Uncovering the effective interval of resolution parameter across multiple community optimization measures

Abstract

The study of community structure is a primary focus of network analysis, which has attracted a large amount of attention. In this paper, we focus on two famous functions, i.e., the Hamiltonian function [Formula: see text] and the modularity density measure [Formula: see text], and intend to uncover the effective thresholds of their corresponding resolution parameter [Formula: see text] without resolution limit problem. Two widely used example networks are employed, including the ring network of lumps as well as the ad hoc network. In these two networks, we use discrete convex analysis to study the interval of resolution parameter of [Formula: see text] and [Formula: see text] that will not cause the misidentification. By comparison, we find that in both examples, for Hamiltonian function [Formula: see text], the larger the value of resolution parameter [Formula: see text], the less resolution limit the network suffers; while for modularity density [Formula: see text], the less resolution limit the network suffers when we decrease the value of [Formula: see text]. Our framework is mathematically strict and efficient and can be applied in a lot of scientific fields.