Introduction

Abstract

High-order correlations among data exist widely in various practical applications. Compared with the simple graph which can only model the pairwise relationship between two subjects, hypergraph is a flexible and representative model to formulate high-order correlations. Based on the hypergraph model, there have been many efforts to design the computation framework and analyze the high-order correlations. In this chapter, we briefly introduce the hypergraph computation, including its background, definition, history, recent challenges, and objectives.

1.1 Background

The basic elements of many natural and artificial systems have dependencies on each other and call for correlation modeling and analytic methods to study these. The graphs are all around us from different perspective, and in general all the objects in the real world are defined based on their connections with other objects. These connections can be described as a graph, which is a common data structure in many cases. For example, graphs can depict the path in a city, where each path is represented with an edge to show the spatial connections between two locations. Graphs are also employed in the airline route map, in which each vertex is an airport and each edge is an airline.

Recently, the most challenging data processing problem comes from the connected data, not just from the discrete ones. How to exploit the underneath connections behind the data has become an urgent and important task in many applications. Generally, graph has been used to formulate such correlations among data. A graph is a nonlinear data structure which is composed of a group of vertices and edges. Here, the vertices in a graph represent the subjects to be analyzed, and the edges in a graph are the lines connecting two vertices in the graph. Figure 1.1 shows an example of a graph.

Fig. 1.1

An example of a graph

As a common way to model pairwise correlations among data, the components in a system can be represented by the vertices of a graph, and the associations between components are described by the edges. In this way, the association pattern is abstracted by the topological structure of the graph. In the past decades, it was not easy to apply graph theory in practice because of the limitation of computing power. In recent years, with the advancement of information technology and computing power, graph theory has demonstrated its practical values. As scales of data grow, scientists have come up with the concept of network science. The study of network science can be applied in various fields. For example, by studying the connection relationship between terminals on the Internet, the efficiency of data transmission in a network can be estimated. The study of interpersonal relationships can help understand the way people communicate with each other, disseminate information, and generate community. Studying the transmission chain of infectious diseases can help predict risks in time, thus interrupt transmission, and prevent their spread. People have also found that many biological, social, information, and other real networks have nontrivial structural patterns in the connections among their elements. These patterns reflect meaningful features of the whole network. For example, the small-world phenomenon (the average path length in the network does not increase significantly with the increase of the network size) widely exists in social networks [1]. Another example is scale-free network [2], in which the vertex degree distribution follows a power-law distribution, and this phenomenon is known in some biological metabolic networks [3].

It is noted that the world is far more complex than just pairwise connections. Typical examples include social networks, protein–protein interaction networks, and brain networks. In social networks, the individual characteristics of users are related to the interactive patterns among users. The users with similar characteristics are more likely to connect with each other to form a social group. The social relationships of users also affect their profiling portraits. We notice that the correlations among these uses are not just pairwise connections but also group-like connections, which are more complex than these pairwise connections. Figure 1.2 shows an example of social connections, in which each user could have different types of connections with two or more other users or items.

Fig. 1.2

An illustration of pairwise correlations and high-order correlations between/among users

In human brain networks, the cerebral cortex contains more than 10¹¹ neurons and a cluster of neurons with similar functions and connections forms a nucleus. The nuclei can be further divided into different brain regions, resulting in a multilevel and multi-scale complex brain network. For example, the whole brain map includes Insula and Cingulate Gyri, Frontal Lobe, Occipital Lobe, Parietal Lobe, and other regions, which can be further divided into 90 brain regions that are provided in AAL atlas [4], such as Hippocampus and Parahippocampus. Each neuron can have more than 10,000 synapses, which can connect the neurons in the brain to other neurons in the rest of the body or connect the neurons to the muscles. The connections among the neurons are complex and hard to be formulated in a graph, although graph is a typical way to model such correlations in the brain.

Such complex correlations, i.e., the high-order correlation rather than pairwise ones, are very common in real-world data. To study these complex systems, it is necessary to characterize and analyze high-order relationships between their elements. Empirical studies have shown that the correlation patterns of a system often play an important role in functions of the system. In recent years, more researchers have begun to pay attention to this field and apply high-order correlation modeling and analytic methods.

