Galois lattice and positional dominance

Abstract

Galois lattices can be applied to analybinary two-mode networks, containing actors and affiliations. Of particular interest is how they help visualize hierarchical structures in the data by using algebraic set theory. This paper aims to outline the concept and definitions as well as to give insights into the inner working of Galois lattices. Furthermore, we discuss how Galois lattices are related to the recently introduced concept of positional dominance, which describes a relation between two nodes based on their neighborhoods. We demonstrate that by utilizing a reduced labeling approach, all paths from the global lower bound to the global upper bound of a Galois lattice determine exactly the definition of positional dominance applied on two-mode networks. By comparing path lengths starting at either of the bounds, hierarchical levels can be identified. Hence, we conclude that a Galois lattice describes positional dominance and hierarchical levels among actors and respectively among affiliations. We propose two algorithms, one to build a Galois lattice and another to extract the positional dominance from a reduced labeled Galois lattice.