13.1 Summary of This Book

Hypergraph computation has attracted much attention and shown apparent advantages in many application fields, such as computer vision, social networks, and biomedicine. In this book, we systematically introduce the basic knowledge, algorithms, and applications of hypergraph computation in three parts and discuss some recent progress in this direction.

In the first part, we mainly introduce the basic knowledge and main concepts of hypergraphs, including the definitions and symbols of common terms and the classification of hypergraphs. More importantly, we discuss the differences between hypergraphs and graphs from several aspects. Following, we introduce three hypergraph computation paradigms, namely, intra-hypergraph computation, inter-hypergraph computation, and hypergraph structure computation. In this part, we can have a general view of the different objectives in hypergraph computation.

In the second part, we specifically introduce a series of algorithms from hypergraph modeling to hypergraph neural networks. In hypergraph modeling sections, we show how to build a hypergraph structure from the collected data. As a typical and fundamental learning framework, label propagation on hypergraph describes how to derive the labels for unknown data from the labels for known data on the structure of a hypergraph. Other typical hypergraph computation tasks, including data clustering, cost-sensitive learning, and link prediction, are also introduced. Regarding the potential inaccurate hypergraph structure, we present the hypergraph structure evolution methods, which optimize the hypergraph structure on the basis of the initial structure. We further introduce the hypergraph neural network, which integrates the neural network framework into the hypergraph computation framework. The large scale hypergraph chapter discusses how to deal with large scale data for classification and clustering applications.

In the third part, we introduce practical examples of hypergraph computation in social media analysis, medical and biological applications, and computer vision, including specific tasks such as recommender system, sentiment analysis, computer-aided diagnosis, and image classification. In these examples, we show how to use hypergraph for high-order correlation modeling and select computation paradigms for different objectives. We further introduce the DeepHypergraph library for hypergraph computation.

13.2 Future Work

Although there have been many efforts to promote the development of hypergraph computation, there are still many open issues that need deep exploration, for instance, the mathematical foundations of hypergraph computation, the interpretability issues, and the temporal hypergraph modeling:

  1. 1.

    At present, the theory of hypergraph modeling and optimization is still far from completeness. As a flexible modeling method for high-order complex correlations, the hypergraph’s main components, i.e., the number and degree of hyperedges on hypergraph, are not fixed, and how to measure the complexity of hypergraph structure is a problem worth further exploring. Previous investigation has shown superior performance of hypergraph computation in various applications, while the fundamental reason for this improvement and how much gain we can have from such high-order correlation modeling are still without a clear answer. In many tasks such as hypergraph matching, it is necessary to define the metrics in the hypergraph space. However, the problem is computationally expensive when the scale of hypergraph is very large. Therefore, efficient hypergraph matching and other algorithms are in immediate need. Existing hypergraph modeling methods still lack an evaluation of the quality of high-order correlation modeling and therefore lack credibility. It is needed to further explore the relationship between task complexity and structural complexity considering both the input data and the downstream tasks. It is expected that the hypergraphs can be generated according to the complexity of specific tasks and data, so as to achieve more reliable hypergraph modeling and optimization performance.

  2. 2.

    Interpretability is also an important research area of neural network models, and its purpose is to complete the explanation of black-box models through techniques such as feature masking and visualization. Since the hypergraph structure provides additional topological information, it brings out new opportunities to interpretability of hypergraph neural networks. Although there has been some work on the explanation problems of deep graph models in recent years, it is still in the infant stage. Interpretability of deep hypergraph model could be a potential road toward better deep neural network interpretability. There are two feasible paths to explanation techniques for hypergraph neural networks: instance-level explanation methods and model-level explanation methods. For example, it is possible to use different mask generation algorithms to obtain masks corresponding to vertices, edges, or the incidence matrix and then apply the masks as disturbances to cover the original structure information to study the effects of different disturbances to the original structure. In addition, for hypergraphs in biochemistry, neurobiology, ecology, and engineering, of which the structure is highly correlated with their functions, how to combine domain knowledge to improve model interpretability is also an important issue. Finally, for text or image data, humans can easily understand the semantic information. However, it is difficult to intuitively understand the information of hypergraph structure. How to visualize the high-order complex correlations for intuitive understanding remains a challenge.

  3. 3.

    The combination of the temporal sequences and the hypergraph neural networks is also worth exploring. Recent research mainly focuses on static data and static hypergraph, where the data and the structure are kept fixed. However, in real-world applications, data may vary over time, which is called the temporal sequence, as well as the topology among the data. Therefore, the temporal information should be considered, and the temporal hypergraph neural networks aim to combine the temporal and spatial information. According to the variation of the data and the structure, there exist two main scenarios:

    • Time sequence data with static structure. This is a common scenario in the field of traffic forecasting, action recognition, and anomaly detection.

    • Time sequence data with evolving structure. This scenario mostly appears in the field of stock prediction and video relation detection.

    Under the above application circumstances, temporal hypergraph modeling is worth study. There are multiple challenges for the tasks mentioned above. For the sensor data, different types of the data are raised by different types of the sensors, while the typical hypergraph neural networks treat the data of the vertices equally. Both temporal and spatial high-order relationships vary over time, which makes the message passing procedure complex. New vertices/hyperedges emerge, and old vertices/hyperedges dissolve during the variation of the structure, which makes it complex to continuously model the varying correlation and aggregate the messages. The vertices/hyperedges may even be completely different at different time steps, which makes the representation-based method questionable. In order to model the temporal information, the vertex representations should be dynamic, and therefore, the representation should be learned on a functional space, rather than on the common vector space. The temporal information from both the vertex representation and the structure topology defies extraction. Considering these challenges, the temporal hypergraph still has a long way to go and needs further exploration.

Besides the above research directions, there are also several other interesting topics, such as big hypergraph model, hypergraph database, and distributed hypergraph, which have not been introduced in detail in this book and deserved further study.