Weiyu Huang

Graph Frequency Analysis of Brain Signals

This paper presents methods to analyze functional brain networks and signals from graph spectral perspectives. The notion of frequency and filters traditionally defined for signals supported on regular domains such as discrete time and image grids has been recently generalized to irregular graph domains and defines brain graph frequencies associated with different levels of spatial smoothness across the brain regions. Brain network frequency also enables the decomposition of brain signals into pieces corresponding to smooth or rapid variations. We relate graph frequency with principal component analysis when the networks of interest denote functional connectivity. The methods are utilized to analyze brain networks and signals as subjects master a simple motor skill. We observe that brain signals corresponding to different graph frequencies exhibit different levels of adaptability throughout learning. Further, we notice a strong association between graph spectral properties of brain networks and the level of exposure to tasks performed and recognize the most contributing and important frequency signatures at different levels of task familiarity.

Diffusion and Superposition Distances for Signals Supported on Networks

We introduce the diffusion and superposition distances as two metrics to compare signals supported in the nodes of a network. Both metrics consider the given vectors as initial temperature distributions and diffuse heat through the edges of the graph. The similarity between the given vectors is determined by the similarity of the respective diffusion profiles. The superposition distance computes the instantaneous difference between the diffused signals and integrates the difference over time. The diffusion distance determines a distance between the integrals of the diffused signals. We prove that both distances define valid metrics and that they are stable to perturbations in the underlying network. We utilize numerical experiments to illustrate their utility in classifying signals in a synthetic network as well as in classifying ovarian cancer histologies using gene mutation profiles of different patients. We also utilize diffusion as part of a label propagation method in semi-supervised learning to classify handwritten digits.

Brain network efficiency is influenced by the pathologic source of corticobasal syndrome

Objective: To apply network-based statistics to diffusion-weighted imaging tractography data and detect Alzheimer disease vs non-Alzheimer degeneration in the context of corticobasal syndrome. Methods: In a cross-sectional design, pathology was confirmed by autopsy or a pathologically validated CSF total tau-to-β-amyloid ratio (T-tau/Aβ). Using structural MRI data, we identify association areas in fronto-temporo-parietal cortex with reduced gray matter density in corticobasal syndrome (n = 40) relative to age-matched controls (n = 40). Using these fronto-temporo-parietal regions of interest, we construct structural brain networks in clinically similar subgroups of individuals with Alzheimer disease (n = 21) or non-Alzheimer pathology (n = 19) by linking these regions by the number of white matter streamlines identified in a deterministic tractography analysis of diffusion tensor imaging data. We characterize these structural networks using 5 graph-based statistics, and assess their relative utility in classifying underlying pathology with leave-one-out cross-validation using a supervised support vector machine. Results: Gray matter density poorly discriminates between Alzheimer disease and non-Alzheimer pathology subgroups with low sensitivity (57%) and specificity (52%). In contrast, a statistic of local network efficiency demonstrates very good discriminatory power, with 85% sensitivity and 84% specificity. Conclusions: Our results indicate that the underlying pathologic sources of corticobasal syndrome can be classified more accurately using graph theoretical statistics derived from patterns of white matter network organization in association cortex than by regional gray matter density alone. These results highlight the importance of a multimodal neuroimaging approach to diagnostic analyses of corticobasal syndrome.

Persistent homology lower bounds on network distances

High order networks are weighted complete hypergraphs collecting relationships between elements of tuples. Valid metric distances between high order networks have been defined but they are difficult to compute when the number of nodes is large. We relate high order networks to the filtrations of simplicial complexes and show that the distance between networks can be lower bounded by the difference between the homological features of their respective filtrations. Practical implications are explored by comparing the coauthorship networks of engineering and mathematics academic journals. The lower bounds succeed in discriminating engineering communities from mathematics and in differentiating engineering communities with different research interests.

Metrics in the Space of High Order Networks

This paper presents methods to compare high-order networks, defined as weighted complete hypergraphs collecting relationship functions between elements of tuples. They can be considered as generalizations of conventional networks where only relationship functions between pairs are defined. Important properties between relationships of tuples of different lengths are established, particularly when relationships encode dissimilarities or proximities between nodes. Two families of distances are then introduced in the space of high-order networks. The distances measure differences between networks. We prove that they are valid metrics in the spaces of high-order dissimilarity and proximity networks modulo permutation isomorphisms. Practical implications are explored by comparing the coauthorship networks of two popular signal processing researchers. The metrics succeed in identifying their respective collaboration patterns.

Persistent Homology Lower Bounds on High-Order Network Distances

High-order networks are weighted hypergraphs collecting relationships between elements of tuples, not necessarily pairs. Valid metric distances between high-order networks have been defined but they are difficult to compute when the number of nodes is large. The goal here is to find tractable approximations of these network distances. The paper does so by mapping high-order networks to filtrations of simplicial complexes and showing that the distance between networks can be lower bounded by the difference between the homological features of their respective filtrations. Practical implications are explored by classifying weighted pairwise networks constructed from different generative processes and by comparing the coauthorship networks of engineering and mathematics academic journals. The persistent homology methods succeed in identifying different generative models, in discriminating engineering and mathematics communities, as well as in differentiating engineering communities with different research interests.

Diffusion filtering of graph signals and its use in recommendation systems

This paper presents diffusion filtering as a method to smooth signals defined on the nodes of a graph or network. Diffusion filtering considers the given signals as initial temperature distributions in the nodes and diffuses heat through the edges of the graph. The filtered signal is determined by the accumulated temperatures over time at each node. We show multiple other interpretations of diffusion filtering and describe how it can be generalized to encompass a wide class of networks making it suitable for real-world applications. We prove that diffused signals are stable to perturbations in the underlying network. Further, we demonstrate how diffusion filtering can be applied to improve the performance of recommendation systems by considering the problem of predicting ratings from a signal processing perspective.

Metrics in the space of high order proximity networks

This paper presents two families of distances in the space of high order proximity networks. The distances measure differences between networks and are shown to be valid metrics in the space of high order proximity networks modulo permutation isomorphisms. Practical implications are explored by comparing the coauthorship networks of two popular signal processing researchers. The metrics succeed in identifying their respective collaboration patterns.

Persistent homology approximations of network distances

This paper presents methods to compare high order networks using persistent homology. High order networks induce well-founded homological features and the difference between networks is measured by the difference between the homological features. This is a reasonable approximation to a valid metric in the space of high order networks modulo permutation isomorphisms. The approximations succeed in discriminating engineering research communities from mathematics communities.

Axiomatic hierarchical clustering for intervals of metric distances

This paper considers metric spaces where distances between a pair of nodes are represented by distance intervals. The goal is to study methods for the determination of hierarchical clusters, i.e., a family of nested partitions indexed by a resolution parameter, induced from the given distance intervals of the metric spaces. Our construction is based on defining admissible methods to be those methods that abide to the axioms of value and transformation. Two admissible methods are constructed and are shown to provide upper and lower bounds in the space of admissible methods. Practical implications are explored by clustering networks representing brain structural connectivity using the lower and upper bounds of the network distance. The proposed clustering methods succeed in differentiating brain connectivity networks of patients from those of healthy controls.