Entropy of random graph ensembles constrained with generalised degrees

Abstract

Generalised degrees provide a natural bridge between local and global topological properties of networks. We define the generalised degree to be the number of neighbours of a node within one and two steps respectively. Tailored random graph ensembles are used to quantify and compare topological properties of networks in a systematic and precise manner, using concepts from information theory. We calculate the Shannon entropy of random graph ensembles constrained with a specified generalised degree distribution. We find that the outcome has a natural connection with the degree-degree correlation which is implied by specifying a generalised degree distribution. We demonstrate how generalised degrees can be used to qualitatively and quantitatively describe a network.