Introduction

All graphs we consider are finite, simple, unweighted and undirected. They are also bipartite, which means their vertices can be partitioned into two partite sets so that the endpoints of each edge lie in different sets. In such a graph, a biclique is a complete bipartite subgraph, that is, a subgraph in which every subgraph vertex in one partite set is adjacent to every subgraph vertex in the other partite set. A biclique with p vertices in one partite set and q vertices in the other is denoted by Kp,q. A biclique is maximum if it is of largest size, with size measured by either its number of vertices (vertex-maximum) or its number of edges (edge-maximum). Finding a vertex-maximum biclique is NP-hard [1], while identifying an edge-maximum biclique can be accomplished in polynomial time [2]. A biclique is maximal if no vertex can be added to it to form a larger biclique.

The problem of enumerating all maximal bicliques has found utility in a host of applications. In the biological sciences, for example, it has been used for biclustering microarray data [3,4,5], modeling proteome-transcriptome relationships [6], identifying discriminating genotype patterns [7], optimizing phylogenetic tree reconstructions [8], discovering epidemiological patterns [9], identifying common gene-set associations [10], and integrating heterogeneous functional genomics data [11]. This problem is difficult in large part due to its combinatorial nature. A bipartite graph with n vertices may contain as many as 2n/2 maximal bicliques [12].

In previous work [13], we presented a fast, general-purpose algorithm for this task. We dubbed it the Maximal Biclique Enumeration Algorithm, MBEA, and presented along with it an improved version we termed iMBEA. In this paper, we describe a publicly available implementation of both algorithms wrapped in R [14]. Simply called Biclique, this R package invokes efficient implementations of MBEA and iMBEA written in C. Our goal is to provide the scientific community with a practical, convenient and efficient tool for finding all maximal bicliques in bipartite graphs.

Main text

Implementation

Biclique consists of four R functions. The core function, bi.clique, invokes an efficient algorithm to enumerate maximal bicliques. Three utility functions, bi.format, bi.print, and bi.degree, provide formatting and output support.

The bi.clique function takes five arguments, four of which have default values. These five are: an input file name, an input file format (either an edge list (the default) or a binary matrix), two arguments, one for each partite set, that specify the minimum number of vertices required for a maximal biclique to be reported (the default is 3), and an argument specifying the algorithm to use, either MBEA or iMBEA (the default is iMBEA). Pseudocode for MBEA and iMBEA is shown in Algorithm 1. Because iMBEA differs from MBEA by only a handful of additional steps, the two algorithms are presented jointly, with starred lines denoting the steps unique to iMBEA. On dense graphs, iMBEA will usually be the faster algorithm, while on sparse graphs, both algorithms are apt to take about the same amount of time. We therefore recommend the use of iMBEA in most cases. See [13] for a thorough discussion of the two methods.

The three utility functions operate as follows. The bi.print function generates a visual histogram of the distribution of sizes of the maximal bicliques enumerated by the most recent call to bi.clique. The bi.format function augments a list of edges with a header line declaring the number of vertices and edges the list contains, as is required by bi.clique. The bi.degree function reads a bipartite graph and outputs the degree of each vertex.

figure a

Application

Biclique is invoked in R as follows:

bicliques = bi.clique(filename, left_least, right_least, version, filetype)

This function generates a list of bicliques, which in the above example are assigned to the bicliques variable. The filename argument is the name of the input file. Using “left” to denote the first partite set and “right” to denote the second, the left_least and right_least arguments specify the minimum number of vertices required from each respective partite set in order for a maximal biclique to be reported. The version argument specifies whether to use MBEA or iMBEA.

The filetype argument can be a little more complicated. It specifies the input file format, which must be either an edge list (0) or a binary matrix (1). The default value is edge list. Such a list is tab-separated, with the first line declaring the number of vertices in each partite set, followed by the number of edges in the graph. Each subsequent line contains a pair of text labels for an edge, with the edge’s left endpoint listed first and its right endpoint second. The binary matrix format is also tab-separated. Example input files are provided with the package.

A sample bipartite graph is depicted in Fig. 1, where vertices u1, u2, u3, u4 and u5 are in the left partite set, while v1, v2, v3 and v4 are in the right. This graph is encoded as graph.el, shown in Table 1.

Fig. 1
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A sample bipartite graph

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Table 1 The encoding of graph.el, stored in edge list format
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The use of bi.clique is exemplified in Sample invocation 1, where graph.el denotes the sample graph just illustrated and encoded. Since neither left_least nor right_least is specified, all maximal bicliques with at least one edge will be reported. Similarly, since no version argument is declared, iMBEA will be invoked by default. And since no filetype argument is provided, graph.el is assumed to be in edge list format. Summary information returned by bi.clique comprises a listing of the input’s biclique distribution, its total number of bicliques, and its vertex- and edge-maximum biclique sizes.

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Biclique is available on CRAN at https://cran.r-project.org/web/packages/biclique/index.html. Included is an R-style reference manual with detailed descriptions of all arguments and options. This stable, CRAN-ready version can be installed in R with the command install.packages(“biclique”). The latest version of Biclique can be obtained via devtools::install_github(“YupingLu/biclique”). Questions or bugs can be submitted to the GitHub webpage. Included in the package are several example bipartite graphs, most of which we obtained from the Koblenz Network Connection [15].

Tests

All tests were conducted on a Dell server with an Intel Xeon E3-1220 v5 3.0 GHz processor under the Red Hat Enterprise Linux 7 operating system, with 16 GB DDR4 SDRAM, using. R 3.4.2. C code compiled with gcc 4.8.5. Eight bipartite graphs obtained from [15] were studied. As shown in Table 2, timings on them ranged from 0.005 s to 21.094 s. These tests were not meant to be comprehensive, but instead merely to demonstrate that this software can handle affiliation graphs, authorship graphs, interaction graphs and others in addition to the various biological and random graphs tested in [13].

Table 2 Timings on eight sample bipartite graphs
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Conclusions

Biclique provides convenient access, through R, to cutting-edge algorithms for maximal biclique enumeration in bipartite graphs. It provides users with a means to extract relationships between pairs of heterogeneous entities, without a need to worry about implementations of complex codes such as MBEA/iMBEA. Biclique also produces extremal information, including the sizes of vertex-maximum and edge-maximum bicliques. Biclique has been tested on a variety of graphs, and is available on both CRAN and GitHub.

Availability and requirements

Project name: Biclique. Project home page: https://github.com/YupingLu/biclique. Operating system(s): Platform independent. Programming language: R. Other requirements: R version 3.4.0 or later is recommended. License: GNU General Public License version 2.0 (GPL-2). Any restrictions to use by non-academics: None.

Limitations

Biclique enumeration can be output bound. The number of bicliques in large, dense graphs can exceed machine memory limitations.