Related papers: Detectability of the spectral method for sparse gr…
Investigating the performance of different methods is a fundamental problem in graph partitioning. In this paper, we estimate the so-called detectability threshold for the spectral method with both unnormalized and normalized Laplacians in…
We consider three distinct and well studied problems concerning network structure: community detection by modularity maximization, community detection by statistical inference, and normalized-cut graph partitioning. Each of these problems…
We introduce a new spectral method for image segmentation that incorporates long range relationships for global appearance modeling. The approach combines two different graphs, one is a sparse graph that captures spatial relationships…
We study networks that display community structure -- groups of nodes within which connections are unusually dense. Using methods from random matrix theory, we calculate the spectra of such networks in the limit of large size, and hence…
We relate two important notions in graph theory: expanders which are highly connected graphs, and modularity a parameter of a graph that is primarily used in community detection. More precisely, we show that a graph having modularity…
We consider the minimum-cut partitioning of a graph into more than two parts using spectral methods. While there exist well-established spectral algorithms for this problem that give good results, they have traditionally not been well…
Spectral Clustering is one of the most traditional methods to solve segmentation problems. Based on Normalized Cuts, it aims at partitioning an image using an objective function defined by a graph. Despite their mathematical attractiveness,…
Let \phi(G) be the minimum conductance of an undirected graph G, and let 0=\lambda_1 <= \lambda_2 <=... <= \lambda_n <= 2 be the eigenvalues of the normalized Laplacian matrix of G. We prove that for any graph G and any k >= 2, \phi(G) =…
Hypergraph partitioning lies at the heart of a number of problems in machine learning and network sciences. Many algorithms for hypergraph partitioning have been proposed that extend standard approaches for graph partitioning to the case of…
Graph partitioning problems emerge in a wide variety of complex systems, ranging from biology to finance, but can be rigorously analyzed and solved only for a few graph ensembles. Here, an ensemble of equitable graphs, i.e. random graphs…
One of the most widely used methods for community detection in networks is the maximization of the quality function known as modularity. Of the many maximization techniques that have been used in this context, some of the most conceptually…
Graph-based clustering methods like spectral clustering and SpectralNet are very efficient in detecting clusters of non-convex shapes. Unlike the popular $k$-means, graph-based clustering methods do not assume that each cluster has a single…
A wide variety of application domains are concerned with data consisting of entities and their relationships or connections, formally represented as graphs. Within these diverse application areas, a common problem of interest is the…
One of the most useful measures of cluster quality is the modularity of a partition, which measures the difference between the number of the edges joining vertices from the same cluster and the expected number of such edges in a random…
We solve the graph bi-partitioning problem in dense graphs with arbitrary degree distribution using the replica method. We find the cut-size to scale universally with <k^1/2>. In contrast, earlier results studying the problem in graphs with…
Spectral clustering is a fundamental method for graph partitioning, but its reliance on eigenvector computation limits scalability to massive graphs. Classical sparsification methods preserve spectral properties by sampling edges…
This paper presents a novel meta algorithm, Partition-Merge (PM), which takes existing centralized algorithms for graph computation and makes them distributed and faster. In a nutshell, PM divides the graph into small subgraphs using our…
A graph is unipolar if it can be partitioned into a clique and a disjoint union of cliques, and a graph is a generalised split graph if it or its complement is unipolar. A unipolar partition of a graph can be used to find efficiently the…
For random graphs distributed according to a stochastic block model, we consider the inferential task of partioning vertices into blocks using spectral techniques. Spectral partioning using the normalized Laplacian and the adjacency matrix…
We prove identifiability of parameters for a broad class of random graph mixture models. These models are characterized by a partition of the set of graph nodes into latent (unobservable) groups. The connectivities between nodes are…