Friday, October 6, 2017

Reading Notes: Community Detection in Bipartite Graphs

A bipartite graph consists of two sets $V_a$ and $V_b$ that unite to the set of vertices, and a set of edges $E\subseteq V_a\times V_b$. The bipartite adjacency matrix $B$ contains a 1 in position $(a,b)$ iff $(a,b)\in E$; the adjacency matrix $A$ of the graph is given by

$$A = \left[\begin{array}{cc} 0 & B \\ B^T & 0 \end{array}\right].$$

A classic paper is the one by Freeman, "Finding Social Groups: A Meta-Analysis of the Southern Women Data (2003)", who compared and analyzed 21 approaches to community detection.

Block Models

• A block model approach has been proposed by Doreian, Bagatelj, and Ferligolj in their paper "Generalized Blockmodeling of Two-mode Network Data (2004)". Their approach is based on modeling blocks via specific attributes (all-ones, all-zeros, row-regular, etc.) and then inferring the position of these blocks in $B$. They do so by proposing a sequential heuristic in which they minimize the difference between the ideal block model and the actual adjacency matrix (more details are here). Hence, they require not only to specify the number of communities, but also the type of blocks used in the model. Different choices yield different solutions, as they illustrate for the Southern Women dataset. Their approach can also be used for weighted graphs.
Stochastic block models assume that the number of edges between two nodes is drawn from some probability model, and that the probabilities are different for edges within blocks and between blocks. Usually, the block probabilities are estimated, and the blocks are inferred by maximizing the (log-)likelihood of the graph given the block model.
• The paper "Efficiently inferring community structure in bipartite networks (2014)" from Larremore, Clauset, and Jacobs proposes a block model in which the number of edges are assumed to be drawn from a Poisson distribution, admitting different Poisson parameters for every block and every pair of blocks. They introduced a second model which admits modelling bipartite graphs with broad degree distributions. The authors suggested a sequential algorithm with a heuristic to escape local minima. The algorithm is implemented for MATLAB and R. The algorithm seems to work also for weighted graphs, but requires the numbers of blocks (i.e., communities) as input.

Spectral/Modularity-based Methods

Modularity is a measure of how well a graph can be split into modules. It is based on a null model (typically one in which the edges are randomized such that the vertex degrees remain constant) and the comparison of a given partition of the network to a partition of this null model. Modularity is usually maximized by spectral methods. Although these methods seem to work also for weighted graphs, the derivation of the null model is only justified for unweighted graphs (i.e., for $B$ comprising of 0s and 1s).
• Barber proposed a modularity measure for bipartite networks in "Modularity and Community Detection in Bipartite Networks (2007)". He proposed optimizing this measure recursively: Fixing the row partition, the column partition can be obtained easily, and vice-versa. He hence alternatively optimizes these two partitions, calling his algorithm BRIM. Barber also suggested a bisection approach to determine the number of communities. A fundamental drawback is that his method requires the same number of clusters for both rows and columns (see eq. (21) in the paper). The cost function has been used in a MATLAB implementation, which is described in this paper.
• A modularity measure was also proposed by Guimera, Sales-Pardo, and Nunes Amaral in "Module Identification in Bipartite and Directed Networks (2007)". They optimize their cost function via simulated annealing, but show that their results do not differ from modularity-based methods applied to $B^TB$.
• Dormann and Strauss proposed a simulated-annealing Monte Carlo method for maximizing modularity in "A method for detecting modules in quantitative bipartite networks (2014)". They measure modularity using Barber's definition. Their algorithm, QuanBiMo, is based on fitting a tree to a graph, a method proposed here. An R package is available.

Projection Methods

Projection Methods usually work on $B^TB$ (weighted projection) or on $\max\{1,B^TB\}$ (unweighted projection) rather than on $A$. In that sense, they work on less information than is available in the original model, but admit using community detection methods for unipartite graphs. If $B$ is binary, $B^TB$ and $BB^T$ contain all "relevant" information about $A$ (thus, $B$), hence a dual-projection approach could potentially work as well as an approach targeted at $A$. This, at least, is argued by Everett and Borgatti in "The dual-projection approach for two-mode networks (2013)".