Fig 1.
Samples of different types of networks.
(A) A sample network. (B) A sample bipartite network. (C) The bi-adjacent matrix of the bipartite networks in Panel B.
Fig 2.
An example of the working flow of the MF-based bipartite link prediction.
In the bi-adjacent matrices, present links, absent links, and unobserved links are represented with crosses, empty cells, and question marks, correspondingly. The bi-adjacent matrix of the original network is converted into a numerical matrix A, which is then decomposed into the product of P and Q. In the reconstructed matrix PQ, the confidence values of two node pairs with unobserved links (v1, v4) and (v1, v7) are 0.1 and 0.9 separately, so finally, we only predict a new link between (v1, v7).
Fig 3.
The comparison of the working flow of the raw MF method and our joint method.
The upper row depicts the working flow of the raw MF method, while the lower row depicts that of our method.
Fig 4.
An example of a bi-clique and a maximal bi-clique.
The sub-network framed out in red from the bipartite network on the left is a bi-clique, which corresponds to the rectangle framed with red dash lines in the bi-adjacent matrix shown on the right. However, it is not a maximal bi-clique because it is a sub-network of another bi-clique framed in blue. The latter bi-clique is a maximal bi-clique, for it corresponds to the rectangle framed with solid blue lines on the right, which is a maximal rectangle box filled with crosses with rows and columns permutable.
Fig 5.
An sample pair of overlapping bi-cliques and their structure hole.
Left: A bipartite network with two overlapping maximal bi-cliques framed out in red and blue, separately. Middle: The bi-adjacent matrix of the network on the left. The aforementioned maximal bi-cliques correspond to the red and blue rectangles, separately. The gray cells filled with dots represent their structure hole. Right: the concept lattice representing all maximal bi-cliques from the network. The aforementioned bi-cliques correspond to the concepts marked out in blue and red, separately.
Fig 6.
An example of the iceberg-shaped sections of the concept lattice corresponding to the bi-cliques with component sizes exceeds a threshold 3.
Fig 7.
An example of a chain-shaped cluster corresponding to a group of non-trivial overlapping bi-cliques.
The structure hole of each pair of concepts is marked with cells in different colors. It is clear that the structure hole of the maximum and minimum concept, i.e., concept a and d contains that of any other pair of concepts from the group.
Table 1.
The comparison of the time complexity of different bipartite link prediction methods.
Table 2.
The statistics of the experiment on the MovieLens and HetRec datasets.
Table 3.
The statistics of the experiment on CTD chemical-disease database.
Fig 8.
The AUC and AUPR scores with different sample rates.