Narrow Results

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 $H$ and the modularity density measure $D$, and intend to uncover the effective thresholds of their corresponding resolution parameter $\gamma$ 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 $H$ and $D$ that will not cause the misidentification. By comparison, we find that in both examples, for Hamiltonian function $H$, the larger the value of resolution parameter $\gamma$, the less resolution limit the network suffers; while for modularity density $D$, the less resolution limit the network suffers when we decrease the value of $\gamma$. Our framework is mathematically strict and efficient and can be applied in a lot of scientific fields.

Detecting communities in weighted networks is becoming a challenging and interesting work. In this paper, a novel quantitative measure called weighted normalized modularity density is proposed and optimized to detect communities in weighted networks. Both theoretical certifications on simple schematic examples and numerical experiments on a suit of simulated networks and real-world networks show that the proposed quantitative measure not only improves the resolution limit in optimizing weighted modularity, but also avoids the emergence of negative communities in optimizing weighted modularity density.