LAUSR

k -core is important in many graph mining applications, such as community detection and clique finding. As one generalized concept of k -core, ( i ,  j )-core is more suited for bipartite graph analysis since it can identify the functions of two different types of vertices. Because ( i ,  j )-cores evolve as edges are inserted into (removed from) a dynamic bipartite graph, it is more economical to maintain them rather than decompose the graph recursively when only a few edges change. Moreover, many applications (e.g., graph visualization) only focus on some dense ( i ,  j )-cores. Existing solutions are simply insufficiently adaptable. They must maintain all ( i ,  j )-cores rather than just a subset of them, which requires more effort. To solve this issue, we propose novel maintenance methods for updating expected ( i ,  j )-cores. To estimate the influence scope of inserted (removed) edges, we first construct quasi-( i ,  j )-cores, which loosen the constraint of ( i ,  j )-cores but have similar properties. Second, we present a bottom-up approach for efficiently maintaining all ( i ,  j )-cores, from sparse to dense. Thirdly, because certain applications only focus on dense ( i ,  j )-cores of top- n layers, we also propose a top-down approach to maintain ( i ,  j )-cores from dense to sparse. Finally, we conduct extensive experiments to validate the efficiency of proposed approaches. Experimental results show that our maintenance solutions outperform existing approaches by one order of magnitude.