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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understanding random graphs with specified degree distribution

I was self-reading this note Random Graphs as models of networks by Newman; Could anyone explain briefly what is happening in section $2$? I did not get the algorithm he described too. This may be due to my poor English understanding. It would be…
Myshkin
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3-cycle enumeration on random dot product graph

The graph is generated with $p=n^{-0.5}$,which $p$ means the probability the two random vector dot product is large enough to add an edge. Brute force algorithm run in expect time $O(n^2)$.I think the expect number of 3-cycles are O(n).Is there an…
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High probability of $\sqrt{\log n}$ degree in random graph

I’m stuck on a problem I’ve found in a book on random graphs. Appreciate any help: Let $G_{n,p}$ be a random graph with $p=\frac{c}{n}$ for some positive $c$. Show that with high probability there is a vertex with degree at least $\sqrt{\log n}$.
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Binomial converging to Poisson in branching process view of $G(n,p)$

I'm trying to understand this claim on page 205 of "The Probabilistic Method" by Alon and Spencer: Set $p=c/n$. A key observation is that $Z_1 \sim Bin[n-1, c/n]$ approaches (in $n$) the Poisson distribution with mean $c$. Further, in a more…
theQman
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Largest size of a complete bipartite sub-graph in a random graph

Let $G\in G(n,\frac{1}{2})$ be a random graph. What is the maximum number of edges of a complete bipartite graph that can appear as a subgraph in $G$ almost surely? Let's give an estimate in the following. Please first note that: 1) It is not hard…
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Behaviour of $G(n,M)$ as $n$ tends to infinity

In a random graph $G(n,M)$ where $n$ is the number of vertices and $M$ the number of edges, prove that as $n$ tends to infinity $P(G(n,M))=H(n,M))=1$ where $H$ is a graph with $M$ independent edges and $n-2M$ isolated vertices. This is a problem…
Koushik
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Erdos-Renyi conditioned on node degrees

Has anyone worked out what the distribution of an Erdos-Renyi graph/adjacency matrix is conditioned on the degree sequence of the graph? More precisely, if $A$ is the adjacency matrix of $G(n,p)$, that is $A_{ij} \sim \text{Bernoulli}(p)$ for $ 1…
passerby51
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Number of $K_4$ in $G(n, \sqrt{n})$

What can you say about the number of copies of $K_4$ in $G(n, \sqrt{\frac{1}{n}})$? What is the asymptotic probability that it is greater than $n$? Let me recall the binomial random graph model here. In the $G(n, p)$ model, a graph is constructed by…
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Branching process interpretation of giant component

I'm trying to understand the following intuitive interpretation of the relationship between branching processes and connected components in random graphs: What is a bit confusing to me is how to imagine the survival of a vertex in a random graph.…
theQman
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Neighbours in random graphs

In any random graph on n vertices for each pair of vertices i and j, we include the edge {i,j} in the graph with probability 1/2. Show that with high probability every 2 vertices have at least n/4 - √nlogn common neighbors. I do not really know…
TedMosby
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Generating a weighted random graph with a correlation between degrees per edge and edge weight

I would like to generate using a single, coherent mathematical formalism a weighted random graph, either a small-wolrd or a scale free graph, with a probabilistic distribution of weights and vertex properties, s.t. vertices with similar properties…
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Erdös-Rényi random graphs: Binomial approximation

I am working on extending the Erdös-Rényi paper "On the evolution of random graphs" (http://www.renyi.hu/~p_erdos/1960-10.pdf). In theorem 2a they calculate the number of isolated trees of order k. I cannot understand equation…
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Question about analyzing greedy algorithm for the max cut problem in random graphs

In https://lucatrevisan.github.io/teaching/bwca17/lectures/lecture02.pdf (Lemma 6), the professor claimed that: "With high probability over the choice of $G$ from $G_{n,\frac{1}{2}}$, the greedy algorithm finds a cut of size…
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why is this equation involving landau symbols true?

I am not understanding the following "equation": $$\left(N\left(1-O\left(\frac{kt}{n}\right)\right)\right)^{m-(k-1)t}=N^{m-(k-1)t}\left(1-O\left(\frac{mkt}{n}\right)\right)$$ $k$ and $t$ are constant positive integers, $N= {{n}\choose{2}}$. For $m$…
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Why a graph generated from a graphon is said to be dense?

Now, I'm studying about a Graphon proposed by Lovasz. Why a graph generated by a graphon is said to be dense? I cannot image this.