views
limiting empirical spectral distribution of the Laplacian matrix on an Erdos-Renyi graph?
Let $G$ be an Erdos-Renyi random graph (i.e. an edge ($ij$) exists with probability $0 < p < 1$ and all edges are independent). Let $L$ be the Laplacian matrix of this graph (i.e $L=D-A$, where ...
3
votes
0answers
92 views
Node covering in a random graph
Given $N$ nodes randomly placed in a $D\times D$ area, i.e., the position of each node is randomly chosen. Assume that both $N$ and $D$ are sufficiantly large.
An agent can move in the area at ...
0
votes
2answers
87 views
different way of selecting a random graph
Consider having a 'base' graph $G=(V,E)$ and selecting each vertex with independent probability $p$ and having the induced subgraph of $G$ with all 'selected' points as your random graph. Has this ...
6
votes
2answers
144 views
Convergence on a random graph
Assume a directed graph $G = (V,E)$ is drawn from a random graph distribution, for instance Erdős–Rényi's $G(n,p)$ (but with directed edges). Let $S:V\rightarrow\mathcal{P}(V)$ be the direct ...
4
votes
1answer
193 views
Expected number of connected components in a random graph
For a random graph G(n,p) what is the expected number of connected components? What is the probability distribution of this value?
I'm specially interested in what happens for small values of p, ...
1
vote
0answers
51 views
Simulation of disassortative random graphs
Recently I have been trying to find a succinct algorithm for generation of disassortative networks. The best I have found is the algorithm by Newman described in his paper "Mixing patterns in ...
7
votes
0answers
95 views
Longest induced cycles in random geometric graphs near criticality
We make a random geometric graph $X(n;r)$ as follows. Choose $n$ points uniformly, independently, in the unit square $[0,1]^2$, for vertices, and then connect a pair of vertices $\{ p,q \}$ by an edge ...
4
votes
0answers
56 views
Cycle removal process
Consider the following stochastic process for generating a forest: start from a complete graph on $n$ vertices and proceed to repeatedly remove the edges of uniformly chosen cycles. Formally, let ...
3
votes
0answers
172 views
Hitting edges in graphs at random and let them die with honor
Let $G=(V,E)$ be a finite simple 2-connected graph. Let $B\subseteq E$ be a set of bad edges of size $k:=|B|$. For each edge $e\in E$ we toss a fair coin ($p=1/2$) and if the outcome is head, we hit ...
7
votes
1answer
167 views
Erdős-Renyi graph restricted to largest connected component
Suppose we have an instance of Erdős-Renyi $G(n,p)$ graph with $p = d/n$. Thus the expected node degree is $d$ which we will fix, while letting $n \to \infty$. Then, there will be more than one ...
5
votes
6answers
430 views
Random planar, bipartite graphs
I have a need to generate random planar graphs none of which have an odd cycle,
i.e., bipartite graphs.
I know there is a substantial two-decade literature on random planar graphs, little with which I ...
3
votes
0answers
56 views
Using Crump-Mode-Jagers processes to get logarithmic bound on a random tree height
I am currently pursuing my PhD degree and in my research I came across a family of random trees. I need to prove a logarithmic asymptotic bound for the heights of such trees as their size grows. I ...
3
votes
1answer
272 views
Expected Value for a Connected Graph
Consider a connected graph of N nodes.
Assign randomly to each node a distinct number from 1 to N.
For each node consider the maximum adjacent value or itself if all adjacent values are smaller.
...
3
votes
3answers
361 views
random networks and prime numbers
I have been studying networks recently and accidentally came up with an heuristic approach towards the distribution of prime numbers. The prime number theorem states that
$\pi(n)\sim n/log(n)$ for ...
4
votes
3answers
175 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 ...
0
votes
0answers
60 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 ...
3
votes
0answers
63 views
Quasi-isometry of giant components in Erdos-Renyi graphs
Take two independent random graphs $G_1, G_2$ in the $G(n,p)$ model for $p = \frac{c}{n}$, $c > 1$ (the question probably makes sense also for $c=1$). Each of them will have a unique giant ...
3
votes
0answers
132 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 $i,j$, ...
5
votes
0answers
134 views
Behavior of eigenspaces of adjacency matrices of random graphs (not via perturbation theory)
For the sake of discussion, let us say that we have the adjacency matrix $A$ of a graph, on $n$ nodes, from a stochastic block model with 2 blocks. Another name for this (usually used in computer ...
7
votes
0answers
131 views
Can two random graphs be metrically embedded into one another?
Let $X, Y$ be two random graphs on $n$ vertices (say, in $G(n, p)$ model for some $p$). Can anything (expectation, value with high probabiity, ...) be said about $D(X, Y)$, where $D$ is the minimal ...
1
vote
1answer
543 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 ...
-3
votes
1answer
179 views
adjacency matrix of random geometric graphs [closed]
Consider a graph with N nodes. All nodes are distributed as a Poisson point process with density of λ in a L*L area. There is an edge between two nodes if and only if the distance between them is less ...
2
votes
0answers
92 views
Reachability in dynamic random graphs
There has been some advice on how to ask questions. So I reask my question.I have a problem related to graph theory, random graphs or random geometric graphs in special that really confuses me. ...
11
votes
3answers
607 views
What fraction of n-point sets in the unit ball have diameter smaller than 1?
This question is inspired by a recent talk by Matt Kahle on random geometric complexes.
Some simple notation: let $\mathcal{B} \subset \mathbb{R}^d$ be the unit ball in $d$-dimensional Euclidean ...
2
votes
2answers
262 views
Random graphs require O(n log(n)) edges until they are almost certainly fully connected - what are more concrete boundaries ?
As far as I understand my literature, the probability of a random graph being fully connected tends toward one as the number of edges approaches a value of size $O(n \log(n))$
The way I read this, ...
1
vote
0answers
61 views
Can this application of the moment closure method to epidemics on networks be made rigorous?
Short version: Consider the SI model of infectious disease spread on a random graph $G$ with a given degree sequence. Let $j$ be a vertex and let $i$ and $k$ be two of its neighbors. If $G$ ...
2
votes
1answer
269 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 ...
0
votes
1answer
61 views
Degree of a node by geometric random undirected graph
suppose nodes with radius R are distributed randomly in Area of size A, then how can we calculate the degree of each node by geometric random graph.
4
votes
1answer
565 views
How many distinct eigenvalues does a random graph have?
It is well-known that a random graph a.e. has diameter 2. It is also well-known that the number of distinct eigenvalues of a graph is at least the diameter plus one.
But what is known about the ...
5
votes
1answer
195 views
The structure of small components in random graphs with a given degree sequence
Background and definitions
Consider a random graph on $n$ vertices with a nicely behaved degree sequence. That is, letting $d_i(n)$ denote the number of vertices of degree $i$, suppose that for all ...
3
votes
0answers
66 views
Is there precedent in the literature for a variant of a random geometric graph where vertices are the centroids of discs placed by random sequential adsorption (RSA)?
Imagine I form a random graph by simulating random sequential adsorption (RSA) of discs (each with the same radius $r$) on $[0,1]^2$ until I cover the plane at a density $\leq U$, where $U \leq ...
0
votes
1answer
115 views
Is there a proper way to define a threshold vertex density for a random graph s.t. the graph is fully connected?
Imagine one generates some form of random graph (e.g. a random geometric graph) and via simulation, calculates the probability that there exists an edge-wise path between all vertices in the graph as ...
