Dynamic Structural Clustering on Graphs

Structural Clustering (StrClu) is one of the most popular graph clustering paradigms. In this paper, we consider StrClu under two commonly adapted similarities, namely Jaccard similarity and cosine similarity on a dynamic graph, 𝐺 = ⟨𝑉 , 𝐸 ⟩, subject to edge insertions and deletions (updates). The goal is to maintain certain information under updates, so that the StrClu clustering result on 𝐺 can be retrieved in 𝑂 ( |𝑉 | + |𝐸 |) time, upon request. The state-of-the-art worst-case cost is 𝑂 ( |𝑉 |) per update; we improve this update-time bound significantly with the 𝜌-approximate notion. Specifically, for a specified failure probability, 𝛿 * , and every sequence of 𝑀 updates (no need to know 𝑀's value in advance), our algorithm, DynELM, achieves 𝑂 (log 2 |𝑉 | + log |𝑉 | • log 𝑀

𝛿 * ) amortized cost for each update, at all times in linear space. Moreover, DynELM provides a provable "sandwich" guarantee on the clustering quality at all times after each update with probability at least 1 -𝛿 * . We further develop DynELM into our ultimate algorithm, DynStrClu, which also supports cluster-group-by queries. Given 𝑄 ⊆ 𝑉 , this puts the non-empty intersection of 𝑄 and each StrClu cluster into a distinct group. DynStrClu not only achieves all the guarantees of DynELM, but also runs cluster-group-by queries in 𝑂 ( |𝑄 | • log |𝑉 |) time. We demonstrate the performance of our algorithms via extensive experiments, on 15 real datasets. Experimental results confirm that our algorithms are up to three orders of magnitude more efficient than state-of-the-art competitors, and still provide quality structural clustering results. Furthermore, we study the difference between the two similarities w.r.t. the quality of approximate clustering results.