Fuzzy Modularity Research Articles

Comparing the Clique Percolation algorithm to other overlapping community detection algorithms in psychological networks: A Monte Carlo simulation study

In psychological networks, one limitation of the most used community detection algorithms is that they can only assign each node (symptom) to a unique community, without being able to identify overlapping symptoms. The clique percolation (CP) is an algorithm that identifies overlapping symptoms but its performance has not been evaluated in psychological networks. In this study, we compare the CP with model parameters chosen based on fuzzy modularity (CPMod) with two other alternatives, the ratio of the two largest communities (CPRat), and entropy (CPEnt). We evaluate their performance to: (1) identify the correct number of latent factors (i.e., communities); and (2) identify the observed variables with substantive (and equally sized) cross-loadings (i.e., overlapping symptoms). We carried out simulations under 972 conditions (3x2x2x3x3x3x3): (1) data categories (continuous, polytomous and dichotomous); (2) number of factors (two and four); (3) number of observed variables per factor (four and eight); (4) factor correlations (0.0, 0.5, and 0.7); (5) size of primary factor loadings (0.40, 0.55, and 0.70); (6) proportion of observed variables with substantive cross-loadings (0.0%, 12.5%, and 25.0%); and (7) sample size (300, 500, and 1000). Performance was evaluated through the Omega index, Mean Bias Error (MBE), Mean Absolute Error (MAE), sensitivity, specificity, and mean number of isolated nodes. We also evaluated two other methods, Exploratory Factor Analysis and the Walktrap algorithm modified to consider overlap (EFA-Ov and Walk-Ov, respectively). The Walk-Ov displayed the best performance across most conditions and is the recommended option to identify communities with overlapping symptoms in psychological networks.

Fuzzy Community Detection Based on Elite Symbiotic Organisms Search and Node Neighborhood Information

In the last decade, fuzzy community detection has received increasing attention, since it cannot only uncover the community structure of a network, but also reflect the membership degrees of each node to multiple communities. Although some pioneers proposed a few algorithms for finding fuzzy communities, there is still room for further improvement in the quality of detected fuzzy communities. In this study, a metaheuristic-based modularity optimization algorithm, named symbiotic organisms search fuzzy community detection (SOSFCD) is proposed. On the one hand, an improved bioinspired metaheuristic algorithm, elite symbiotic organisms search, is designed as optimization strategy to improve the global convergence of fuzzy modularity optimization. On the other hand, a neighbor-based membership modification operation is proposed to intensify exploitation ability and speed up convergence, by efficiently utilizing local information (i.e., node neighborhood) of network topology. Experimental results on both of synthetic and real-life networks with different scales and characteristics show that SOSFCD can find max-modularity fuzzy partitions and coverings, and outperforms many state-of-the-art algorithms in terms of accuracy and stability.

Soft Overlapping Community Detection in Large-Scale Networks via Fast Fuzzy Modularity Maximization

Soft overlapping clustering is one of the notable problems of community detection. Extensive research has been conducted to develop efficient methods for nonoverlapping and crisp-overlapping community detection in large-scale networks. In this article, fast fuzzy modularity maximization (FFMM) for soft overlapping community detection is proposed. FFMM exploits novel iterative equations to calculate the modularity gain associated with changing the fuzzy membership values of network vertices. The simplicity of the proposed scheme enables efficient modifications, reducing computational complexity to a linear function of the network size, and the number of communities. Moreover, to further reduce the complexity of FFMM for very large networks, multicycle FFMM (McFFMM) is proposed. The proposed McFFMM reduces complexity by breaking networks into multiple subnetworks and applying FFMM to detect their communities. Performance of the proposed techniques is demonstrated with real-world data and the Lancichinetti-Fortunato-Radicchi benchmark networks. Moreover, the performance of the proposed techniques is evaluated versus some state-of-the-art soft overlapping community detection approaches. Results show that the McFFMM produces a remarkable performance in terms of overlapping modularity with fuzzy memberships, computational time, number of detected overlapping nodes, and overlapping normalized mutual information.

Ulam Type Stability of ?-Quadratic Mappings in Fuzzy Modular ∗-Algebras

In this paper, we find the solution of the following quadratic functional equation n∑1≤i<j≤nQxi−xj=∑i=1nQ∑j≠ixj−(n−1)xi, which is derived from the gravity of the n distinct vectors x1,⋯,xn in an inner product space, and prove that the stability results of the A-quadratic mappings in μ-complete convex fuzzy modular ∗-algebras without using lower semicontinuity and β-homogeneous property.

Detecting Overlapping Communities in Modularity Optimization by Reweighting Vertices.

On the purpose of detecting communities, many algorithms have been proposed for the disjointed community sets. The major challenge of detecting communities from the real-world problems is to determine the overlapped communities. The overlapped vertices belong to some communities, so it is difficult to be detected using the modularity maximization approach. The major problem is that the overlapping structure barely be found by maximizing the fuzzy modularity function. In this paper, we firstly introduce a node weight allocation problem to formulate the overlapping property in the community detection. We propose an extension of modularity, which is a better measure for overlapping communities based on reweighting nodes, to design the proposed algorithm. We use the genetic algorithm for solving the node weight allocation problem and detecting the overlapping communities. To fit the properties of various instances, we introduce three refinement strategies to increase the solution quality. In the experiments, the proposed method is applied on both synthetic and real networks, and the results show that the proposed solution can detect the nontrivial valuable overlapping nodes which might be ignored by other algorithms.

