A cycle in a graph is a subset of a connected component with redundant additional connections. If there are many cycles in a connected component, the connected component is more densely connected. While the number of connected components represents the integration of the brain network, the number of cycles represents how strong the integration is. However, enumerating cycles in the network is not easy and often requires brute force enumerations. In this study, we present a new scalable algorithm for enumerating the number of cycles in the network. We show that the number of cycles is monotonically decreasing with respect to the filtration values during graph filtration. We further develop a new statistical inference framework for determining the significance of the number of cycles. The methods are applied in determining if the number of cycles is a statistically significant heritable network feature in the functional human brain network.