$(k,q)$-core decomposition of hypergraphs

Recently, structural and dynamical properties of complex networks are being reconsidered in the framework of a hypergraph and simplicial complex to reflect individual-group relationships in a real situation. Here, the group consists of more than two elements, and the interaction among them is called higher-order interaction (HOI). Examples include the coauthorship of a research paper, the synapse of neurons in the human brain, and functional complexes in biological networks such as metabolic and protein interaction networks. Such a group interaction differs from a linear combination of pairwise interactions. For instance, if a protein is missing in a functional complex, then that functional complex does not work at all [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16].

A hypergraph consists of nodes (elements) and hyperedges (such as social groups, synapse, and functional complexes). The number of elements in a group is called hyperedge size and is denoted as $n_{i}$ with hyperedge index $i$. A node can relate to more than one hyperedge. Therefore, hyperedges are connected, as shown in Fig. 1. In contrast, a graph consists of nodes and links representing pairwise interactions.

Emerging phenomena created in the HOI representation can be different from those in the graph representation. For instance, oscillators governed by the Kuramoto model with HOIs can exhibit an explosive synchronization transition. However, oscillators with pair-wise interactions exhibit a continuous synchronization transition [9, 10, 11]. The conventional epidemic models such as susceptible-infected-susceptible (SIS) and susceptible-infected-recovered (SIR) models exhibit continuous transitions when interactions are pairwise. However, they can exhibit discontinuous transitions when interactions are higher-order [4, 5, 6, 7]. Moreover, percolation transitions with HOI can be discontinuous [17, 18]. Therefore, due to HOIs, various phase transitions in hypergraphs can have different types of transitions such as discontinuous and hybrid transitions.

$k$-core in the graph is the maximal subgraph in which all nodes have degrees of at least $k$ [19]. The simplest way to obtain a $k$-core is recursively pruning all nodes with degrees less than $k$ [20]. This pruning process decomposes a graph into a core subgraph in a hierarchical manner. This $k$-core decomposition has been applied to various issues [21]. For example, it serves as an efficient way to identify influential spreaders in epidemic contagion [22] and vulnerable (or robust) nodes in an electrical power grid system [23]. Additionally, it is applied in many areas such as protein interactions [24], evolution [25], neuron systems. [26].

$k$-core percolation was studied in random graphs: Once an Erdős-Rényi (ER) random network of size $N$ with mean degree $z$ is generated, all nodes with degrees less than $k$ are deleted recursively until no more nodes with degrees less than $k$ remain in the system. The fraction of nodes remaining in the largest $k$-core subgraph is defined as the order parameter $m$. The mean degree $z$ plays a role as the control parameter. The order parameter $m$ is of $O(1)$ for $z>z_{c}$, where $z_{c}$ is a transition point and decreases continuously with $z$. As $z$ approaches $z_{c}$, when $k\geq 3$, the deletion of a node from the ER network can lead to collapse of the giant $k$-core subgraph. Thus, the order parameter is expressed as

where $m_{0}$ and $a$ are constants, and $\beta=1/2$. The order parameter is discontinuous at the transition point and exhibits critical behavior. Thus, $k$-core percolation exhibits a hybrid phase transition [27, 28, 29, 30, 31, 32].

In this study, we extend the $k$-core decomposition in the graph into $(k,q)$-core decomposition in the hypergraph. With this, previous researchers considered the following: In Ref. [17], $k$-core percolations for vertex and hyperedge are considered separately. This corresponds to the $(k,1)$- or $(1,q)$-core decomposition in our case below. Ref. [18] proposed another type of decomposition process by removing vertices with degree one and their neighbors repeatedly. However, this core cannot be considered a variant of the $(k,q)$-core we consider here.

In hypergraph, $(k,q)$-core decomposition can be proposed easily by converting a hypergraph into a bipartite network comprising nodes and hyperedges. In this bipartite representation, each hyperedge is connected to nodes that are the elements of the hyperedge. We call the connection between the pair of a node and a hyperedge bipartite link. Intraedges between two nodes or two hyperedges are not permitted. $(k,q)$-core is the subgroup of the given hypergraph in which every node (hyperedge) has more than or equal to $k$ ($q$) degrees. $(k,q)$-core was initially proposed in computer science [33, 34, 35] as a visualization tool for large-scale hypergraph data. However, those studies did not deal with phase transitions. $(k,q)$-core percolation is similarly defined to $k$-core percolation. We will consider the phase transition of $(k,q)$-core percolation.

A $(k,q)$-core substructure in the hypergraph is not similar to the corresponding $k$-core in the graph. We will show that the $(k,q)$-core decomposition is more efficient than $k$-core decomposition to identify an influential group such as leading scholar and super spreader groups. Moreover, the structure of the $(k,q)$-core enables us to understand what extent the information propagates or disease spreads across hyperedges [36, 37]. This information is helpful for setting up a vaccination strategy.

This paper is organized as follows: In Sec. 2, we first introduce and formulate $(k,q)$-core decomposition of random hypergraphs, in which node degrees and hyperedge sizes are not correlated and their distributions have finite moments. Additionally, we determine the transition type of $(k,q)$-core percolation analytically and numerically. It exhibits a hybrid phase transition. The critical behavior of the hybrid transition is calculated. In Sec. 3, the relaxation dynamics of the fraction of vertices with a given degree are considered at a transition point. The relaxation dynamics decays in a power-law manner. The associated exponent values are determined for degree and hyperedge size. In Sec. 4, we show that a redundant modular structure can be selected and eliminated using $(k,q)$-core decomposition in a real coauthorship hypergraph. In Sec. 5, the summary and discussion are presented.

