Extensional Confluences and Local Closure Operators

Abstract

This work is motivated by knowledge discovery in attributed graphs. Our approach consists in extending the methodology of frequent closed pattern mining, as developed in Formal Concept Analysis (FCA), to the case where the objects in which attribute patterns may occur are the vertices of a graph, typically representing a social network. For that purpose we extend the framework of abstract concept lattices, in which the extensional space is a pointed join-subsemilattice of the powerset $X$ of the object set, by considering as the extensional space a weaker structure called a confluence of $X$. Confluences were recently investigated as intensional spaces in FCA. In this article we show that when the intensional space is a lattice $L$ and the extensional space is a confluence $F$ of $X$, that leads to a set of closure operators, called local closure operators, whose union form the set of intensions of $F$. We investigate the structure of the set of (extension,intension) pairs, i.e. the set of local concepts built on $(L,F)$ and related local implications. As an example, we consider the detection of all frequent k-communities in an attributed network.