At the beginning of the development of machine learning on graph and network science, only graph has been used to model the network or the correlations, and the associations between the elements of the system were generally described by the topological structure of the graph. As a result, the pairwise connections can be described in the graph, while a large amount of semantic information in the system could be lost, and descriptive features in the network could not be extracted. Some well-discussed network properties, such as degree centrality, semi-locality centrality, and closeness centrality, were all based on such a static single network model. The underneath high-order information behind the data has to be degenerated to pairwise ones for processing, which may lead to serious information loss. With the development of big data, the explosive growth of data demonstrates their complexity and diversity, which calls for more complex data modeling methods. The network modeling methods for complex data types, complex topological structures, and complex connection patterns emerge. For example, the social closeness between individuals in a social network can be strong or weak, and a system with weight distribution for the association between vertices can be modeled using a weighted network [5]. Also, the power network and the communication network are inter-dependent in infrastructure construction. The vertices of the communication network provide control signals to the vertices of the power network, whereas the vertices of the power network supply power to the vertices of the communication network. The interdependence between different networks can be modeled using an inter-dependent graph [6]. Another example is the air transportation network, where the routes between the vertices may belong to different airlines. For the heterogeneity of object types and association relationships, the concept of multi-layer network or graph has been proposed [7]. The last example is that the ecological food chain in the species network changes with the change of seasonal environmental conditions. For dynamic systems, the concept of temporal network has been introduced [8] to formulate the correlation among the subjects.

Although graph-based methods have been developed for decades and great progresses have been achieved, they still have limitations. These graph models can better formulate the binary relationships between the elements in the system, while they may ignore the high-order correlations among three or more elements. In recent years, many studies have shown that modeling and optimizing high-order correlations are even more important in most of the applications [9,10,11]. For example, in the biosphere system, the high-order interactions between species ensure stable diversity of species [10]. The high-order characteristics of different networks can effectively distinguish their fields [11]. With the rapid development of network science, the complexity of data and correlation increase rapidly. In the fields of biological information, social computing, and image processing, there are a large number of multi-modal, heterogeneous, high-level data, and there are needs for effective high-order correlation modeling and optimization methods.

As the subject of interdisciplinary study in many different fields including computer science, physics, and biology, high-order correlation modeling and optimization have attracted much attention in recent decades. There are a large number of high-order relationships in many systems in the real world [12]. For example, in social networks, people form groups of three or more to communicate, and in academic networks, multiple authors cooperate to write an article. Protein interactions in biological networks may occur between multiple proteins, and gene expression is driven by high-order interactions between biomolecules [13]. High-order associations among elements are difficult to be described by the topology of simple graphs. Under such circumstances, the corresponding mathematical expressions have been introduced, such as set systems [14], simplicial complexes, and hypergraphs [15]. However, how to deploy the mathematical expressions in computation paradigm is still an open problem. The complexity of high-order correlations is much higher than that of pairwise correlations, which brings about new challenges to computation paradigms.

Hypergraph, as a generation of graph, which is able to formulate high-order correlations among the data, has been investigated recently. In this book, we introduce recent progress on hypergraph computation, from hypergraph modeling to hypergraph neural networks. Below we first introduce the basic definitions of hypergraph and then show the applications and research history of hypergraph. Finally, we provide the summary of our works in hypergraph computation and the structure of this book.

1.2 The Definition of Hypergraph

The hypergraph is an important concept in discrete mathematics, which is a generalization of the graph. Therefore, many concepts of hypergraphs can be defined related to the well-known definition of graphs. A hypergraph is defined as a pair of hypervertex set and hyperedge set. The hypervertex set, also called the vertex set, is a finite set, whereas the hyperedge represents the subset of the vertex set. As the hyperedge can connect any number of vertices, more general types of relationships could be modeled by hypergraphs rather than graphs. The order and the size of the hypergraph can be defined based on the vertex set and hyperedge set, i.e., the order of the hypergraph represents the cardinality of the vertex set, and the size of the hypergraph denotes the cardinality of the hyperedge set.

