The Consistency of Common Neighbors for Link Prediction in Stochastic Blockmodels
Link prediction and clustering are key problems for network-structured data. While spectral clustering has strong theoretical guarantees under the popular stochastic blockmodel formulation of networks, it can be expensive for large graphs. On the other hand, the heuristic of predicting links to nodes that share the most common neighbors with the query node is much fast, and works very well in practice. We show theoretically that the common neighbors heuristic can extract clusters with high probability when the graph is dense enough, and can do so even in sparser graphs with the addition of a "cleaning" step. Empirical results on simulated and real-world data support our conclusions.
Spectral clustering is a standard tool for identifying link prediction. However, it is expensive for large graphs. A popular heuristic is to consider common neighbors, which is a fast and performs well in practise. This paper proves the consistency of the common neighbor heuristic for different classes of graphs. The authors also propose a cleaning step that improves the quality for sparse graphs. The approach is studied empirically on simulated and real world data.This problem is very relevant and the theoretical analysis seems to be mathematically sound. However, the practical relevance remains unclear, since the experiments fail to explore all strengths and weaknesses: - The proposed algorithm depends on a number of unknown parameters. The authors fail to explain the significance of the link probability of the clusters and the cluster sizes. It is not clear how the algorithm performs if these parameters are not optimally chosen.- Depending on if the graph is semi-sparse or not, the authors propose to conduct a cleaning step and Figure 1 shows that in that scenario it improves recall and precision. Given that it is hard to check the sparsity in practise, it is also important to show how the cleaning step affects the quality when it is not appropriate. - Equal cluster sizes (\pi = n/K) seems to be a strong assumption. You mentioned that this assumption could be easily relaxed. Can you give an idea of how that would influence your analysis? I assume that relaxing this assumption introduces more parameters that are unknown and we need to make this assumption anyhow. I would expect also an experiment showing the quality if the cluster sizes are skewed.Section 3 and 4 are very confusing and a clear structure is missing; they introducing and mixing different concepts where it is unclear how they relate together and for what they are needed. E.g., - the formular (line 182-187) are detached from the text - how do the sections "current approaches", "common neighbors", "dependence", "dependence",... relate to each other?Minor: - Line 97 seems to be a copy&paste error (see Line 126) - Line 163: "eigenvectors of the graph" -> "eigenvectors of the graph Laplacian matrix" - please write out w.h.p (Line 23) and WLOG (Line 222) - Line 214: "when average degree" -> "when the average degree"  After rebuttal: ---------------- Thank you for addressing my concerns about the algorithm's parameters, when the cleaning step is needed, and the number of clusters. However, I find Section 3 and 4 still confusing. I adapted my score accordingly and would like to encourage the authors to clarify the structure and elaborate these issues in the paper.