Probabilistic analysis of cycles in random Tanner graphs

Cycles in bipartite graphs (also called Tanner graphs in channel coding field) are of particular interest in modern coding theory, especially in capacity-achieving low-density parity-check (LDPC) codes. In this paper, the expected number of cycles of various lengths in randomly constructed regular and irregular Tanner graphs are calculated. For a given degree distribution, the expected number of cycles in randomly constructed Tanner graphs has negligible changes with the size of Tanner graphs. Based on tree expanding, we propose a cycle counting algorithm (CCA) for Tanner graphs. Numerical results by counting the average number of short cycles over 200 random ensembles using the CCA give convincing supports to our analysis.