Hypergraph querying using structural indexing and layer-related-closure verification

Abstract

Graph indexing and querying mechanisms have been receiving significant attention due to their importance in analyzing the growing graph datasets in many domains. Although much work has been done in the context of simple graphs, they are not directly applicable to hypergraphs that represent more complex relationships in various applications. The key problem here is to search a given subhypergraph query in a larger hypergraph dataset. This search problem is known to be NP-hard as it is related to graph isomorphism. To solve this search problem in an efficient manner, we first create an index set by extracting the common subhypergraph structures from the given hypergraph dataset. Upon receiving a query, we use the same indexing techniques and create a query index set from the given subhypergraph. Utilizing both indices, we identify the possible locations of the query in the hypergraph dataset. We then start the subhypergraph search to verify whether the query really appears at each location by using an accelerated verification mechanism called layer-related-closure method. Through experiments on a real hypergraph dataset and random datasets, we demonstrate the efficiency and effectiveness of hypergraph indexing and our verification method.

Appendix

The steps for generating the adjacency list from the incidence matrix transpose are shown in Algorithm 9.1.

For the hypergraph in Fig. 1, this algorithm will generate the adjacency list $AJL$ as $\{e1: \langle e_2, e_3 \rangle ; e_2: \langle e_1, e_3 \rangle ; e_3: \langle e_1, e_3 \rangle \}$. For each hyperedge $e_i$, the algorithm goes over each node $v_j$ to determine whether $v_j \in e_i$. If so, the algorithm goes over each other hyperedge $e_k$ to see whether $v_j$ belongs to $e_k$. If that is the case too, $e_i$ and $e_k$ are adjacent hyperedges, and thus, we add $e_k$ to the adjacency list of $e_i$. Accordingly, the time complexity of the algorithm generating the adjacency list would be $O(m^2n)$.

One problem we may need to mention here is that if we have adjacency list $\{e1: e2, e3; e2: e1, e3; e3: e1, e3\}$ as in Fig. 1, HITE algorithm in Sect. 5 would extract the same hypertriangle three times as it goes through all of the hyperedges’ possible neighbors. To eliminate such duplicates of the checking, we can modify the HALG algorithm only on the third “for” loop (form $1 \le k$ to $i \le k$) as shown in Algorithm 9.2 so that HITE algorithm would not generate the same hypertriangles.

Now the adjacency list for Fig. 1 will be $\{e_1: \langle e_2, e_3\rangle ; e_2: \langle e_3\rangle ; e_3:\langle \rangle \}$. Although we do not keep a full set of all the related hyperedges to each hyperedge, this list is easy to generate and allow HITE algorithm to visit only one of the same hypertriangles.