All Questions
9 questions
57
votes
4
answers
15k
views
Connectivity of the Erdős–Rényi random graph
It is well-known that if $\omega=\omega(n)$ is any function such that $\omega \to \infty$ as $n \to \infty$, and if $p \ge (\log{n}+\omega) / n$ then the Erdős–Rényi random graph $G(n,p)$ is ...
- 7,857
10
votes
2
answers
2k
views
Probability of Generating a Connected Graph
$N$ points are generated randomly within a unit square, with a uniform distribution.
What is the probability that the points form a connected graph, given that two points are connected if the distance ...
- 103
4
votes
2
answers
4k
views
Expected global clustering coefficient for Erdős–Rényi graph
What is the expected global clustering coefficient $\mathbb{E}[C_{GC}]$ for the Erdős–Rényi random graph (ER-graph) $\mathcal{G}(n,p)$ (expectation is over the ensemble of all ER-graphs) as $n \...
- 151
4
votes
0
answers
617
views
Expected number of components with multiple cycles in a subgraph of a square lattice
Short version
Is there an understanding of the emergence and subsequent disappearance of components with zero, one, or more cycles in a random subgraph of a square or cubic lattice, as the edge-...
- 1,414
4
votes
1
answer
245
views
Probability of a vertex being a "degree-celebrity" in a random graph
If $G(n,p)$ is a random graph of the Erdös-Rényi model,
what is the probability that $\mathrm{deg}(v)\gt\mathrm{deg}(u)\ \forall u\in\mathrm{adj}(v)$
Please feel free to relate answers to other ...
- 13.2k
3
votes
1
answer
192
views
Discrepancy of random bipartite graphs
This is a crosspost from MathStackExchange (original question).
Fix $k>0$ and let $X, Y$ be two vertex sets of size $n$ a positive integer (we're interested in the limit $n\to \infty$).
Define a ...
- 3,446
3
votes
1
answer
206
views
Component properties in Euclidean graphs with distance threshold
In the context of Euclidean graphs with vertices randomly embedded in either a 2D plane (for instance square with length $L$) or in 3D (similarly, cube of side $L$), where an edge between two given ...
- 649
2
votes
1
answer
426
views
Random subgraph properties
Consider a graph $G$ of $N$ vertices and $M$ edges, and assume $G$ has typical complex network properties: it is not necessarily connected, but it has a high clustering coefficient and a giant ...
- 129
1
vote
1
answer
152
views
Discrepancy of random bipartite graphs (2)
This question is a modification of the one asked here, which turned out to ask for something too strong to be true.
Given $k>0$ and a positive integer $n$, let $X, Y$ be two vertex sets of size $n$ ...
- 3,446