@article{brandes2008modularity,
  abstract = {Modularity is a recently introduced quality measure for graph clusterings. It has immediately received considerable attention in several disciplines, particularly in the complex systems literature, although its properties are not well understood. We study the problem of finding clusterings with maximum modularity, thus providing theoretical foundations for past and present work based on this measure. More precisely, we prove the conjectured hardness of maximizing modularity both in the general case and with the restriction to cuts and give an Integer Linear Programming formulation. This is complemented by first insights into the behavior and performance of the commonly applied greedy agglomerative approach.},
  author = {Brandes, U. and Delling, D. and Gaertler, M. and Gorke, R. and Hoefer, M. and Nikoloski, Z. and Wagner, D.},
  doi = {10.1109/TKDE.2007.190689},
  interhash = {b7195d25a851617a48d4f15bef5ad789},
  intrahash = {9e2e5f9d06d2f83be98083175560c835},
  issn = {1041-4347},
  journal = {Knowledge and Data Engineering, IEEE Transactions on},
  month = {feb. },
  number = 2,
  pages = {172 -188},
  title = {On Modularity Clustering},
  url = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4358966&tag=1},
  volume = 20,
  year = 2008
}

@article{barber2007mac,
  abstract = {The modularity of a network quantifies the extent, relative to a null model network, to which vertices cluster into community groups. We define a null model appropriate for bipartite networks, and use it to define a bipartite modularity. The bipartite modularity is presented in terms of a modularity matrix B; some key properties of the eigenspectrum of B are identified and used to describe an algorithm for identifying modules in bipartite networks. The algorithm is based on the idea that the modules in the two parts of the network are dependent, with each part mutually being used to induce the vertices for the other part into the modules. We apply the algorithm to real-world network data, showing that the algorithm successfully identifies the modular structure of bipartite networks.},
  author = {Barber, M. J.},
  doi = {10.1103/PhysRevE.76.066102},
  interhash = {e1d9f528c49b34ff4a05b2b0060bd653},
  intrahash = {61f9d5839845d5d8fa1883a46a2f7744},
  journal = {Physical Review E},
  number = 6,
  title = {Modularity and community detection in bipartite networks},
  url = {http://arxiv.org/abs/arXiv:0707.1616},
  volume = 76,
  year = 2007
}

@article{guimera2007mib,
  abstract = {Modularity is one of the most prominent properties of real-world complex networks. Here, we address the issue of module identification in two important classes of networks: bipartite networks and directed unipartite networks. Nodes in bipartite networks are divided into two non-overlapping sets, and the links must have one end node from each set. Directed unipartite networks only have one type of nodes, but links have an origin and an end. We show that directed unipartite networks can be conviniently represented as bipartite networks for module identification purposes. We report a novel approach especially suited for module detection in bipartite networks, and define a set of random networks that enable us to validate the new approach.},
  author = {Guimer{\`a}, R. and Sales-Pardo, M. and Amaral, L.A.N.},
  doi = {10.1103/PhysRevE.76.036102},
  interhash = {a87821c7c8e7d5ca89cb369e6215a0f3},
  intrahash = {6145a42fe04aee556fa7a68c7cea7db3},
  journal = {Physical review. E, Statistical, nonlinear, and soft matter physics},
  number = {3 Pt 2},
  pages = 036102,
  publisher = {NIH Public Access},
  title = {Module identification in bipartite and directed networks},
  url = {http://arxiv.org/abs/physics/0701151},
  volume = 76,
  year = 2007
}