Loading [MathJax]/extensions/MathMenu.js
On Computing the Number of Short Cycles in Bipartite Graphs Using the Spectrum of the Directed Edge Matrix | IEEE Journals & Magazine | IEEE Xplore

On Computing the Number of Short Cycles in Bipartite Graphs Using the Spectrum of the Directed Edge Matrix


Abstract:

Counting short cycles in bipartite graphs is a fundamental problem of interest in many fields including the analysis and design of low-density parity-check (LDPC) codes. ...Show More

Abstract:

Counting short cycles in bipartite graphs is a fundamental problem of interest in many fields including the analysis and design of low-density parity-check (LDPC) codes. There are two computational approaches to count short cycles (with length smaller than 2g, where g is the girth of the graph) in bipartite graphs. The first approach is applicable to a general (irregular) bipartite graph, and uses the spectrum {ηi} of the directed edge matrix of the graph to compute the multiplicity Nk of k-cycles with k <; 2g through the simple equation Nk = Σi ηik /(2k). This approach has a computational complexity O(|E|3), where |E| is number of edges in the graph. The second approach is only applicable to bi-regular bipartite graphs, and uses the spectrum {λi} of the adjacency matrix (graph spectrum) and the degree sequences of the graph to compute Nk. The complexity of this approach is O(|V|3), where |V| is number of nodes in the graph. This complexity is less than that of the first approach, but the equations involved in the computations of the second approach are complex and tedious, particularly for k ≥ g+6. In fact, the computational complexity of the equations increases exponentially with k. In this paper, we establish an analytical relationship between the two spectra {ηi} and {λi} for bi-regular bipartite graphs. Through this relationship, the former spectrum can be derived from the latter through simple equations with computational complexity constant in k. This allows the computation of Nk using Nk = Σi ηik /(2k) but with a complexity of O(|V|3) rather than O(|E|3).
Published in: IEEE Transactions on Information Theory ( Volume: 66, Issue: 10, October 2020)
Page(s): 6037 - 6047
Date of Publication: 23 July 2020

ISSN Information:

Funding Agency:


I. Introduction

Bipartite graphs appear in many fields of science and engineering to represent systems that are described by local constraints on different subsets of variables involved in the description of the system. In such a representation, the nodes on one side of the bipartition represent the variables while the nodes on the other side are representative of the constraints. One example is the Tanner graph representation of low-density parity-check (LDPC) codes, where variable nodes represent the code bits and the constraints are parity-check equations. In the bipartite graph representation of systems, the cycle distribution of the graph often plays an important role in understanding the properties of the system. For example, the performance of LDPC codes, both in waterfall and error floor regions, is highly dependent on the distribution of short cycles of the Tanner graph [1]–[12].

References

References is not available for this document.