Simplicial degree in complex networks. Applications of topological data analysis to network science

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]. For example, two triangles are connected if they have an edge in common.…”

“…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 . Also, the Upper adjacency matrix is defined by The Adjacency matrix for k-edges is defined as In this work, as we focused only on 3-point interactions in rs-fMRI data, the lower adjacency matrix coincided with the adjacency matrix for 3-edges (triplets).…”

“…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 .…”

“…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 e ⌣ h f , or simply e ⌣ f . The lower adjacency matrix is defined by (L) ij = 1 e i ⌣ e j 0 otherwise .…”

“…We further utilize the fact that a uniform hypergraph can be represented as a high-order adjacency matrix, accounting for the connectivity between hyperedges [33]. Consequently, multiple algorithms already used for pairwise adjacency matrices such as Eigenvector centrality, modularity, or between-ness centrality could be inherited in the high-order context [33, 34], speeding up the methodological bridge from pairwise to high-order networks.…”