# Hypergraph Analysis of Functional Brain Connectivity During Figurative Attention
## 2 Mathematical Basis
### Main Definitions  
A _plain graph_\(G\) is an abstract mathematical structure composed of a finite set of \(n\) vertices (or nodes) \(V\) and a set of \(m\) edges (or links) \(E\), where each edge represents an interaction between a pair of vertices. Mathematically, the plain graph is defined as  
\[G=(V,E),\;\;V=\{v_{1},\ldots,v_{i},\ldots,v_{n}\},\;\;E=\{v_{1},\ldots,v_{j}, \ldots,v_{m}\}. \tag{1}\]  
The plain graph is commonly represented by its adjacency matrix \(A\), which can be either binary (indicating the presence or absence of a connection between two vertices) or numerical (reflecting the strengths or weights of the connections between edges). A graph with labels assigned to its edges and vertices is referred to as a _labeled graph_. A label of edge \(j\) is its weight \(w(e_{j})\) (or connectivity measure), i.e., \(\text{Lab}(e_{j})=w(e_{j})\), while a label of vertex \(i\) is the sum of weights of all edges which connect this vertex with other nodes, i.e.,  
\[\text{Lab}(v_{i})=\sum_{j}^{m}w(e_{j})_{\{e_{j}\sim v_{i}\}}, \tag{2}\]  
where \(w(e_{j})_{\{e_{j}\sim v_{j}\}}\) is the indicator function, equal to weight \(w(e_{j})\) of edge \(j\) if this edge is incident to vertex \(i\), and 0 otherwise. In contrast to a plain graph, where an edge connects only two vertices, a _hypergraph_ generalizes this concept by allowing a single edge, called a hyperedge, to connect more than two vertices [31, 32]. Formally, a hyperedge is defined as a collection of links that connect two or more edges with significantly correlated temporal or spectral profiles. Thus, a hypergraph is defined by a set of such hyperedges and nodes, offering a powerful framework for modeling and analyzing complex relationships, particularly in systems such as brain networks. To illustrate this, consider a brain neural network that is conventionally partitioned into \(n\) distinct regions, referred to as nodes or vertices. Signals from these regions can be recorded using various neuroimaging modalities, such as EEG, MEG, or fMRI. When the brain is exposed to a stimulus, neuronal activity in specific regions associated with the stimulus is either activated or deactivated, and information about this process is encoded in the corresponding vertices. By capturing the relationships between these regions, a hypergraph \(\mathcal{H}\) is defined by a set of \(n\) vertices \(\mathcal{V}\) and a set of \(m\) hyperedges \(\mathcal{E}\) as  
\[\mathcal{H}=(\mathcal{V},\mathcal{E}),\;\mathcal{V}=\{v_{1},v_{2},\ldots,v_{i },\ldots,v_{n}\},\;\mathcal{E}=\{e_{1},e_{2},\ldots,e_{j},\ldots,e_{m}\}, \tag{3}\]  
where each hyperedge \(e_{j}\in\mathcal{E}\) represents a specific measure of connectivity. In this work, we consider widely used measures of brain connectivity, such as coherence, cross-correlation, and mutual information. These measures can be computed for specific spectral bands and time intervals of interest, allowing for a detailed characterization of functional relationships within the brain. A hypergraph is characterized by its _order_ and _size_. The order refers to the number of vertices \(|\mathcal{V}|=n\), while the size corresponds to the number of hyperedges \(|\mathcal{E}|=m\). Furthermore, both vertices and hyperedges are associated with a degree (Deg), which quantifies their connectivity within the hypergraph. Specifically, the degree of a vertex \(v\in\mathcal{V}\) is defined as the number of hyperedges incident to it, whereas the degree of a hyperedge \(v\in\mathcal{E}\) is defined as the number of vertices incident to it. Mathematically, these degrees can be expressed as follows:  
\[\text{Deg}(v)=\sum_{e\in\mathcal{E}}1_{\{v\in\mathcal{E}\}}\;\;\text{and}\; \;\text{Deg}(e)=\sum_{v\in\mathcal{V}}1_{\{v\in e\}}, \tag{4}\]  
where \(1_{\{\cdot\}}\) is the indicator function, equal to \(1\) if the condition is satisfied and \(0\) otherwise.