We investigate the properties of $(k,q)$-core percolation on a random synthetic hypergraph of different degree distributions. We focus on the properties of vertices. However, the properties of hyperedges can be obtained similarly.

Random hypergraphs can be constructed by generalizing the rule of Erdős–Rényi (ER) random graphs [38]. Given system size $N$, the mean degree $\langle z\rangle^{(v)}$, and the mean hyperedge size $\langle n\rangle^{(e)}$, we consider $N$ vertices and $E=N\langle z\rangle^{(v)}/\langle n\rangle^{(e)}$ empty hyperedges first. Then, we select a pair of vertices and hyperedges randomly and add a bipartite link between the pair if they are not connected yet. This process is repeated until the number of bipartite links reaches $L=N\langle z\rangle^{(v)}=E\langle n\rangle^{(e)}$. Multiple bipartite links are not allowed, i.e., a hyperedge cannot contain the same vertex more than once.

This model generates Poisson distributions of both degrees $P^{(v)}(z)={(\langle z\rangle^{(v)})^{z}}e^{-\langle z\rangle^{(v)}}/{z!}$ and hyperedge size $P^{(e)}(n)={(\langle n\rangle^{(e)})^{n}}e^{-\langle n\rangle^{(e)}}/{n!}$ because each vertex (hyperedge) is selected independently with uniform probability $1/N$ ($1/E$).

Here, we discuss the structural difference between the $(k,q)$-core of a hypergraph and the $k$-core of its corresponding graph representation. Consider a group of $\ell$ vertices forms a hyperedge. In the graph representation, they constitute a clique of $\ell$ fully connected nodes with internal degree $\ell-1$. This clique can be removed completely only by $k$-core decomposition with at least $k\geq\ell$. In contrast, in the hypergraph representation, vertices or hyperedges associated with such a group can be removed by $(k,q)$-core decomposition for $k$ smaller than $\ell$. This difference becomes more pronounced as $\ell$ becomes larger. Decomposing the graph representation eliminates all the nodes with a degree less than $\ell$, resulting that the remaining structure tending to be over-pruned and grossly fragmented. In contrast, the $(k,q)$-core decomposition with $k<\ell$ can extract the group elements more selectively and distort the other parts of the hypergraph significantly less.

For example, we consider a coauthorship hypergraph comprising authors who published papers citing particular papers [40, 41, 42, 43, 44] in the network science field in the early era, during the years 1999–2004 [45]. Here, authors are vertices, and the coauthors of a paper are regarded to form a hyperedge. We selected the network science field because this field is a multidisciplinary field ranging from mathematics and theoretical physics to biological science. Generally, the number of coauthors in the former field is less. In contrast, the number of coauthors in the latter field is routinely more than 10 or even a 100. Therefore, analysis using $k$-core decomposition in graph representation is bound to overvalue the contribution of the researchers from the biological science field.

To avoid this overestimation, we apply the $(k,q)$-core decomposition to this coauthorship hypergraph. Figure 7 demonstrates that even $(2,2)$-core decomposition can perform well by filtering out authors who contributed to only a single “mega-project” paper. For example, the coauthorship hypergraph [45] contains a paper on the Human Genome Project [46] of 274 authors and several papers on yeast genetic interaction [47] and Drosophila melanogaster protein interactions [48]. Many authors of this paper are removed from the $(k,q)$-core decomposition with $k>1$. The $(k,q)$-core of the coauthorship hypergraph indicates that a team composed of $q$ or more coauthors had written more than $k$ papers. This observation implies that the $(k,q)$-core decomposition can be a prototype method for systematically selecting not only large but also active modules. For the coauthorship hypergraph, the core structure is presented in Fig. 8.

It would be interesting to consider the $(k,q)$-core percolation in the coauthoship hypergraph in which degrees are likely to follow a heavy-tailed distribution, whereas hyperedge sizes follow a Poisson distribution [45]. We perform numerical simulations by extending the way proposed in Ref. [32]: Once we obtain a $(k,q)$-core of the coauthorship hypergraph, we remove a node randomly selected. This removal can trigger another cascading pruning process, leading to a new $(k,q)$-core. This process enables us to capture the percolation property from real-world data. All panels in Fig. 9 show that the order parameter, i.e., the fraction of nodes remaining in the ($k,q$) core decreases to zero continuously. Accordingly, the $(k,q)$-core percolation does not exhibit a hybrid phase transition (see Fig. 9). Hence, we need further in-depth theoretical study to uncover the cause of this different type of ($k,q$)-core percolation transition. Because the real coauthorship data contain correlations between degrees and/or degree and hyperedge size, the heterogeneity of degree distribution could not be the main matter.

In this study, we studied $(k,q)$-core decomposition of hypergraphs that encompasses both individual and group units. The evolution equation was derived for uncorrelated hypergraphs to describe the recursive pruning process. The self-consistency equation of the core structure was obtained through the generating functions, and then, the features of $(k,q)$-core percolation transition were elucidated. When $k\geq 3$ or $q\geq 3$ and the second moments of the degree and size distributions are finite, the $(k,q)$-core percolation exhibits a hybrid phase transition. This implies that the order parameter jumps, while critical behavior appears at the transition point. Moreover, the relaxation of the fractions of vertices with degree $z=k-1$ and that of hyperedges of size $n=q-1$ at the transition point exhibits universal behavior $\sim t^{-2}$ for $k\geq 3$ or $q\geq 3$. When $k=q=2$, the novel degree-dependent critical relaxation dynamics was analytically derived and numerically confirmed as $P^{(v)}(z,t)\sim t^{-z}$ and $P^{(e)}(n,t)\sim t^{-n}$ for $z$ and $n\geq 2$, and both decay as $\sim t^{-3}$ for $z=n=1$.