Background and Objectives: Community detection is a critical problem in ‎investigating complex networks. Community detection based on ‎Modularity/general Modularity Density are the popular methods with the ‎advantage of using complex network features and the disadvantage of ‎being NP-hard problem for clustering. Moreover, Non-negative matrix ‎factorization (NMF)-based community detection methods are a family of ‎community detection tools that utilize network topology; but most of ‎them cannot thoroughly exploit network features. In this paper, a hybrid ‎NMF-based community detection infrastructure is developed, including ‎Modularity/ general Modularity Density as more comprehensive indices of ‎networks. The proposed infrastructure enables to solve the challenges of ‎combining the NMF method with Modularity/general Modularity Density ‎criteria and improves the community detection methods for complex ‎networks.‎Methods: First, new representations, similar to the model of symmetric ‎NMF, are derived for the model of community detection based on ‎Modularity/general Modularity Density. Next, these indices are ‎innovatively augmented to the proposed hybrid NMF-based model as two ‎novel models called ‘general Modularity Density NMF (GMDNMF) and ‎mixed Modularity NMF (MMNMF)’. In order to solve these two NP-hard ‎problems, two iterative optimization algorithms are developed.‎Results: it is proved that the Modularity/general Modularity Density-‎based community detection can be consistently represented in the form ‎of SNMF-based community detection. The performances of the proposed ‎models are verified on various artificial and real-world networks of ‎different sizes. It is shown that MMNMF and GMDNMF models ‎outperform other community detection methods. Moreover, the ‎GMDNMF model has better performance with higher computational ‎complexity compared to the MMNMF model.‎Conclusion: The results show that the proposed MMNMF model improves ‎the performance of community detection based on NMF by employing ‎the Modularity index as a network feature for the NMF model, and the ‎proposed GMDNMF model enhances NMF-based community detection by ‎using the general Modularity Density index.‎

Abstract. In this paper, we have introduced and studied the notion of a fuzzy independent pair and obtain some properties of fuzzy α-modular pairs and independent pairs.

In literature, for the four common classes of uninorms, the Modularity equation has been solved except for the kind of ones having continuous underlying functions. This paper is devoted to solving the Modularity equation involving two uninorms with continuous underlying functions.We discuss this Modularity equation in detail by dividing the main section into two parts. The structure characterization of the two uninorms is almost completely obtained and it is found that they are equal in the unit square except in a subdomain.

In the cognitive sciences, two distinct theories have been proposed about the structure of the human mind, both of which are evolutionary but also different. One of them is the theory of evolutionary psychology and its related claim to the massive Modularity hypothesis, which considers the mind as a set of modules. Another is the simple heuristic and its related claim to the existence of an adaptive toolbox in cognitive methods that assigns mind guidance to the existence of a set of heuristics. Both theories seek to explain cultural diversity by applying these modules/ heuristics. Although proponents of each do not routinely mention the existence of another theory, both theories, both the massive Modularity hypothesis and the existing idea of heuristics in the mind, seek to provide explanations not only from an evolutionary perspective but also in comparative psychology (Which compares the behavior of non-human species) are also acceptable. They also want to explain how cognitive processes are processed in our minds. However, at first, there are reasons to think that these theories offer explanations of human cognition that are incompatible with each other or undermine each other. What is challenged in this article is how a human being, who has always been influenced by a variety of heuristics, cognitive biases, and irrationality in reasoning, judgment, and decision-making during the process of evolution, can have a massive modular cognitive structure in its mind. And be organized to have a logical function