Similar to graphs, two specific types of hypergraphs can be defined, including the empty hypergraph and the trivial hypergraph.

• The empty hypergraph is the hypergraph with empty vertex set and empty hyperedge set.

• The trivial hypergraph is the hypergraph with nonempty vertex set and empty hyperedge set.

Generally speaking, unless stated otherwise, hypergraphs have a nonempty vertex set and nonempty hyperedge set and do not contain empty hyperedges.

The isolated vertex denotes the vertex which is not contained in any of the hyperedges. Two vertices are adjacent if there exists a hyperedge containing both of these two vertices. Two hyperedges are incident if they have a nonempty intersection.

The sub-hypergraph and partial hypergraph can be defined as follows:

• An induced sub-hypergraph of given hypergraph is the hypergraph whose vertex set is the subset of the given hypergraph, and the hyperedges have only one element or the intersection of the vertex set no less than two.

• A sub-hypergraph of the given hypergraph is the hypergraph whose both the vertex set and the hyperedge set are the subset of that of the given hypergraph.

• A partial hypergraph is a hypergraph whose hyperedge set is the subset of the given hypergraph.

Two special types of the hypergraph can be defined based on the degree:

• A regular hypergraph is the hypergraph in which all of the vertices have the same degree.

• A uniform hypergraph is the hypergraph in which all of the hyperedges have the same degree.

The concept of connectivity is defined as follows. The loop denotes the hyperedge with only one element. The path is a vertex–hyperedge alternative sequence, where the vertex belongs to the consecutive hyperedge in the sequence. The cycle is a path whose first vertex is the same as the last vertex. The length of a path is the number of vertices in the path. A path connects two vertices if these two vertices are in the path. A hypergraph is connected if any pair of vertices is connected, otherwise it is disconnected. The distance between two vertices is the minimum length of the path connecting these two vertices. The diameter of the hypergraph is the maximum distance among all pairs of vertices.

Here, we provide an example of a hypergraph in Fig. 1.3. In this hypergraph, there are 11 vertices and 5 hyperedges. In this hypergraph, the hyperedge e ₁ connects vertices x ₁, x ₂, x ₃, and x ₄. The hyperedge e ₂ connects x ₄, x ₆, x ₇, and x ₈. The hyperedge e ₃ connects x ₅ and x ₆. The hyperedge e ₄ connects x ₁, x ₅, and x ₈. The hyperedge e ₅ is a loop, which only connects vertex x ₁₀ itself. Vertices x ₉ and x ₁₁ are two isolated vertices. The hypergraph is disconnected since x ₁₁ is not connected with any other vertex. x ₃ → e ₁ → x ₁ → e ₃ → x ₈ → e ₂ → x ₇ is a path from x ₃ to x ₇, with length 4. The distance between x ₄ and x ₅ is 3 since the shortest path from x ₄ to x ₅ is x ₄ → e ₂ → x ₈ → e ₄ → x ₅.

Fig. 1.3

An example of a hypergraph

Besides Fig. 1.3, there are also other typical illustrations of hypergraph, which are shown in Fig. 1.4. In Fig. 1.4a, each circular represents a hyperedge. In Fig. 1.4b, all the lines with the same color represent a hyperedge, which connect the vertices in the hyperedge. In Fig. 1.4c, each hollow circle indicates a hypergraph and the lines with the same color link the vertices in the hyperedge.

Fig. 1.4

Three typical hypergraph illustrations

It is noted that the hypergraph-type structures may be not explicit in many applications and they are hidden behind the data which can be observed directly. In some cases, we may only capture some pairwise correlations among the data, while the high-order correlation is needed to the regenerated based on these observations. For example, some popular citation networks, such as Cora, Citeseer, and PubMed [16], are widely used for analysis, while all these datasets only contain graph-type data, which treat the articles as vertices and the citation relationships as links. Under such circumstances, to exploit the high-order correlation among these data, we need to transform these data to a hypergraph. As a typical method, a co-authorship hypergraph can be generated, which formulates the articles as vertices, and articles with the same authors are connected by a hyperedge. In a similar way, a co-citation hypergraph can be generated, which treats the articles as vertices as well, and the articles with the same citation are treated as a hyperedge.

