Detecting communities plays a vital role in studying group level patterns of a social network and it can be helpful in developing several recommendation systems such as movie recommendation, book recommendation, friend recommendation and so on. Most of the community detection algorithms can detect disjoint communities only, but in the real time scenario, a node can be a member of more than one community at the same time, that leads to overlapping communities. A novel approach is proposed to detect such overlapping communities by extending the definition of newman’ s Modularity for overlapping communities. The proposed algorithm is tested on LFR benchmark networks with overlapping communities and on real-world networks. The performance of the algorithm is evaluated using popular metrics such as ONMI, Omega Index, F-score and Overlap Modularity and the results are compared with its competent algorithms. It is observed that Extended Modularity Gain can detect highly modular structures in complex networks with overlapping communities.

Massive Modularity Hypothesis of Mind assumes that the mind is composed largely, or perhaps even entirely, of modules. Since the publication of Modularity of the Mind (Fodor, 1983) Modularity has occupied a central role in the studies of mind and cognitive science. But in philosophy there is no agreement about this role. In this paper, first I propose that many of these problems are the effects of the Fodorian base of Massive Modularity Hypothesis, and then I suggest taking Simon’s works on complexity as a proper base for Modularity. In the last section of this paper, I consider two common characteristics of mind, i.e. domain- specificity and encapsulation, and argue that in shadow of Massive Modularity Hypothesis these characteristics can be explained better.

Nonlinearity is one of the main behaviors of systems in the real world. Therefore, it seems necessary to introduce a method to determine the stability margin of these systems. Although the Gain and phase margins are established criteria for the analysis of linear systems, finding a specific way to determine the true value of these margins in nonlinear systems in general is an ongoing research in the literature. The main goal of this paper is to introduce a new method to determine these margins more precisely for particular type of nonlinearities. Based on Lyapunov theorem, stability conditions are obtained and the stability margins of Lur'e systems are determined by applying the Extended circle criterion. By providing numerical examples, the correctness of the obtained results is checked and by comparing them with other methods, the accuracy of the obtained results is evaluated.

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 this paper, relying on the clustering of complex networks that can determine large scale features of ‎the network, we study 48 financial markets across the world. To this end, we develop a Modularity ‎maximization method for directed and weighted networks. According to the linear correlation measure, ‎we construct the adjacency matrix, and by using the theory of random matrices, we divide the space of ‎eigenvalues of our matrix into two irrelevant and relevant fragments. By considering the temporal ‎window and its evolution over time series, our results demonstrate that in the vicinity of so-called ‎financial crisis clusters, which are often affected by geographical characteristics, are formed and from the ‎perspective of complex networks, they show more random behavior‎.‎‎

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.

The focus of this paper is to investigate the Modularity equation involving uninorms and Mayor's aggregation operators.  Necessary and sufficient conditions are established for this equation. And, it finds that theModularity equation of a Mayor's aggregation operator over a uninorm is reduced to the Modularityof a commutative semi-t-norm over a uninorm and the Modularity equation of a uninorm over a Mayor's aggregation operator is reduced to the Modularity equation of a uninorm over a commutative semi-t-conorm. Among them, the cases of a uninorm which is locally internal on the boundary are studied in \cite{Su_Riera_Aguilera_Torrens_2019}. In this paper, we consider whether the neutral element $e$ of the uninorm is idempotent element of the Mayor's aggregation operator in Modularity equation to get solutions in the corresponding cases.