Sublinear-Time Clustering Oracle for Signed Graphs
This provides a practical tool for applications needing clustering information for only a few vertices in signed social networks, generalizing previous work from unsigned graphs.
The paper tackles the problem of efficiently identifying community membership in signed graphs, which model social networks with positive/negative interactions, by developing a local clustering oracle that answers membership queries in sublinear time with preprocessing and query times of ~O(√n poly(1/ε)), correctly classifying a (1-ε)-fraction of vertices.
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.