Entropy is a classical measure to quantify the amount of information or complexity of a system. Various entropy-based measures such as functional and spectral entropies have been proposed in brain network analysis. However, they are less widely used than traditional graph theoretic measures such as global and local efficiency because they are not well-defined on a network or difficult to interpret the biological meaning. In this talk, I will present a new entropy-based network invariant, called volume entropy, for brain network analysis. It measures the exponential growth rate of the number of network paths, based on the intuition that information flows through a network forever. When the proposed method was applied to various networks such as artificial regular, small-world, random, scale-free, and hyperbolic networks, the simulation showed that the volume entropy distinguished the underlying network topology and geometry better than the existing network measures, efficiencies and entropy-based invariants. The volume entropy method was also applied to two real datasets, resting state fMRI and FDG PET data of 38 normal controls between 20s and 60s and FDG PET data in Alzheimer`s Disease Neuroimaging Initiative (ADNI). The results showed that the volume entropy could be used as a biomarker to detect the change of the network structure due to normal aging or disease progression.