Bottom-up k-Vertex Connected Component Enumeration by Multiple Expansion

Bottom-up k-vertex connected component (k- VCC) enumeration methods, referred to as VCCE-BU, have exhib-ited better efficiency compared to the exact top-down k- VCC enumeration method (VCCE-TD). However, VCCE-BU has been found to have surprisingly low detection accuracy, that it may detect fewer k- VCC vertices than VCCE-TD. This raises the question of what causes VCCE-BU to have a low k-VCC enumeration quality. This paper investigates the reason and proposes that the local expansion should be reformulated as a Multiple vertex collaborative Expansion problem instead of the traditional Unitary Expansion (UE). A Multiple Expansion (ME) approach, which allows to expand multiple neighboring vertices jointly and collaboratively is proposed, which is proven exact in local expansion. However, the exact ME-based local expansion needs to explore large neighborhoods in each step, which is time-consuming. To address the efficiency issue, a Ring-based Multiple Expansion (RME) is proposed to conduct ME within one-hop neighbors. A maximum flow-based merging algorithm FBM is proposed for effective merging. A maximal clique and breath-first-search-based quick seeding algorithm QkVCS is proposed to generate k-VCC seeds efficiently. As a result, RIPPLE which integrates QkVCS+FBM+RME is presented as a new accurate and efficient bottom-up approach. Extensive verifications in real large-scale graph datasets demonstrate that even the single-thread RIPPLE is much more accurate and a magnitude faster than the state-of-the-art VCCE-BU method. We also demonstrate the effective speeding up to run RIPPLE in parallel.