Some Properties of Topology Fuzzy Modular Space

In the present paper , the authors have introduced and studied fuzzy modular space. they have investigated some properties of this space in the open and closed balls. Also the authors discussed the convex set and the locally convex in fuzzy modular space. The result obtained are correct and the methods used are interesting.

Quadratic Program-Based Modularity Maximization for Fuzzy Community Detection in Social Networks

One of the most important elements of social network analysis is community detection, i.e., finding groups of similar people based on their traits. In this paper, we present the fuzzy modularity maximization (FMM) approach for community detection, which finds overlapping—that is, fuzzy—communities (where appropriate) by maximizing a generalized form of Newman's modularity. The first proposed FMM solution uses a tree-based structure to find a globally optimal solution, while the second proposed solution uses alternating optimization to efficiently search for a locally optimal solution. Both of these approaches are based on a proposed algorithm called one-step modularity maximization (OSMM), which computes the optimal cluster memberships for one person in the social network. We prove that OSMM can be formulated as a simplified quadratic knapsack optimization problem, which is $O(n)$ time complexity. We then propose a tree-based algorithm, called FMM/Find Best Leaf Node (FMM/FBLN), which represents sequences of OSMM steps in a tree-based structure. It is proved that FMM/FBLN finds globally optimal solutions for FMM; however, the time complexity of FMM/FBLN is $O(n^d)$ , $d\ge 2$ ; thus, it is impractical for most real-world networks. To combat this inefficiency, we propose five heuristic-based alternating optimization schemes, i.e., FMM/H1–H5, which are all shown to be $O(n^2)$ time complexity. We compare the results of the FMM/H solutions with those of state-of-the-art community detection algorithms, MULTICUT spectral FCM (MSFCM) and GALS, and with those of two fuzzy community detection algorithms called GA and vertex-similarity based gradient-descent method (VSGD) on ten real-world datasets. We conclude that one of the five heuristic algorithms (FMM/H2) is very competitive with GALS and much more effective than MSFCM, GA, and VSGD. Furthermore, all of the FMM/H schemes are at least two orders of magnitude faster than GALS in run time. Finally, FMM/H, unlike GALS (which only produces crisp partitions) and MSFCM (which always finds fuzzy partitions), is the only fuzzy community detection algorithm to date that can find the max-modularity partition, fuzzy or crisp.

A Soft Modularity Function For Detecting Fuzzy Communities in Social Networks

We discuss a new formulation of a fuzzy validity index that generalizes the Newman-Girvan (NG) modularity function. The NG function serves as a cluster validity functional in community detection studies. The input data is an undirected weighted graph that represents, e.g., a social network. Clusters correspond to socially similar substructures in the network. We compare our fuzzy modularity with two existing modularity functions using the well-studied Karate Club and American College Football datasets.

Some Analogies of the Banach Contraction Principle in Fuzzy Modular Spaces

We established some theorems under the aim of deriving variants of the Banach contraction principle, using the classes of inner contractions and outer contractions, on the structure of fuzzy modular spaces.

Anomaly-based detection system in computer networks

Effective detection of new types of cyber-attacks, such as 'zero-day', as well as attacks which have no signatures associated with the widespread use of appropriate means of protection. The existing intrusion detection systems use the mathematical models for security solution, which require a lot of resources and expenses of different nature, for example, connected with sample of statistical data, training systems, its adaptation, etc. More effective in this regard are peer-based approaches and knowledge of the subject area specialists. To meet this challenge, based on the anomaly detection technique based on fuzzy logic it was suggested the structural solution of anomaly-based detection system in computer networks, created by the unsignature attacks. The system is composed of primary processing subsystems, generation of fuzzy standards and heuristic rules, as well as Fuzzy Modular Arithmetic, linguistic corollary and visualization, that make possible to identify the abnormal condition in the network traffic, generated by a specific type of cyber-attacks. The result of the system is represented in linguistic and graphic forms.

Quantification of fuzzy modularity in biologic networks

Quantification of fuzzy modularity in biologic networks

Fuzzy modularity and fuzzy community structure in networks

To find the fuzzy community structure in a complex network, in which each node has a certain probability of belonging to a certain community, is a hard problem and not yet satisfactorily solved over the past years. In this paper, an extension of modularity, the fuzzy modularity is proposed, which can provide a measure of goodness for the fuzzy community structure in networks. The simulated annealing strategy is used to maximize the fuzzy modularity function, associating with an alternating iteration based on our previous work. The proposed algorithm can efficiently identify the probabilities of each node belonging to different communities with random initial fuzzy partition during the cooling process. An appropriate number of communities can be automatically determined without any prior knowledge about the community structure. The computational results on several artificial and real-world networks confirm the capability of the algorithm.