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.

860 questions
4
votes
1 answer

Distribution of a random vertex's neighbours' degrees

We choose a graph distribution $\mathcal D$. We draw a graph $g$ from $\mathcal D$. We choose a vertex $v$ uniformly at random. We look at its degree. What is the distribution of the degrees of the neihgbours of $v$ (conditioned on the degree of…
tst
  • 1,415
  • 9
  • 18
4
votes
1 answer

Mean cut size in generated graphs

Assume an undirected simple graph $G=(V,E)$ is created randomly: an edge $e=(u,v)$ is in $E$ with probability $p$, independent of other edges. Assume we select a random cut $(S,T = V\setminus S)$ in the graph. What is the expected value of its size?…
TheNotMe
  • 4,841
3
votes
1 answer

Bernoulli random graph as a special case of exponential random graph

Consider a Bernoulli random graph where we have $n$ vertices and we fill in each edge independently with probability $p$. Let $X$ denote a random variable which uniformly draws one such Bernoulli graph. The value of $X$ is an adjacency matrix. How…
rehband
  • 1,921
3
votes
1 answer

Probability of having a girlfriend in a school with groups

A school has $r$ groups. Each group has $n$ girls and $n$ boys. Any boy and girl know each other with probability $p$ if they belong to the same group, and with probability $q$ if they belong to different groups. In random order, a boy marries a…
fox
  • 679
  • 7
  • 18
3
votes
2 answers

Expected number of isolated vertices for random graph G(n, N)

For a random graph G(n,p), I can understand how to calculate the E(X) where X is the number of isolated vertices. However, in the case of fixed N edges (N = cn), I am not sure how to proceed in the same calculation of E(X). I don't think finding p…
2
votes
1 answer

How to compute the average number of wedges and triangles present in the Erdős-Rényi random graph?

Let $ER_{n} (p)$ be the Erdős-Rényi random graph (see the second model in the link), where $p=\frac{\lambda}{n}$. Furthermore, let $W_{ER_{n}}$ be the number of wedges in the graph, and $\Delta_{ER_{n}} $ be the number of triangles in the graph. I…
Max Muller
  • 7,006
2
votes
0 answers

Expected path length in a Random Geometric Graph

Random Geometric graphs (graphs where n points are placed at random in the unit square, and two nodes are connected with probability 1 if $r \leq r^*$) are known to percolate iff: $$\pi r^2 = \frac{\log{n} + c\left(n\right)}{n}$$ This implies that…
Tom Kealy
  • 282
2
votes
1 answer

Number of spanning trees in random graph

Let $G$ be a graph in $G(n, p)$ (Erdős–Rényi model). What is the (expected) number of different spanning trees of $G$?
Croq
  • 35
2
votes
3 answers

Probability of not having a path between two certain nodes, in a random graph

Suppose we construct an Erdős–Rényi graph $G(n, p)$. Fix two nodes $u$ and $v$. What is the probability that there is no path connecting the two nodes? My take: I tried to model the problem as $P(\text{no path between } i \text{ and } j) = 1 -…
Daniel
  • 2,630
2
votes
2 answers

How many paths of length 2 in a general random graph?

Suppose you have several random graphs. Each one has $n$ nodes, connected among them with probability $p$. There are $r$ random graphs. Now, each node is connected to nodes of another random graph with probability $q$, in general different from $p$.…
fox
  • 679
  • 7
  • 18
2
votes
1 answer

"Non random graphs"

In my recent study about random graphs, I learned that there are numerous models:Bernoulli, Binomial, Watts-Strogatz, Barabasi-Albert, exponential models...etc. Now I have an idea about random graphs, what I have a problem with are "non random…
2
votes
1 answer

Confused about this notation about random graphs in Bollobas' book

After Theorem 2.1 on page 36 (second edition) he states Note that the argument above shows that if $Q$ is a nontrivial monotone increasing property then $$P_{p_2}(Q) \ge P_{p_1}(Q) + \{1 - P_{p_1}(Q)\}P_p(Q) \ge P_{p_1}(Q) + P_{p_1}(E^n)P_p(K^n)…
Fequish
  • 709
2
votes
2 answers

Max matching size in a random graph

Consider the random graph $G(n,\frac{1}{n})$. I'm trying to estimate the size of the maximum matching in $G$. If we look at one vertex, the expected value of its degree is $\frac{n-1}{n}$ so it seems like with high prob it should be 1. So if I can…
Snufsan
  • 2,125
1
vote
2 answers

Highest degree vertices in random graph with hidden clique

We create a random graph $G(n,1/2)$. Next, we choose randomly $k$ vertices, and form a clique out of these $k$ vertices. My question is: how do you prove (with high probability) that when $$k > C {\sqrt{n{\log(n)}}}$$ for some sufficiently large…
1
vote
0 answers

What is the expected distribution of sphere sizes in random graphs with constant degree?

Let $G=(V,E)$ be a random undirected, unweighted graph where each vertex $v\in V$ has degree $n>1.$ Let the $k$-sphere $S(v_0,k)=\{\,v\mid v\in V,\;d(v,v_0)=k\,\}$ of $v_0$ be the set of vertices at distance $k$ from $v_0$. How can we estimate the…
FUZxxl
  • 9,307
1
2 3