Efficient processing of coverage centrality queries on road networks

World Wide Web (2024) 27:25 https://doi.org/10.1007/s11280-024-01248-5 Efficient processing of coverage centrality queries on road networks Yehong Xu1 · Mengxuan Zhang2 · Ruizhong Wu3 · Lei Li1,3 · Xiaofang Zhou1,3 Received: 11 January 2023 / Revised: 28 August 2023 / Accepted: 21 November 2023 / Published online: 12 April 2024 © The Author(s) 2024 Abstract Coverage Centrality is an important metric to evaluate vertex importance in road networks. However, current solutions have to compute the coverage centrality of all the vertices together, which is resource-wasting, especially when only some vertices centrality is required. In addition, they have poor adaption to the dynamic scenario because of the computation inefficiency. In this paper, we focus on the coverage centrality query problem and propose a method that efficiently computes the centrality of single vertices without relying on the underlying graph being static by employing the intra-region pruning, inter-region pruning, and top-down search. We further propose the bottom-up search and mixed search to improve efficiency. Experiments validate the efficiency and effectiveness of our algorithms compared with the state-of-the-art method. Keywords Coverage centrality · Road networks · Shortest paths 1 Introduction Centrality computation serves as a fundamental operation in a range of applications within road networks, including traffic monitoring and prediction [1], network maintenance and B B Yehong Xu Mengxuan Zhang Ruizhong Wu Lei Li Xiaofang Zhou 1 The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong 2 Australian National University, 2601 Canberra, ACT, Australia 3 The Hong Kong University of Science and Technology (Guangzhou), Guangzhou, China 0123456789().: V,-vol 123 25 Page 2 of 24 World Wide Web (2024) 27:25 assessment [2]. Compared to various metrics of centrality evaluation, coverage centrality of a vertex s (denoted as CC(s)) [3–5] has a particularly high correlation with s’s transportation ability and surrounding traffic condition. This is because CC is defined based on the shortest paths in a road network. A road network is an undirected weighted graph G(V , E, W ) with the vertex set V (i.e., road intersections), the edge set E ⊆ V × V (i.e., road segments), and cost function W : E → R+ that assigns a non-negative travel cost to each edge (u, v) ∈ E. We denote n = |V |, m = |E|, and N (v) for the neighbors of v ∈ G. A path p = v1 , ..., vk  is a sequence vertices where (vi , vi+1 ) ∈ E, vi ∈ V . The length of a path p is defined as of k−1 l( p) = i=1 w(vi , vi+1 ). Let ps,t denote any path between a vertex pair (s, t). The shortest path p̂s,t is a path among all ps,t with the minimum length. Coverage centrality (CC) of one vertex is defined as the number of vertex pairs that have at least one shortest path passes it (as shown in (1)).  CC(v) = δs,t (v) (1) s,t∈V ,s=t=v where δs,t (v) is equal to 1 if at least one shortest path between s and t passing through v, otherwise 0. In the context of the road network G, the travel cost associated with each edge can be interpreted as the dynamic travel time required to traverse that edge. In this scenario, a high value of CC(s) indicates the importance of vertex s in terms of transportation within the graph G. In other words, the blockage of s can have a profound impact on the overall travel costs within G. Moreover, an increase in CC(s) suggests an improvement in travel time when passing through vertex s, and vice versa. By closely monitoring CC(s), we can obtain valuable insights into the traffic conditions related to vertex s, allowing us to implement more precise traffic management strategies. Therefore, its quick measurement is essential, especially for those urgent situations. Existing solutions CC has been extensively studied, mainly including two branches of methods. One branch orders CC roughly in proportion with the vertex degree [6] in a hypergraph, which is constructed by the sampled vertices in the original graph. But it does not calculate the centrality directly, so it is out of this paper’s consideration. Another branch focuses on Betweenness Centrality (BC). The definition of BC is quite similar to that of CC except that δst (v) is defined as the ratio of shortest paths between s, t passing through v, relative to all shortest paths between s, t. The Brandes algorithm [7] is the fundamental BC algorithm, whose time complexity is O(nm + n 2 logn) where n = |V |, m = |E| are the vertex and edge number, respectively. Brandes is computationally expensive for large graphs and thus cannot support real-time query answering. Subsequently, there come other strategies that aim at improving the scalability of Brandes [8–13]. However, these strategies that either distribute or parallelize the computation could hardly apply to road networks, as analyzed in Section 2.2. Therefore, only Brandes could be used and extended to CC computation by ignoring the path number [5]. Motivation The network is dynamic with traffic conditions keep changing [14–19], obtaining vertices’ CC values in real-time is quite useful. Typically, we are only interested in monitoring a small set of critical vertices over time, e.g., those that connect different parts of the road network. For other vertices, it is unnecessary to maintain their CC. However, existing BC-based methods that either maintain CC of all vertices or none of them are computationally expensive and cost-ineffective. Then a question comes naturally: why not focus on developing an online CC-answering method for single vertices that can be easily adapted to dynamic 123 World Wide Web (2024) 27:25 Page 3 of 24 25 road networks? Therefore, we aim to propose Coverage Centrality Query Answering Framework that relieves the heavy computation involved in Brandes-based methods. Instead, it is lightweight and can efficiently answer most CC queries in real-time. Challenges Our problem is how to efficiently answer CC queries given a underlying road network G(V , E, W ), where a CC query is denoted as q(s) (s ∈ V ). According to formula (1), to answer q(s), we need to check every vertex pair (a, b) in G to see whether it has a shortest path that passes through v. We term the checking as the dependency check for (a, b). Thus, it takes t = (n − 1)(n − 2)τ time in total, where τ is the time of one dependency check. This naive calculation is obviously time-consuming. Approximately, t is around 25 hours for medium-size road network (with around 300, 000 vertices like New York City and Beijing) with τ in microsecond level. There comes our challenge: how to calculate accurate CC values efficiently. Our idea We focus on speeding up CC calculation through reducing the number of vertex pairs that require dependency checks. Given a CC query q(v), we initially need to check (n − 1)(n − 2) vertex pairs. However, we find that some vertex pairs can be pruned: vertex pairs (a, b) that (...truncated)