Modularity gradients: Measuring the contribution of edges to the community structure of a brain network

Modularity measures the quality of a particular division of a brain network, and it tends to have high values for networks with strong community structure. In this paper we show that the derivative of modularity with respect to each edge in a network quantifies the contribution of the edge to the global modular structure of the network. We derive analytical forms for the derivative of modularity and investigate its properties. Our approach focuses on the significance of edges on modularity and deviates from standard node-centric network measures, such as hub analysis.