text: Clustering is a fundamental task in data analysis, and spectral clustering has been recognized as a promising approach to it. Given a graph describing the relationship between data, spectral clustering explores the underlying cluster structure in two stages. The first stage embeds the nodes of the graph into real space, and the second stage groups the embedded nodes into several clusters. The use of the @math -means method in the grouping stage is currently standard practice. We present a spectral clustering algorithm that uses convex programming in the grouping stage, and study how well it works. The concept behind the algorithm design lies in the following observation. The nodes with the largest degree in each cluster may be found by computing an enclosing ellipsoid for embedded nodes in real space, and the clusters may be identified by using those nodes. We show that the observations are valid, and the algorithm returns clusters to provide the conductance of graph, if the gap assumption, introduced by Peng el al. at COLT 2015, is satisfied. We also give an experimental assessment of the algorithm's performance.
  @cite_41: Networks or graphs can easily represent a diverse set of data sources that are characterized by interacting units or actors. Social ne tworks, representing people who communicate with each other, are one example. Communities or clusters of highly connected actors form an essential feature in the structure of several empirical networks. Spectral clustering is a popular and computationally feasi ble method to discover these communities. The Stochastic Block Model (, 1983) is a social network model with well defined communities; each node is a member of one community. For a network generated from the Stochastic Block Model, we bound the number of nodes "misclus- tered" by spectral clustering. The asymptotic results in th is paper are the first clustering results that allow the number of clusters in the model to grow with the number of nodes, hence the name high-dimensional. In order to study spectral clustering under the Stochastic Block Model, we first show that under the more general latent space model, the eigenvectors of the normalized graph Laplacian asymptotically converge to the eigenvectors of a "population" normal- ized graph Laplacian. Aside from the implication for spectral clustering, this provides insight into a graph visualization technique. Our method of studying the eigenvectors of random matrices is original. AMS 2000 subject classifications: Primary 62H30, 62H25; secondary 60B20.
  @cite_9: Problems such as bisection, graph coloring, and clique are generally believed hard in the worst case. However, they can be solved if the input data is drawn randomly from a distribution over graphs containing acceptable solutions. In this paper we show that a simple spectral algorithm can solve all three problems above in the average case, as well as a more general problem of partitioning graphs based on edge density. In nearly all cases our approach meets or exceeds previous parameters, while introducing substantial generality. We apply spectral techniques, using foremost the observation that in all of these problems, the expected adjacency matrix is a low rank matrix wherein the structure of the solution is evident.
  @cite_0: We analyze the performance of spectral clustering for community extraction in stochastic block models. We show that, under mild conditions, spectral clustering applied to the adjacency matrix of the network can consistently recover hidden communities even when the order of the maximum expected degree is as small as @math , with @math the number of nodes. This result applies to some popular polynomial time spectral clustering algorithms and is further extended to degree corrected stochastic block models using a spherical @math -median spectral clustering method. A key component of our analysis is a combinatorial bound on the spectrum of binary random matrices, which is sharper than the conventional matrix Bernstein inequality and may be of independent interest.
summary: There is also a considerable amount of research on spectral clustering on a random graph. In a planted partition model, we assume that the node set is partitioned into several clusters and edges connecting the nodes are stochastically generated: any two nodes in the same cluster have an edge with probability @math , and any two nodes in different clusters have an edge with probability @math . McSherry @cite_9 showed that spectral clustering can extract the clusters with high probability if @math and @math lie within some range. Rohe @cite_41 and Lei @cite_0 studied KSC on a stochastic block model.