Community detection in bipartite networks is a popular topic. Two widely used methods to capture community structures in bipartite networks are the method of modularity and the method of graph partitioning. Our analytical results show that the modularity maximization problem can be reformulated as a spectral problem after relaxing the discreteness constraints. This means that the method of modularity and the method of graph partitioning are essentially equivalent. As an application, a spectral algorithm of modularity is devised for identifying community structures in bipartite networks. Experimental results on synthetic networks and real-world networks indicate that our algorithm performs better than those algorithms of modularity local maximization, such as BRIM (bipartite recursively induced moduls) and bLP (bipartite label propagation). Therefore, our results shed light on the methods of community detection in bipartite networks.

Investigating the community structures of bipartite networks is a frequent topic of discussion in the ecological and social fields. The most widely used methods, as proposed by numerous academics from varying perspectives, include spectral graph partitioning, modularity, non-negative matrix factorization, and stochastic block model. In this paper, we demonstrate three equivalences among these four methods. i) Both Dhillon spectral graph partitioning and Barber modularity clustering are equivalent to solving for the matrix's left and right singular vectors after relaxing the discrete constraints. ii) The non-negative matrix factorization clustering is equivalent to the Dhillon spectral graph partitioning. iii) The bipartite stochastic block model is equivalent to the constraint-based NMF that uses K-L divergence as its cost function. These equivalences, obtained through rigorous mathematical derivations, will aid in the future development of efficient algorithms for community detection in bipartite networks.

Cascaded coded distributed computing (CDC) reduces Shuffle communication through redundant Map computations and replicated output assignments. We propose Map–Shuffle–Reduce arrays (MSRAs), a unified framework for describing file placement, output assignment, transmitter selection, and one-shot XOR delivery. For fixed placement, assignment, and equal-length packetization, minimizing communication load is equivalent to minimizing the number of distinct array labels. We establish communication lower bounds for two settings: coded transmissions involving different file–output pairs, and the case where every output function is assigned to all nodes. We develop two cyclic MSRA constructions that attain the corresponding bounds under their respective conditions. In particular, the packetized construction achieves a communication load of $L_{\mathrm{comm}}=(K-r)/(K-1)$ when $2\le r<K$ and $K-r$ divides $K-1$, using only $N=K$ files and $Q=1$ output function. For partial output assignment, we introduce a group–helper construction that delivers complete intermediate values without splitting them into smaller packets. Under regular file storage and direct one-shot decoding using locally computed intermediate values, this construction attains a receiver-gain lower bound for its parameter family. Finally, we demonstrate how existing CDC schemes can be represented within the MSRA framework and compare their communication and implementation tradeoffs.

Graph Neural Networks (GNNs) have achieved remarkable progress in community detection, which is an essential topic in network analysis with the aim of dividing a network into multiple subgraphs to mine potential information. However, most existing GNN-based community detection approaches adopt static loss function weights during the training process. In this paper, we propose an unsupervised end-to-end community detection framework and define an adaptive loss weighting layer within this framework, named QALW, which is capable of learning an optimal combination of loss weights during model training to balance reconstruction loss and clustering loss. Experimental results obtained based on three real-world benchmark datasets (Cora, Citeseer, and Pubmed) demonstrate that QALW achieves effective and stable community detection performance compared with eight representative baseline methods. In particular, QALW improves ACC by 5.7% over the strongest baseline on the Cora dataset, and achieves ACC values of 64.9%, 61.7%, and 63.9% on Cora, Citeseer, and Pubmed, respectively. Furthermore, the results verify that the proposed dynamic scheduling mechanism effectively alleviates gradient conflicts and enables more stable optimization than fixed-weight strategies. Overall, QALW demonstrates promising competitiveness and good robustness for unsupervised community detection in attributed networks.