Ranking of vertices in a graph for different objectives is one of the most fundamental tasks in computer science. It is known that traditional ranking algorithms can generate unbalanced ranking when the graph has underlying communities, resulting in loss of information, polarised opinions, and reduced diversity (Celis, Straszak \& Vishnoi [ICALP 2018]). In this paper, we focus on *unsupervised ranking* on graphs and observe that popular centrality measure based ranking algorithms such as PageRank may often generate unbalanced ranking here as well. We address this issue by coining a new approach, which we term *relative centrality*. Our approach is based on an iterative graph-dependent local normalization of the centrality score, which promotes balancedness while maintaining the validity of the ranking. We further quantify reasons behind this unbalancedness of centrality measures on a novel structure that we propose is called multi-core-periphery with communities (MCPC). We also provide theoretical and extensive simulation support for our approach towards resolving the unbalancedness in MCPC. Finally, we consider graph embeddings of $11$ single-cell datasets. We observe that top-ranked as per existing centrality measures are better separable into the ground truth communities. However, due to the unbalanced ranking, the top nodes often do not contain points from some communities. Here, our relative-centrality-based approach generates a ranking that provides a similar improvement in clusterability while providing significantly higher balancedness.

Principal component analysis (PCA) is one of the most fundamental tools in machine learning with broad use as a dimensionality reduction and denoising tool. In the later setting, while PCA is known to be effective at subspace recovery and is proven to aid clustering algorithms in some specific settings, its improvement of noisy data is still not well quantified in general. In this paper, we propose a novel metric called *compression ratio* to capture the effect of PCA on high-dimensional noisy data. We show that, for data with *underlying community structure*, PCA significantly reduces the distance of data points belonging to the same community while reducing inter-community distance relatively mildly. We explain this phenomenon through both theoretical proofs and experiments on real-world data. Building on this new metric, we design a straightforward algorithm that could be used to detect outliers. Roughly speaking, we argue that points that have a *lower variance of compression ratio* do not share a *common signal* with others (hence could be considered outliers). We provide theoretical justification for this simple outlier detection algorithm and use simulations to demonstrate that our method is competitive with popular outlier detection tools. Finally, we run experiments on real-world high-dimension noisy data (single-cell RNA-seq) to show that removing points from these datasets via our outlier detection method improves the accuracy of clustering algorithms. Our method is very competitive with popular outlier detection tools in this task.

Community detection in the stochastic block model is one of the central problems of graph clustering. Since its introduction by Holland, Laskey, and Leinhardt (Social Networks, 1983), many subsequent papers have made great strides in solving and understanding this model. However, despite the long history of study, there are still unsolved challenges.

The stochastic block model (SBM) is a fundamental model for studying graph clustering or community detection in networks. It has received great attention in the last decade and the balanced case, i.e., assuming all clusters have large size, has been well studied. However, our understanding of SBM with unbalanced communities (arguably, more relevant in practice) is still limited. In this paper, we provide a simple SVD-based algorithm for recovering the communities in the SBM with communities of varying sizes. We improve upon a result of Ailon, Chen and Xu [ICML 2013; JMLR 2015] by removing the assumption that there is a large interval such that the sizes of clusters do not fall in, and also remove the dependency of the size of the recoverable clusters on the number of underlying clusters. We further complement our theoretical improvements with experimental comparisons. Under the planted clique conjecture, the size of the clusters that can be recovered by our algorithm is nearly optimal (up to poly-logarithmic factors) when the probability parameters are constant. As a byproduct, we obtain an efficient clustering algorithm with sublinear query complexity in a faulty oracle model, which is capable of detecting all clusters larger than $\tilde{\Omega}({\sqrt{n}})$, even in the presence of $\Omega(n)$ small clusters in the graph. In contrast, previous efficient algorithms that use a sublinear number of queries are incapable of recovering any large clusters if there are more than $\tilde{\Omega}(n^{2/5})$ small clusters.