Theory of preference modelling for communities in scale-free networks

Detecting a community structure on networks is a problem of interest in science and many other domains. Communities are special structures which may consist nodes with some common features. The identification of overlapping communities can clarify not so apparent features about relationships among the nodes of a network. A node in a community can have a membership in a community with a different degree. Here, we introduce a fuzzy based approach for overlapping community detection. A special type of fuzzy operator is used to define the membership strength for the nodes of community. Fuzzy systems and logic is a branch of mathematics which introduces many-valued logic to compute the truth value. The computed truth can have a value between 0 and 1. The preference modelling approach introduces some parameters for designing communities of particular strength. The strength of a community tells us to what degree each member of community is part of a community. As for relevance and applicability of the community detection method on different types of data and in various situations, this approach generates a possibility for the user to be able to control the overlap regions created while detecting the communities. We extend the existing methods which use local function optimization for community detection. The LFM method uses a local fitness function for a community to identify the community structures. We present a community fitness function in pliant logic form and provide mathematical proofs of its properties, then we apply the preference implication of continuous-valued logic. The preference implication is based on two important parameters ν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu$$\end{document} and α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}. The parameter ν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu$$\end{document} of the preference-implication allows us to control the design of the communities according to our requirement of the strength of the community. The parameter α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document} defines the sharpness of preference implication. A smaller value of the threshold for community membership creates bigger communities and more overlapping regions. A higher value of community membership threshold creates stronger communities with nodes having more participation in the community. The threshold is controlled by δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta$$\end{document} which defines the degree of relationship of a node to a community. To balance the creation of overlap regions, stronger communities and reducing outliers we choose a third parameter δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta$$\end{document} in such a way that it controls the community strength by varying the membership threshold as community evolves over time. We test the theoretical model by conducting experiments on artificial and real scale-free networks. We test the behaviour of all the parameters on different data-sets and report the outliers found. In our experiments, we found a good relationship between ν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu$$\end{document} and overlapping nodes in communities.