Random regular graph

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A random d-regular graph is a graph from $\mathcal{G}_{n,d}$ , which denotes the probability space of all d-regular graphs on vertex set [n], where nd is even and [n]:={1,2,3,...,n-1,n}, with uniform distribution.

1 Pairing Model
2 Properties of Pairing Model
3 Properties of Random Regular Graphs
4 References

[edit] Pairing Model

Pairing Model, also called Configuration Model, is one of the most commonly used model to generate regular graphs uniformly. Suppose we want to generate graphs from $\mathcal{G}_{n,d}$ , where d is a constant, the pairing model is described as follows.

Consider a set of nd points, where nd is even, and partition them into n buckets with d points in each of them. We take a random matching of the nd points, and then contract the d points in each bucket into a single vertex. The resulting graph is a d-regular multigraph on n vertices.

Let $\mathcal{P}_{n,d}$ denote the probability space of random d-regular multigraphs generated by the pairing model as described above. Every graph in $\mathcal{G}_{n,d}$ corresponds to (d!)^n copies in $\mathcal{P}_{n,d}$ , So all simple graphs, namely, graphs without loops and multiple edges, in $\mathcal{P}_{n,d}$ occur uniformly at random.

[edit] Properties of Pairing Model

The following result gives the probability of a multigraph from $\mathcal{P}_{n,d}$ being simple. It is only valid up to $d = o (n 1 / 3)$ .

$P(\mathcal{P}_{n,d} \hbox{ is simple}) \sim \exp((1-d^2)/4)$ .

Since the expected time of generating a simple graph is exponential to d^2, the algorithm will get slow when d gets large.

[edit] Properties of Random Regular Graphs

All local structures but trees and cycles appear in a random regular graph with probability o(1) as n goes to infinity. For any given subset of vertices with bounded size, the subgraph induced by this subset is a tree a.a.s.

The next theorem gives nice property of the distribution of the number of short cycles.
For d fixed, let X_i=X_i,n (i>=3) be the number of cycles of length i in a graph in $\mathcal{G}_{n,d}$ . For fixed k>=3, X₃,...,X_k are asymptotically independent Poisson random variables with means $\lambda_i=\frac{(d-1)^i}{2i}$ .