Dynamic Structural Clustering on Graphs
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Structural Clustering (DynClu) 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| ·logM/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.
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