1.3 Applications of Hypergraph

Hypergraph has been applied across several disciplines, including biology, economics, and sociology, due to its superiority in complex correlations modeling, which has promoted intelligent applications. In this part, we introduce several typical applications of hypergraphs to help understand this powerful tool.

One representative application is social computing. The social media data have been increasing rapidly over the past couple of decades, which can provide potential population-level insights. The hypergraph [17] is a useful tool for discovering the complex and hidden correlations from the data, in which the hypergraph structure can be used to formulate the high-order correlation in social networks.

In recommender system, the hypergraph is used to model the user–item network, to profile the user, and to further predict the preferences (future interactions). Given the raw user–item network without other information than the historical interactions between users and items, hypergraph [17] can be used to discriminatively formulate the high-order connectivities among users and items separately and conduct the collaborative filtering task. Sometimes the users and the items may be attached with different attributes or properties. For example, the user-side information may include the gender, age, and personality, and the item-side information may contain the category, text description, and image. This attribute information can help capture the user’s preference. Therefore, another application of hypergraphs in recommender system is attribute modeling and inference.

Another popular yet challenging social media computing application is sentiment analysis, with the goal of recognizing the real emotions and attitudes of people in social media contexts. Nevertheless, the multi-modality and complexity of social media data have made the task more difficult. For example, the text, images, and videos may coexist in one tweet. Additionally, there are intricate relationships between posts, such as in the dimensions of time, location, and user preferences. Therefore, how to find out the complex relationship between tweets and analyze the user sentiment has become an urgent issue. To this end, hypergraph [18] can be used to formulate the correlation among each sample and conduct robust and accurate multi-modal sentiment prediction, taking into consideration different moods having their own characteristics, and that sentiment analysis should be based on the joint analysis of multiple information. As far as social event detection is concerned, exploring a set of highly related posts becomes more important because of noise and insufficient content in a single post that fails to convey clear and comprehensive information. Hypergraph [19] can be used to characterize the relationship between heterogeneous data among different tweets for its superiority in modeling high-order correlations between data of various posts, modalities, and times, therefore enabling real-time social event detection. Specifically, each microblog is connected with its several textual-related and visual-related microblogs and forms two hyperedges. Next, the microblog clique, a basic unit consisting of a set of highly related tweets, is produced by using the hypergraph cut method to put together microblogs that are about the same subject.

Hypergraph has also shown its advantage in medical and biological applications. In the past few decades, massive amounts of biological and medical data have been produced. The data is complex, heterogeneous, and multi-modal, with interwoven inter- and intra-data correlations. By concatenating hyperedge groups, the hypergraph [20,21,22] can naturally accommodate multi-modal or heterogeneous data. Moreover, in doing so, it can discriminatively use the complementary information among these data. The pipeline below can be used to describe how hypergraph computation is used in biological and medical tasks: (1) modeling the medical image, patches, or biological entities as vertices and connecting them with hyperedges based on their feature similarity or high-order topological links and (2) learning high-order correlations between data using a series of hypergraph computation methods. In this type of applications, hypergraph has been used for mild cognitive impairment (MCI) identification using magnetic resonance imaging (MRI) [23], COVID-19 identification using CT imaging [24], ASD identification using brain functional networks [25], medical image retrieval [26], etc.

The aforementioned examples are just a small part of hypergraph applications. Hypergraph computation techniques can be used in any cases where there exist high-order and complex correlations among data, such as computer vision, knowledge graph, and so on.

