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This paper considers metric the exact dissimilarities between pairs of points are not unknown but known to belong to some interval. 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 dissimilarities. Our construction of hierarchical clustering methods is based on defining admissible methods to be those methods that satisfy the axioms of value—nodes in a metric space with two nodes are clustered together at the convex combination of the upper and lower bounds determined by a parameter—and transformation—when both distance bounds are reduced, the output may become more clustered but not less. Two admissible methods are constructed and are shown to provide universal bounds in the space of admissible methods. Practical implications are explored by clustering moving points via snapshots and by clustering coauthorship networks representing collaboration between researchers from different communities. The proposed clustering methods succeed in identifying underlying hierarchical clustering structures via the maximum and minimum distances in all snapshots, as well as in differentiating collaboration patterns in journal publications between different research communities based on bounds of network distances. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM PUBLISHER]