Maximum core spanning tree maintenance for large dynamic graphs

With the increase in network scale and online applications, the maintenance problem of cohesive structures in large graphs has attracted great attention. The Maximum Core Spanning Tree (MCST) is a representative cohesive structure generated based on k-core, which is the maximum edge weight spanning tree indicating the “staired coreness hierarchy” in each connected component. The edge weight here is defined as $w_{uv}=\min \left\{core\right(u),core(v\left)\right\}$, and $core\left(x\right)$ is the corness of vertex x. Unlike the maintenance problem of Maximum Spanning Tree (MST) which has known efficient algorithms, MCST maintenance raises special challenges, which is mainly due to the cascaded vertex coreness changes after single-edge insertion or deletion. In this paper, we show a series properties of MCST and MCST maintenance problems and propose an OrderPassed method and a LoopFree method to maintain the MCST efficiently. In particular, the time complexity for MCST maintenance for edge insertion and deletion is bounded by $O\left(\right|E^{⁎}|+|V\left|\right)$ and $O\left(\right|E^{⁎}|+{\sum }_{i=1}^K|O_i\left|\right)$ respectively, where $E^{⁎}$ is the edge set whose edge weight changes after insertion/deletion and $\left|O_i\right|$ denotes the number of edges whose edge weight is i. Through extensive evaluations, we show the proposed MCST maintenance algorithms have good efficiency, scalability and stability on real-world datasets.