1.4 The History of Studies on Hypergraph

1.4.1 Topology and Coloring on Hypergraph

The studies of utilization on hypergraph have a long history. In 1943, Prenowitz et al. [27] first illustrated several kinds of geometries (projective, descriptive, and spherical) as hypergroup or multigroup. Prenowitz et al. [28] created Geometries on Join Spaces, a unique hypergroup that has been proven to be a valuable tool in the study of a variety of topics, including graphs, hypergraphs, binary relations, fuzzy sets, and rough sets. In 1996, Rosenberg et al. [29] first addressed the relationships between Hyperstructures (hypergraphs) and Binary Relations in the broadest sense. Later, they were also studied by Corsini and Leoreanu [30]. Rosenberg et al. [29] first developed join spaces related to fuzzy sets in 1996. Corsini, Leoreanu, and Tofan [31] have all reexamined these structures. Zahedi et al. [32] also advanced the concepts of linking a hypergraph with a fuzzy set and examining algebraic structures equipped with a fuzzy structure.

Hypergraph coloring is a typical and important task, which has attracted much attention since last century. It is fundamental to combinatorics and can be used to determine bounds for the chromatic number of some graphs as described by Kierstead et al. [33]. Lu et al. [34] suggested these algorithms to solve different optimization problems, such as divide and conquer and partition problems, in which hypergraph coloring can also be used to find monochromatic paths and cycles. Voloshin et al. [35, 36] described how to color mixed hypergraphs, which are divided into hyperedge and anti-hyperedge families. In such a case, they further applied it to energy supply problem.

The problem of finding large matches is closely related to the problem of bounding the chromatic index of a hypergraph (notice that the color classes of a proper edge-coloring form a matching). As a classical subject in the study of graphs, matching theory is very well developed and goes back to the work [37] in the 1930s. Tutte’s theorem [38] is a characterization of graphs that contains perfect matchings. Edmonds et al. [39] proposed the Blossom algorithm, which uncovers a maximum matching in a graph in a polynomial amount of time for graphs containing a perfect matching. The above methods are early works on hypergraph-related research.

1.4.2 Hypergraph Partitioning, Clustering, and Machine Learning

Hypergraph partition is another important problem on hypergraph. It is defined in the Encyclopedia of Parallel Computing¹ that hypergraph partitioning involves dividing a hypergraph into two or more roughly equal parts in such a way that the cost function of the hyperedge connecting vertices in the different parts is minimized. In many cases, this definition is too restrictive and requires more than two parts. Karypis et al. [40] proposed the hMetis algorithm, which is based on multilevel coarsening of hypergraphs. The method iteratively bisections coarsened hypergraphs, starting with the smallest. George et al. [41] further developed the hMeTiS-Kway algorithm, which directly constructs a K-way partitioning of a hypergraph with coarse–uncoarse paradigm to solve the K-way hypergraph partitioning problem.

Besides, Papa et al. [42] provided several methods of partitioning hypergraphs and defines clustering as “the process of merging vertices into larger groups of vertices known as clusters to compute a coarser hypergraph from an input hypergraph.” A number of applications of partitioning and clustering are also given, including VLSI design, numerical linear algebra, automated theorem proving, and formal verification. Several applications and methods have been described in the literature. For more details, a survey of clustering ensemble techniques has been published in [43], which includes hypergraph partitioning techniques as well. Multilevel strategies are often required in clustering and partitioning, which have been well studied in previous works. It has been extensively used in VLSI design [40], parallel scientific computing [44,45,46], image categorization [47], and social networks [48, 49].

In this century, hypergraph has been used in machine learning. Transductive hypergraph learning [48] is introduced to give the basic mathematical formulation of the objective function for predicting labels of vertices on a hypergraph. Since the performance of hypergraph learning is related to the modeling quality of the hypergraph, there are some efforts to further assign weights to the components in the hypergraph, including hyperedges, vertices, and hyperedge-dependent vertex weights [50, 51]. To accelerate the label propagation process on hypergraph, the cross diffusion on multiple hypergraphs is further introduced to model the high-order correlations among multi-modal data and conduct multi-modal information fusion [52].

1.4.3 Deep Learning on Hypergraph

Research on high-order representations of hypergraph structures has also been inspired by deep learning’s powerful learning and modeling abilities. Generally speaking, most deep learning methods on hypergraph can be divided into spectral-based methods and spatial-based methods.

