Sublinear-Time Clustering Oracle for Signed Graphs (2206.13813v1)

Abstract: Social networks are often modeled using signed graphs, where vertices correspond to users and edges have a sign that indicates whether an interaction between users was positive or negative. The arising signed graphs typically contain a clear community structure in the sense that the graph can be partitioned into a small number of polarized communities, each defining a sparse cut and indivisible into smaller polarized sub-communities. We provide a local clustering oracle for signed graphs with such a clear community structure, that can answer membership queries, i.e., "Given a vertex $v$, which community does $v$ belong to?", in sublinear time by reading only a small portion of the graph. Formally, when the graph has bounded maximum degree and the number of communities is at most $O(\log n)$, then with $\tilde{O}(\sqrt{n}\operatorname{poly}(1/\varepsilon))$ preprocessing time, our oracle can answer each membership query in $\tilde{O}(\sqrt{n}\operatorname{poly}(1/\varepsilon))$ time, and it correctly classifies a $(1-\varepsilon)$-fraction of vertices w.r.t. a set of hidden planted ground-truth communities. Our oracle is desirable in applications where the clustering information is needed for only a small number of vertices. Previously, such local clustering oracles were only known for unsigned graphs; our generalization to signed graphs requires a number of new ideas and gives a novel spectral analysis of the behavior of random walks with signs. We evaluate our algorithm for constructing such an oracle and answering membership queries on both synthetic and real-world datasets, validating its performance in practice.