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.
A robust and informative local shape descriptor plays an important role in mesh registration. In this regard, spectral descriptors that are based on the spectrum of the Laplace-Beltrami operator have gained a spotlight among the researchers for the last decade due to their desirable properties, such as isometry invariance. Despite such, however, spectral descriptors often fail to give a correct similarity measure for non-isometric cases where the metric distortion between the models is large. Hence, they are in general not suitable for the registration problems, except for the special cases when the models are near-isometry. In this paper, we investigate a way to develop shape descriptors for non-isometric registration tasks by embedding the spectral shape descriptors into a different metric space where the Euclidean distance between the elements directly indicates the geometric dissimilarity. We design and train a Siamese deep neural network to find such an embedding, where the embedded descriptors are promoted to rearrange based on the geometric similarity. We found our approach can significantly enhance the performance of the conventional spectral descriptors for the non-isometric registration tasks, and outperforms recent state-of-the-art methods reported in literature.
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