The Distribution of Cycle Lengths in Graphical Models

This paper analyzes the distribution of cycle lengths in turbo decoding and low-density parity check (LDPC) graphs. The properties of such cycles are of signi cant interest in the context of iterative decoding algorithms which are based on belief propagation or message passing. We estimate the probability that there exist no simple cycles of length less than or equal to k at a randomly chosen node in a turbo decoding graph using a combination of counting arguments and independence assumptions. For large block lengths n, this probability is approximately e 2k 1 4 n ; k 4. Simulation results validate the accuracy of the various approximations. For example, for turbo codes with a block length of 64000, a randomly chosen node has a less than 1% chance of being on a cycle of length less than or equal to 10, but has a greater than 99:9% chance of being on a cycle of length less than or equal to 20. The e ect of the \S-random" permutation is also analyzed and it is shown that while it eliminates short cycles of length k < 8, it does not signi cantly a ect the overall distribution of cycle lengths. Similar analyses and simulations are also presented for graphs for LDPC codes. The paper concludes by commenting brie y on how these results may provide insight into the practical success of iterative decoding methods.