As for the spectral-based methods, Feng et al. [53] proposed Hypergraph Neural Networks (HGNNs) to model non-pairwise relations based on the hypergraph Laplacian. Multi-modal data can be naturally modeled using the proposed methods. It is also possible to classify images using hypergraph neural networks[54]. Using tools from the spectral theory of hypergraphs, Yadati et al. [55] proposed HyperGCN to train a GCN for semi-supervised learning on hypergraphs using graph convolutional networks (GCNs). As for the spatial-based method, by extending the dynamic hypergraph learning, Jiang et al. [56] proposed a dynamic hypergraph neural network, which can adaptably change the hypergraph structure at each layer. As opposed to hypergraph convolution, where the underlying structure is defined beforehand, Bai et al. [57] proposed a hypergraph attention mechanism strategy to learn a dynamic connection of hyperedges, which propagates and gathers information in the task-relevant parts of the graph, thereby generating more discriminative vertex embeddings. Moreover, Gao et al. [58] proposed a general hypergraph neural network framework, which can be applied to multiple types of hypergraphs like undirected hypergraph, directed hypergraph, probabilistic hypergraph, vertex/hyperedge weighted hypergraph, etc.

For homogeneous and heterogeneous hypergraphs, Zhang et al. [59] proposed a self-attention-based hypergraph neural network (Hyper-SAGNN). By mapping the hypergraph to a weighted attribute line graph, Bandyopadhyay et al. [60] achieved a bi-injective hypergraph structure. Huang et al. [61] proposed UniGNN, which can generalize general GNN models into hypergraphs by interpreting the message passing process in graph and hypergraph neural networks. These neural network methods on hypergraph enable the representation learning by incorporating high-order correlation in process.

1.5 Hypergraph Computation: Challenges and Objectives

Hypergraph has its advantage on high-order correlation modeling compared with graph and other structures. To take this advantage in practice, hypergraph can be used to formulate such correlations and the conduct computing task accordingly. In this part, we summarize the objective of hypergraph computation, especially the main challenges and the tasks inside.

Below we give the definition of hypergraph computation: hypergraph computation is to formulate the high-order correlations underneath the data using hypergraph and then conduct semantic computing on the hypergraph for different applications.

The main challenges and objectives in hypergraph computation are from three parts, including how to generate a hypergraph, how to deal with large scale data, and how to conduct learning on hypergraph.

• 1. How to generate a hypergraph. In most cases, the hypergraph structure is not explicitly existed. What can be observed could be non-structure data, such as images, videos and discrete signals, and pairwise relationships between two subjects. To reveal the underneath high-order correlation as a hypergraph, it is needed to define how to generate it. More importantly, the observed data could be noisy, missing, and tend to be multi-modal. How to describe these data is also challenged. Under such circumstances, it is difficult to generate an accurate hypergraph structure based on these data. Therefore, how to generate a hypergraph, especially a good hypergraph structure for specific task, is the first challenge in practice.

• 2. How to deal with large scale data. Computational complexity is a major issue for graph data, which is also very serious for hypergraph. The data in many applications, such as social media and brain neurons, are in million level or more. Confronting such large scale data, how to effectively and efficiently conduct storage and computing on hypergraph require further research.

• 3. How to conduct learning on hypergraph. Given a hypergraph, learning task can be conducted on the hypergraph structure, and it is important to design label propagation method on hypergraph. Besides traditional feature representation methods, the connections can also be used as representation. Given such high-order correlation by hypergraph, it is useful to learn new representations on hypergraph. Therefore, how to conduct representation learning on hypergraph is an important topic.

