# Hypergraph Analysis of Functional Brain Connectivity During Figurative Attention
## 2 Mathematical Basis
### Main Definitions
These degree measures provide insights into the topological structure of the hypergraph, highlighting the centrality of vertices and the complexity of hyperedges in capturing higher-order interactions. If a vertex \(v\) is covered by two or more hyperedges, we say that these hyperedges are adjacent through \(v\) (\(e_{i}\sim v_{j}\)). Similarly, the vertices covered by a hyperedge \(e\) are considered adjacent through \(e\) (\(v_{j}\sim e_{i}\)). A hypergraph \(\mathcal{H}\) can be represented in several equivalent forms, including an incidence graph, an incidence matrix \(\mathcal{H}=\{0,1\}^{|\mathcal{V}|\times|\mathcal{E}|}\), and a hyperedge weight matrix \(\mathcal{W}\). An _incidence graph_ (also known as a bipartite graph representation) is a way to represent a hypergraph by transforming it into a standard graph. In this representation, the hypergraph's vertices and hyperedges are treated as two distinct sets of nodes in a bipartite graph. The _incidence matrix_ is defined as  
\[\mathcal{H}[i][j]= \begin{cases}1,&\text{if }v_{i}\in e_{j},\\ 0,&\text{otherwise},\end{cases}  \tag{5}\]  
where \(v_{i}\in\mathcal{V}\) represents the \(i\)-th vertex and \(e_{j}\in\mathcal{E}\) represents the \(j\)-th hyperedge. This matrix captures the membership of vertices in hyperedges, providing a binary representation of the hypergraph structure. A _hyperedge weight matrix_\(\mathcal{W}\) encodes the significance of each hyperedge in the hypergraph. Specifically, the weight of each hyperedge \(e_{j}\) is located at the corresponding diagonal entry of \(\mathcal{W}\), such that  
\[\text{diag}(\mathcal{W})=[w(e_{1}),w(e_{2}),\ldots,w(e_{|\mathcal{E}|})], \tag{6}\]  
where \(w(e_{j})\) represents the weight of the \(j\)-th hyperedge. In this work, the weight \(w(e_{j})\) is defined as the degree of the hyperedge \(e_{j}\), i.e., \(\text{Deg}(e_{j})\). This weighting scheme reflects the topological importance of each hyperedge within the hypergraph. A hypergraph can be modified by transforming vertices into edges (\(\mathcal{V}\rightarrow\mathcal{E}\)) and vice versa. This transformation is called \"duality of hypergraphs\" and is denoted as \(\mathcal{H}=(\mathcal{V},\mathcal{E})\rightarrow\mathcal{H}^{*}=(\mathcal{E},\mathcal{X})\), where \(\mathcal{X}=\{x_{1},x_{2},\ldots,x_{n}\}\), with \(x_{i}\) representing the set of all edges \(\mathcal{E}\) incident to vertex \(v_{i}\). The degree of vertices in a _dual hypergraph_\(\mathcal{H}^{*}\) is equal to the degree of hyperedges in \(\mathcal{H}\), while the degree of the hyperedges remains unchanged from \(\mathcal{H}\). This relationship can be expressed as follows:  
\[\text{Deg}(v_{i})=\text{Deg}(x_{i}),\;\text{Deg}(e_{i})=\text{Deg}(e_{i}). \tag{7}\]Several hypergraphs can be represented as _multilayer hypergraphs_, which generalize both hypergraphs and multilayer networks to model complex relationships in multi-dimensional or multi-modal data. This framework extends traditional hypergraphs by incorporating multiple layers, where each layer represents a distinct context, domain, or interaction type among entities. A multilayer hypergraph is formally defined as follows:  
\[\mathcal{H}=(\mathcal{V},\mathcal{E},\mathcal{L}), \tag{8}\]  
where \(\mathcal{L}\) is the set of layers. Each layer represents a different context, interaction type, or domain. Each layer represents a different context or interaction type, and hyperedges can be associated with one or more layers, enabling the representation of complex, heterogeneous relationships beyond simple pairwise connections. Key properties of multilayer hypergraphs include the following: (i) cross-layer connectivity--nodes can belong to multiple layers, allowing hyperedges to span across different layers, (ii) layer-specific interactions--each layer captures a distinct type of interaction, such as coherence, correlation, or mutual information, and (iii) heterogeneous hyperedges--hyperedges can vary in size and exist across multiple layers, enabling flexible modeling of higher-order relationships.