# Hypergraph Analysis of Functional Brain Connectivity During Figurative Attention
## 2 Mathematical Basis
### Summarization  
Graphs of real datasets are often very massive. For example, the graph constructed on the basis of the MEG data has 306 nodes, each representing time series recorded by magnetic sensors [30; 43]. The manipulation of such large networks requires high computational capacity and extensive time. To decrease the communicational cost, a graph summarization technique is used [44]. Summarization methods allow one to reduce the graph size by extracting the most important information from the original graph. Then, the resultant summary graph can be queried, analyzed, and understood more efficiently using existing tools and algorithms. The resultant graph can more easily visualize the dataset that is originally too large to load into memory. In addition, real graph data are frequently large scale and considerably noisy, with many hidden, unobserved, or erroneous links and labels. Such noise hinders analysis by increasing the workload of data processing and hiding the more important information. Summarization serves to filter out noise and reveal specific features in the data.  
Summarization is application-dependent and can be made using various methods. The most popular summarization techniques are node grouping and edge grouping. While the node-grouping methods recursively aggregate nodes into \"supernodes\" based on an application-dependent optimization function, which can be based on structure and/or attributes, e.g., clustering techniques and map each densely connected cluster to a supernode, edge-grouping methods aggregate edges into hyperedges. In this paper, we apply the latter approach that compresses neighborhoods around high-degree nodes, accelerating query processing and enabling direct operations on the compressed graph. The edge-grouping method was used by [PERSON] and [PERSON] [45], who introduced \"compressor nodes\", which represent common connections of high-degree nodes. They assumed that high-degree nodes are surrounded by redundant information that can be synthesized and eliminated. To provide global guarantees and reduce the scope of compressor handling during query processing, dedensification only occurs when every node has at most one outgoing edge to a compressor node, and every high-degree node has incoming edges coming only from a compressor node. These guarantees are then used to create query processing algorithms that enable direct pattern matching queries on the compressed graph.  
Other summarization techniques imply simplification or sparsification. These methods streamline an input graph by removing less \"important\" nodes or edges, resulting in a sparsified graph. In the brain connectivity network, some edges are more indicative of predicting cognitive performance. Therefore, the node grouping layer is designed to \"hide\" the non-indicative ('noisy') edges by grouping them into a cluster (supernode), thus highlighting the indicative edges. Finally, there are influence-based approaches that aim to discover a high-level description of the influence propagation in large-scale graphs. Techniques in this category formulate the summarization problem as an optimization process in which some quantity related to information influence is maintained.  
The output of a summarization procedure can take one of two forms: (i) a sparsified graph, which retains only a subset of nodes and/or edges from the original graph, or (ii) a hypergraph composed of selected nodes united by hyperedges. Unlike traditional graphs, hypergraphs allow nodes to belong to multiple hyperedges, enabling the representation of complex, higher-order relationships within the data. In this paper, we integrate both approaches, leveraging sparsified graphs as a foundation to construct hypergraphs that model brain connectivity associated with attention. This combined methodology enables the capture of intricate neural interactions while maintaining computational efficiency, providing a robust framework for analyzing attention-related brain networks.