On the Tanner Graph Cycle Distribution of Random LDPC, Random Protograph-Based LDPC, and Random Quasi-Cyclic LDPC Code Ensembles

In this paper, we study the cycle distribution of random low-density parity-check (LDPC) codes, randomly constructed protograph-based LDPC codes, and random quasi-cyclic (QC) LDPC codes. We prove that for a random bipartite graph, with a given (irregular) degree distribution, the distributions of cycles of different length tend to independent Poisson distributions, as the size of the graph tends to infinity. We derive asymptotic upper and lower bounds on the expected values of the Poisson distributions that are independent of the size of the graph, and only depend on the degree distribution and the cycle length. For a random lift of a bi-regular protograph, we prove that the asymptotic cycle distributions are essentially the same as those of random bipartite graphs as long as the degree distributions are identical. For random QC-LDPC codes, however, we show that the cycle distribution can be quite different from the other two categories. In particular, depending on the protograph and the value of <inline-formula> <tex-math notation="LaTeX">$c$ </tex-math></inline-formula>, the expected number of cycles of length <inline-formula> <tex-math notation="LaTeX">$c$ </tex-math></inline-formula>, in this case, can be either <inline-formula> <tex-math notation="LaTeX">$\Theta (N)$ </tex-math></inline-formula> or <inline-formula> <tex-math notation="LaTeX">$\Theta (1)$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> is the lifting degree (code length). We also provide numerical results that match our theoretical derivations. Our results provide a theoretical foundation for emperical results that were reported in the literature but were not well-justified. They can also be used for the analysis and design of LDPC codes and associated algorithms that are based on cycles.