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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Probability that graph is bipartite in Erdős–Rényi model with constant probability.

Given $G(n,p)$ where $p$ is constant, how could we determine the probability that the random graph is bipartite? I have seen the approach shown here but I don't fully understand why setting $|A| = |B| = n$ is correct (although in my case, they would…
davidaap
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Help with proof that a binomial random graph with m edges is equally likely to be any uniform random graph with m edges

I'm currently working through Introduction to Random Graphs by Frieze & Karonski, and I'm struggling with some of the notation (and quite possibly some background concepts) in a lemma. Please note that I'm quite the newcomer to mathematics. I…
Louis Thibault
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Expectation of degree in Bernoulli graphs

Let $\mathcal{G}(n,p)$ be the Bernoulli graph ditribution of $n$ verteces with edge probability $p$. It is known that the degree distribution of such graphs is the binomial distribution. My question is the following: Let $G$ be a graph drawn from…
tst
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Square Archimedean Spiral

I saw somewhere an image of a “spiral of squares,” and I looked online to see how I could graph it myself. I found the Wikipedia page for the Archimedean Spiral and its equation, but that just gives a general spiral. I would like to know if there is…
MattChris
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Sparse graph seqeunces and deterministic distribtuions.

In these notes on p. 12, a sparse sequence of graphs is defined as follows: A graph sequence $(G_n)_{n \ge 1}$ is sparse if $\lim_{n \to \infty}P^{(n)}_k = p_k \;\;\; (k \ge 0)$, where $P^{(n)}_k$ is the proportion of vertices of degree $k$ in…
theQman
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Number of k-cliques in random graphs is Poisson

How do I show that number of k-cliques in G(n,p) has a Poisson distribution?
Abhratanu
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Theorem 2.6 Random Graphs Bollobas Second Edition (page 44) - Infinite duplicates Despite Isomorphism ??

As part of Theorem 2.6 Bollobas gives an example of a unique graph up to isomorphism for a countable vertex random graph with $P_k$ for each k (i.e. extension axiom) : $G_0$ = ($\mathbb{N}$,E) with E ={ij : i < j, $p_i$ | j}. Presumably this is the…
user239186
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Theorem 2.1 Random Graphs Bollobas Second Edition (page 36)

I am new to random graph theory, and is it correct that the proofs of theorem 2.1 of Random Graphs by Bollobas are brief and miss out several steps ? I am particularly interested in the probabilistic part (ii): Theorem 2.1 Suppose Q is a monotone…
user239186
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