Clustering from Labels and Time-Varying Graphs
We present a general framework for graph clustering where a label is observed to each pair of nodes. This allows a very rich encoding of various types of pairwise interactions between nodes. We propose a new tractable approach to this problem based on maximum likelihood estimator and convex optimization. We analyze our algorithm under a general generative model, and provide both necessary and sufficient conditions for successful recovery of the underlying clusters. Our theoretical results cover and subsume a wide range of existing graph clustering results including planted partition, weighted clustering and partially observed graphs. Furthermore, the result is applicable to novel settings including time-varying graphs such that new insights can be gained on solving these problems. Our theoretical findings are further supported by empirical results on both synthetic and real data.
Clustering from Labels and Time-Varying GraphsThe authors consider graph clustering i.e. nodes are assigned to groups based on dense interconnectivity, with no/sparse connectivity to other nodes not in the same group. As pointed out by the authors, their model is in the vein of the stochastic labelled block model. However, they can cover additional scenarios such as time-varying graphs. The basic scenario is presented in eqn (1). This is restated as a relaxed convex optimisation problem in (2) (incidentally a bug ‘and thus and has nuclear …’). The only concern with (2) is that, though convex, and a linear objective, it may be computational intensive for large graphs. The experiment with n=400 in the Supplementary suggests otherwise but it would have been interesting to know algorithmic time complexity dependence on n.The main theorems in the text give bounds on the estimate of the true cluster matrix Y for the convex program (2). Specifically, Theorem 1 gives sufficient conditions for this convex program to find the true cluster matrix, dependent on label distribution, minimum size of the true clusters (note: I don’t think you would know this in reality?) and the weight function. I wasn’t sure of the significance of c (a universal constant) here. Thus the later Corollary 3 gives a condition on recovering Y which has some support in the experimental Section 5 (below Fig. 1-3), for a particular special case. I was only able to check two lemmas in the Supplementary – though argument looked sound. In section 4 the author(s) move from the general argument of Section 3 to two special cases (e.g. non-uniform edge densities, Corollary 3). This includes Markov sequence and time-varying e.g. Corollary 5 presents sufficient conditions on finding the true cluster matrix based on the same dependency on c, \mu, \nu, K and n for the case of Markov snapshots. Section 5 presents some brief experimental results (there is no Conclusion), though the Supplementary (Section A) gives more detail about the experiments.In short, the authors present a convex program for graph clustering which has broader implantation to time-varying graphs. They present criteria for establishing if the true cluster matrix has been found – which appears sound and has experimental support (see e.g. Figure 4 in the Supplementary). The criterion (Theorem 1, for the general case) appears useable (possibly as a stopping criterion – or if stopped early, whether a sound solution has been achieved). I think the topic of graph clustering is important in terms of real-word applications in bioinformatics, social networks, etc, and this paper looks like a valid and useful contribution. Thus I recommend acceptance of the paper.