Fast and scalable reachability queries on graphs by pruned labeling with landmarks and paths

Abstract

Answering reachability queries on directed graphs is ubiquitous in many applications involved with graph-shaped data as one of the most fundamental and important operations. However, it is still highly challenging to efficiently process them on large-scale graphs. Transitive-closure-based methods consume prohibitively large index space, and onlinesearch-based methods answer queries too slowly. Labelingbased methods attain both small index size and query time, but previous indexing algorithms are not scalable at all for processing large graphs of the day. In this paper, we propose new labeling-based methods for reachability queries, referred to as pruned landmark labeling and pruned path labeling. They follow the frameworks of 2-hop cover and 3-hop cover, but their indexing algorithms are based on the recent notion of pruned labeling and improve the indexing time by several orders of magnitude, resulting in applicability to large graphs with tens of millions of vertices and edges. Our experimental results show that they attain remarkable trade-offs between fast query time, small index size and scalability, which previous methods have never been able to achieve. Furthermore, we also discuss the ingredients of the efficiency of our methods by a novel theoretical analysis based on the graph minor theory.

Coverage centralities for temporal networks

“…In this section, to manage large temporal networks, we give efficient methods for computing TCC and TBCC on the basis of a graphindexing technique developed recently in the database community [42], in particular, the method proposed in reference [31]. The key idea is in how to speed up the computation of τ eat and τ ldt in Algorithms 1 and 2.…”

“…With respect to Lemma 2, to enumerate the number of pairs (u, w) being covered by v (at the boundary, if needed), we want to efficiently answer reachability in the DAG representation G of given temporal network G. To this end, it is beneficial to construct an index of G that computes the reachability between any pair of nodes on the basis of information of the reachability between a fraction of node pairs. Such an index is often called a reachability oracle in the database community [27][28][29][30][31].…”

“…Based on the information, it efficiently computes the reachability for the whole network. The method proposed in reference [31], which we will use for the numerical experiments in Section 5, computes a small table for each temporal vertex that stores reachability from (and to) a smaller number of other certain temporal vertices than the number of all the temporal vertices. It depends on the structure of each temporal network how small the table becomes.…”

“…This method resolves these issues, so that it can handle directed networks of millions of edges with the query time of less than a microsecond on average (see Ref. [31] for further technical details).…”

“…2.2 for details). The latter problem is well studied in the database community [27][28][29][30][31], and it can be solved by constructing an index of the directed network, which computes the reachability between any pair of nodes by using information of the reachability between a fraction of node pairs. If it suffices to use approximations to the TCC and TBCC values, we only need to query the index at most O(log 2 N ) times, where N is the total number of vertices in the network (see Appendix A).…”