Community detection is one of the most important and challenging problems in network analysis. However, real-world networks may have very different structural properties and communities of various nature. As a result, it is hard (or even impossible) to develop one algorithm suitable for all datasets. A standard machine learning tool is to consider a parametric algorithm and choose its parameters based on the dataset at hand. However, this approach is not applicable to community detection since usually no labeled data is available for such parameter tuning. In this paper, we propose a simple and effective procedure allowing to tune hyperparameters of any given community detection algorithm without requiring any labeled data. The core idea is to generate a synthetic network with properties similar to a given real-world one, but with known communities. It turns out that tuning parameters on such synthetic graph also improves the quality for a given real-world network. To illustrate the effectiveness of the proposed algorithm, we show significant improvements obtained for several well-known parametric community detection algorithms on a variety of synthetic and real-world datasets.
The standard random graph model assumed by likelihood maximization methods is the stochastic block model (SBM) or its simplified version --- planted partition model (PPM). In these models, the probability that two vertices are connected by an edge depends only on their community assignments. Recently, the degree-corrected stochastic block model (DCSBM) together with the degree-corrected planted partition model (DCPPM) were proposed @cite_11 . These models take into account the observed degree sequence of a graph, and, as a result, they are more realistic. It was also noticed that if we fix the parameters of DCPPM, then likelihood maximization based on this model is equivalent to modularity optimization with some @math @cite_12 . Finally, in a recent paper @cite_1 the independent LFR model (ILFR) was proposed and analyzed. It was shown that ILFR gives a better fit for a variety of real-world networks @cite_1 . In this paper, to illustrate the generalization ability of the proposed hyperparameter tuning strategy, in addition to the Louvain algorithm, we also use parametric likelihood maximization methods based on PPM and ILFR.
@cite_1 @cite_12 @cite_11
1562488694 1467346733 2508909698 2262666886
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 stochastic block model (SBM) provides a popular framework for modeling community structures in networks. However, more attention has been devoted to problems concerning estimating the latent node labels and the model parameters than the issue of choosing the number of blocks. We consider an approach based on the log likelihood ratio statistic and analyze its asymptotic properties under model misspecification. We show the limiting distribution of the statistic in the case of underfitting is normal and obtain its convergence rate in the case of overfitting. These conclusions remain valid when the average degree grows at a polylog rate. The results enable us to derive the correct order of the penalty term for model complexity and arrive at a likelihood-based model selection criterion that is asymptotically consistent. Our analysis can also be extended to a degree-corrected block model (DCSBM). In practice, the likelihood function can be estimated using more computationally efficient variational methods or consistent label estimation algorithms, allowing the criterion to be applied to large networks. We develop an information-theoretic view of the stochastic block model, a popular statistical model for the large-scale structure of complex networks. A graph G from such a model is generated by first assigning vertex labels at random from a finite alphabet, and then connecting vertices with edge probabilities depending on the labels of the endpoints. In the case of the symmetric two-group model, we establish an explicit ‘single-letter’ characterization of the per-vertex mutual information between the vertex labels and the graph, when the graph average degree diverges. The explicit expression of the mutual information is intimately related to estimation-theoretic quantities, and -in particular- reveals a phase transition at the critical point for community detection. Below the critical point the per-vertex mutual information is asymptotically the same as if edges were independent of the vertex labels. Correspondingly, no algorithm can estimate the partition better than random guessing. Conversely, above the threshold, the per-vertex mutual information is strictly smaller than the independent-edges upper bound. In this regime there exists a procedure that estimates the vertex labels better than random guessing. 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.