Generating Random Graphs with Large Girth

Abstract

We present a simple and efficient algorithm for randomly generating simple graphs without small cycles. These graphs can be used to design high performance Low-Density Parity-Check (LDPC) codes. For any constant k, α ≤ 1/2k(k + 3) and m = O(n 1+α ), our algorithm generates an asymptotically uniform random graph with n vertices, m edges, and girth larger than k in polynomial time. To the best of our knowledge this is the first polynomial algorithm for the problem.Our algorithm generates a graph by sequentially adding m edges to an empty graph with n vertices. Recently, this type of sequential process has been very successful for efficiently counting and generating random graphs [35,18,11,7,5,6].

The random k‐matching‐free process

“…For example, it is natural to ask about the k ‐path‐free process, or the restricted‐girth process where we greedily add edges keeping the girth above some value k . (The restricted‐girth process has already been studied for fixed k by Osthus and Taraz ; see also the work of Bayati, Montanari and Saberi on a slightly different process.) There are also natural generalizations of these processes to hypergraphs.…”

Generating Random Networks Without Short Cycles

“…Finally, we note that a preliminary and weaker version of our main result has appeared in proceedings of annual ACM-SIAM Symposium on Discrete Algorithms (Bayati et al 2009b). In particular, Theorem 3.1 of Bayati et al (2009b) only shows that the total variation distance between the output distribution (for a different version) of RandGraph and the uniform distribution converges to 0 as size of the graphs goes to ∞. But here, we characterize size of the total variation distance for any finite n, that is of order n −1/2+k(k+3)α .…”

A Sequential Algorithm for Generating Random Graphs

“…Consider one iteration of Algorithm GenerateGraph from step (1) to step (5). Let Ev 1 be the event that at least one of the fractions…”

“…This means Ev c 2 is when "Otherwise go to step (3)" is called. Therefore, P(Ev 1 ) ≤ 2δ < 0.5 and P(Ev 1 ) + P(Ev 2 |Ev For each graph G ∈ L(d) let Ev 2 (G) be the event that G is reported in step (5). Each graph G is reported with probability P(Ev 2 (G)|Ev…”

“…In a follow up work, Bayati et al [5] uses similar ideas to generate random graphs with large girth. These graphs are useful for designing high performance Low-Density Parity-Check (LDPC) codes (cf.…”