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The study of probability distributions over graphs. For example, the Erdős–Rényi model where each edge occurs independently with equal probability.

3 votes
0 answers
83 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 component, …
Michal Kotowski's user avatar
10 votes
3 answers
1k views

Eigenfunctions of random graphs

Consider a random $d$-regular graph on $n$ vertices. What can be said about its nontrivial (i.e. orthogonal to the constant) eigenfunctions? For example, I'm interested whether there are "nodal zones" …
Michal Kotowski's user avatar
8 votes
2 answers
977 views

Spectrum of the Laplacian on G(n, p) and G(n, M)

A random graph in $G(n, p)$ model is a graph on $n$ vertices in which for each of the $n\choose{2}$ edges we independently flip a coin, then take the edge with probability $p$ or remove it with $1 - p …
Michal Kotowski's user avatar
10 votes
3 answers
4k views

Random bipartite graphs

Consider the following situation: I have a set $A$ of $n$ vertices and a set $B$ of $N = n^2$vertices. I consider the bipartite graph $(A, B)$ and put at random $M = n^{1 + \varepsilon}$ edges (or I c …
Michal Kotowski's user avatar