Questions tagged [random-graphs]

A random graph is a graph - a set of vertices and edges - that is chosen according to some probability distribution. In the most common model, $G_{n, p}$, a graph has $n$ vertices, and edges are present independently with probability $p$. Use (graphing-functions) instead if your question is about graphing or plotting functions.

A random graph is a graph that is chosen according to some probability distribution. The most common model is $G_{n, p}$. A graph with this distribution has $n$ vertices, and edges are present independently with probability $p$. A closely related model is $G_{n,m}$. In this model, a graph on $n$ vertices is chosen uniformly among all graphs with $m$ edges.

The standard references for this area are Random Graphs by Bollobás and Random Graphs by Janson, Luczak, and Ruciński.

This tag refers to graphs in the sense of the tag: collections of vertices, some of which are connected by edges. Use instead if your question is about graphing or plotting functions.

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How likely is it not to be anyone's best friend?

A teenage acquaintance of mine lamented: Every one of my friends is better friends with somebody else. Thanks to my knowledge of mathematics I could inform her that she's not alone and $e^{-1}\approx 37\%$ of all people could be expected to be in…
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Why "One cannot construct more than countably many independent random variables"?

I'm reading the book "Large Networks and Graph Limits" by László Lovász. On the page 18 he said the following: One cannot construct more than countably many independent random variables (in a nontrivial way, neither of them concentrated on a …
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Size of connected regions on a randomly-colored infinite chessboard

Consider an infinite chessboard where each square is colored white with probability $p$ and black with probability $1-p$. Suppose without loss of generality that the square at $(0,0)$ is white. We can consider the entire connected region $W$ of…
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How many edges does an Erdős-Rényi graph have to have, to almost surely have a component with multiple cycles?

An Erdős-Rényi graph is a random graph, selected according to the distribution obtained one where we have some number $n$ of nodes, and some probability $p$ of each potential edge being present. One often asks questions of this distribution when $p$…
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What is the Probability of Transmission Between Two Nodes in a Neural Network?

I have a network which is an Erdős–Rényi graph. It is a simple neural network with degree 0.7N where N is the number of nodes. Each weight between neurons is $\frac{1}{N}$, meaning that if node n has fired the probability that any connected node k…
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Probability that a random graph is planar

I've been attempting to solve the following challenge problem from a combinatorics class but am getting absolutely nowhere. Prove: For sufficiently large $n$, the probability a random graph $G=(V,E)$ with $n$ vertices is planar is less than…
rm95
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Probability of the existence of a path of a specified length between any tw0 vertices in a random graph

Let $G$ be a graph with $n$ vertices, whose average degree is $k$. What is the probability that between any two vertices, there exists a path of length at most $l$? NOTE: For the above problem the random graphs follow a $G(n, p)$ model, i.e., in a…
S Maity
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Random walk on thin ice?

My Question: Is the stochastic process which is described below a (special case of a) well-studied model? What kind of properties are known under which assumptions? Let's think of a random walker on a frozen lake. Obviously, this lake is a subset…
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Expectation number of cycles in a Erdős–Rényi random directed graph $G(n,p)$

Let $G \sim G(n,p)$ be a directed Erdős–Rényi random graph with $n$ vertices and the probability $p$ that there is a directed edge between any two ordered pairs of vertices. What is the expected number of cycles in $G$? Is there an exact formula…
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Random graph probability lemma

I'm trying to prove a fiddly lemma for homework, but getting absolutely nowhere with it. Here, $G_{n,p}$ and $G_{n,m}$ represent, respectively, random graphs on $n$ vertices where the number of edges is binomially distributed with probability $p$…
T. Hughes
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What is the average weight of a minimal spanning tree of $n$ randomly selected points in the unit cube?

Suppose we pick $n$ random points in the unit cube in $\mathbb{R}_3$, $p_1=\left(x_1,y_1,z_1\right),$ $p_2=\left(x_2,y_2,z_2\right),$ etc. (So, $x_i,y_i,z_i$ are $3n$ uniformly distributed random variables between $0$ and $1$.) Let $\Gamma$ be a…
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What is the probability that a random $n\times n$ bipartite graph has an isolated vertex?

By a random $n\times n$ bipartite graph, I mean a random bipartite graph on two vertex classes of size $n$, with the edges added independently, each with probability $p$. I want to find the probability that such a graph contains an isolated…
Spook
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Literature recommendation on random graphs

I'm looking for introductory references on random graphs (commonly mentioned as Erdős–Rényi graphs), having previous acquaintance with basic graph theory. I know that Bela Bollobas' book on random graphs is the used reference, as are all his books…
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The probability of having a perfect matching in a bipartite graph

Say we have a bipartite graph $G$ with two sets, $\{x_1,\dotsc,x_n\}$ and $\{y_1,\dotsc,y_n\}$. For each pair $xy$, there is an edge with probability $p$. Then, what is the probability of having a perfect matching in $G$?
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Expected number of triangles in a random graph of size $n$

Consider the set $V = \{1,2,\ldots,n\}$ and let $p$ be a real number with $0
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