Efficient ($$\alpha $$, $$\beta $$)-core computation in bipartite graphs

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The problem of computing ( $$\alpha , \beta $$ )-core in a bipartite graph for given $$\alpha $$ and $$\beta $$ is a fundamental problem in bipartite graph analysis and can be used in many applications such as online group recommendation and fraudsters detection Existing solution to computing ( $$\alpha , \beta $$ )-core needs to traverse the entire bipartite graph once and ignore the fact that real-world graphs are often dynamic. Considering the real bipartite graph can be very large and dynamically updated, and the requests to compute ( $$\alpha , \beta $$ )-core can be issued frequently in real applications, the existing solution is too expensive to compute the $$(\alpha ,\beta )$$ -core. In this paper, we present an efficient algorithm for ( $$\alpha , \beta $$ )-core computation based on a novel index such that the algorithm runs in linear time regarding the result size (thus, the algorithm is optimal since it needs at least linear time to output the result). We prove that the index only requires O(m) space where m is the number of edges in the bipartite graph. We also devise an efficient algorithm with time complexity $$O (\delta \cdot m)$$ for index construction where $$\delta $$ is bounded by $$\sqrt{m}$$ and is much smaller than $$\sqrt{m}$$ in practice. Moreover, we discuss efficient algorithms to maintain the index when the bipartite graph is dynamically updated. We show that we can decide whether a node in the index should be updated or not by visiting its neighbors. Based on this locality property, we propose an efficient index maintenance algorithm which only needs to visit a local subgraph near the inserted or removed edge. Finally, we show how to implement our index construction and maintenance algorithms in parallel. The experimental results on real and synthetic graphs (more than 1 billion edges) demonstrate that our algorithms achieve up to 5 orders of magnitude speedup for computing $$(\alpha ,\beta )$$ -core, up to 3 orders of magnitude speedup for index construction and up to 4 orders of magnitude speedup for index maintenance, respectively, compared with existing techniques.