With the recent popularity of graphical clustering methods, there has been an increased focus on the information between samples. We show how learning cluster structure using edge features naturally and simultaneously determines the most likely number of clusters and addresses data scale issues. These results are particularly useful in instances where (a) there are a large number of clusters and (b) we have some labeled edges. Applications in this domain include image segmentation, community discovery and entity resolution. Our model is an extension of the planted partition model and our solution uses results of correlation clustering, which achieves a partition O(log(n))-close to the log-likelihood of the true clustering.
The work most closely related to ours extends the stochastic block model edge weights to other parametric distributions. Motivated by observations that Bernoulli random variables often do not capture the degree complexity in social networks, Karrer & Newman Karrer2011 , Mariadassou2010 and Ball2011 each used Poisson distributed edge weights. This may also be a good choice because the Bernoulli degree distribution is asymptotically Poisson @cite_3 . considered an SBM with edge weights drawn from the exponential family distribution Aicher2013 . Like Thomas & Blitzstein Thomas2011 , he also showed better results than thresholding to binary edges. Lastly, Balakrishnan2011 consider Normally distributed edge weights as a method of analyzing spectral clustering recovery with noise.
@cite_3
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The stochastic block model (SBM) with two communities, or equivalently the planted bisection model, is a popular model of random graph exhibiting a cluster behaviour. In the symmetric case, the graph has two equally sized clusters and vertices connect with probability @math within clusters and @math across clusters. In the past two decades, a large body of literature in statistics and computer science has focused on providing lower-bounds on the scaling of @math to ensure exact recovery. In this paper, we identify a sharp threshold phenomenon for exact recovery: if @math and @math are constant (with @math ), recovering the communities with high probability is possible if @math and impossible if @math . In particular, this improves the existing bounds. This also sets a new line of sight for efficient clustering algorithms. While maximum likelihood (ML) achieves the optimal threshold (by definition), it is in the worst-case NP-hard. This paper proposes an efficient algorithm based on a semidefinite programming relaxation of ML, which is proved to succeed in recovering the communities close to the threshold, while numerical experiments suggest it may achieve the threshold. An efficient algorithm which succeeds all the way down to the threshold is also obtained using a partial recovery algorithm combined with a local improvement procedure. The binary symmetric stochastic block model deals with a random graph of @math vertices partitioned into two equal-sized clusters, such that each pair of vertices is connected independently with probability @math within clusters and @math across clusters. In the asymptotic regime of @math and @math for fixed @math and @math , we show that the semidefinite programming relaxation of the maximum likelihood estimator achieves the optimal threshold for exactly recovering the partition from the graph with probability tending to one, resolving a conjecture of Abbe14 . Furthermore, we show that the semidefinite programming relaxation also achieves the optimal recovery threshold in the planted dense subgraph model containing a single cluster of size proportional to @math . In this paper, we consider the cluster estimation problem under the stochastic block model. We show that the semidefinite programming (SDP) formulation for this problem achieves an error rate that decays exponentially in the signal-to-noise ratio. The error bound implies weak recovery in the sparse graph regime with bounded expected degrees as well as exact recovery in the dense regime. An immediate corollary of our results yields error bounds under the censored block model. Moreover, these error bounds are robust, continuing to hold under heterogeneous edge probabilities and a form of the so-called monotone attack. Significantly, this error rate is achieved by the SDP solution itself without any further pre- or post-processing and improves upon existing polynomially decaying error bounds proved using the Grothendieck’s inequality. Our analysis builds on two key ingredients: 1) showing that the graph has a well-behaved spectrum, even in the sparse regime, after discounting an exponentially small number of edges and 2) an order-statistics argument that governs the final error rate. Both arguments highlight the implicit regularization effect of the SDP formulation. We consider the problem of identifying underlying community-like structures in graphs. Toward this end, we study the stochastic block model (SBM) on k-clusters: a random model on n = km vertices, partitioned in k equal sized clusters, with edges sampled independently across clusters with probability q and within clusters with probability p, p > q. The goal is to recover the initial “hidden” partition of [n]. We study semidefinite programming (SDP)-based algorithms in this context. In the regime (p = (m) m ) and (q = (m) m ), we show that a certain natural SDP-based algorithm solves the problem of exact recovery in the k-community SBM, with high probability, whenever ( - > 1 ), as long as (k=o( n) ). This threshold is known to be the information theoretically optimal. We also study the case when (k= ( (n)) ). In this case however, we achieve recovery guarantees that no longer match the optimal condition ( - > 1 ), thus leaving achieving optimality for this range an open question.