Vector centrality in hypergraphs

Emergence of High-Order Functional Hubs in the Human Brain

“…We follow a procedure to encode the resulting k-uniform hypergraph as an “hyper-adjacency matrix.” Among the multiple alternatives [33], the one used in this work inherits the lower adjacency matrix representation of simplicial complexes into uniform hypergraphs, which was also recently adapted to develop vector centralities on hypergraphs [41]. Put simply, two hyperedges of dimension k are connected if they share an hyperedge of dimension k – 1 [33, 34, 42].…”

“…In this work, we use an extension of the Eigenvector centrality, introduced in [42], to investigate high-order hubs in the human brain. High-order hubs were explored recently in network science in diverse settings [33,43]. In this work, we will rely on the Perron-Frobenius theorem [44] to explore high-order hubs of uniform hypergraphs.…”

“…Notice that for a given hypergraph or simplicial complex, one could have different ways to define such connectivity, e.g., by sharing edges of smaller sizes and so on [33]. This representation was constructed initially for simplicial complexes [33] In this work, as we only explored 3-point interactions in rs-fMRI data, the lower adjacency matrix coincided with, and further explored recently for hypergraphs [43]. Formally, the hyperedge adjacency representation can be defined as follows [33, 34]: We say that two k -edges e, f are lower adjacent if there is a k − 1-edge l such that l ⊂ e, f and we denote , or simply . The lower adjacency matrix is defined by Also, we say that they are upper adjacent if there is a (k + 1)-edge h such that h ⊂ e, f and we denote by , or simply .…”

“…In this work, as we only explored 3-point interactions in rs-fMRI data, the lower adjacency matrix coincided with, and further explored recently for hypergraphs [43]. Formally, the hyperedge adjacency representation can be defined as follows [33, 34]: We say that two k -edges e, f are lower adjacent if there is a k − 1-edge l such that l ⊂ e, f and we denote , or simply .…”

“…Notice that for a given hypergraph or simplicial complex, one could have different ways to define such connectivity, e.g., by sharing edges of smaller sizes and so on [33]. This representation was constructed initially for simplicial complexes [33] In this work, as we only explored 3-point interactions in rs-fMRI data, the lower adjacency matrix coincided with, and further explored recently for hypergraphs [43].…”