Hypergraph modeling can be briefly divided into two categories, i.e., the intra-correlation modeling and the inter-correlation modeling, as shown in Fig. 1.5. Here, the intra-correlation modeling regards the high-order correlations inside the subject. The components of the subject are represented as the vertices, and the correlations among these components are represented as hyperedges in the hypergraph. In these cases, the hypergraph, named intra-hypergraph, aims to represent the subject itself. The inter-correlation modeling concentrates on the high-order correlations among different subjects. A group of subjects is represented as the vertices, and the correlations among these subjects are represented as hyperedges in the hypergraph, named inter-hypergraph. The objective is to learn the representation or connections of the target subject with the help of its correlations to other subjects. Here we take image representation as an example. When an image is selected as the subject, the correlations among the pixels or the patches in the image are intra-correlations, and the corresponding intra-hypergraph can be generated for image representation. On the other side, we can also observe other images for processing. The correlations among the subject image and other images are inter-correlations, and the corresponding inter-hypergraph can be generated for image representation too. That is to say, the intra- and inter-correlations can be regarded as the views from different scales. If we take the subject itself as the target system, the correlations of the subject and other subjects are inter-correlations of the subject, corresponding to an inter-hypergraph. If we take the group of subjects as the target system, the correlations of these subjects are intra-correlations, leading to an intra-hypergraph accordingly.

Fig. 1.5

The intra-hypergraph and the inter-hypergraph based on the intra-correlations and the inter-correlations among components and subjects

1.6 Structure of This Book

This book is composed of 13 chapters and the structure of the remainders is introduced here.

• Chapter 2. Mathematical Foundations of Hypergraph. This chapter introduces the fundamental mathematics of hypergraph and presents the mathematical notations that are used to facilitate deep understanding and analysis of hypergraph structure.

• Chapter 3. Hypergraph Computation Paradigm. This chapter introduces three typical hypergraph computation paradigms, including inter-representation computing, inter-representation computing, and group correlation computing.

• Chapter 4. Hypergraph Modeling. This chapter introduces different hypergraph modeling methods, including implicit hypergraph modeling and explicit hypergraph modeling. Examples on computer vision, recommender system, and other applications are also provided in this chapter.

• Chapter 5. Typical Hypergraph Computation Tasks. This chapter introduces the typical hypergraph computation tasks, including label propagation on hypergraph, data clustering on hypergraph, imbalanced learning on hypergraph, and link prediction on hypergraph.

• Chapter 6. Hypergraph Structure Evolution. This chapter introduces the structure evolution methods on the hypergraph, which optimize the hypergraph structure accordingly, including both the hypergraph component optimization and hypergraph structure optimization. We briefly introduce the incremental learning method on growing data.

• Chapter 7. Neural Networks on Hypergraph. This chapter introduces recent progresses on hypergraph neural networks, including the spectral-based methods and the spatial-based methods. The comparison between graph neural networks and hypergraph neural networks is also provided in this chapter.

• Chapter 8. Large Scale Hypergraph Computation. This chapter introduces how to deal with large scale data. More specifically, two kinds of large scale hypergraph computation methods, i.e., factorization-based hypergraph reduction and hierarchy-based hypergraph learning, are provided in this chapter.

• Chapter 9. Hypergraph Computation for Social Media Analysis. This chapter introduces applications of hypergraph computation on social media analysis, including recommender system, sentiment analysis, and emotion recognition.

• Chapter 10. Hypergraph Computation for Medical and Biological Applications This chapter introduces applications of hypergraph computation on medical and biological applications, including computer-aided diagnosis, survival prediction with histopathological image, drug discovery, and medical image segmentation.

• Chapter 11. Hypergraph Computation for Computer Vision. This chapter introduces applications of hypergraph computation on computer vision, including visual classification, 3D object retrieval, and tag-based social image retrieval.

• Chapter 12. The DeepHypergraph Library. This chapter introduces the DeepHypergraph Library, a hypergraph computation library based on Python.

• Chapter 13. Conclusions and Future Work. This chapter concludes this book and introduces three further research directions of hypergraph computation.

1.7 Summary

In this chapter, we introduce the basic ideas and background of hypergraph computation. We also provide the applications and the related research history on hypergraph. The idea of hypergraph computation is detailed introduced and discussed in this chapter. We also summarize our studies on hypergraph computation and present the organization of this book.