Related papers: The Graph Pencil Method: Mapping Subgraph Densitie…
For a set $S$ of vertices of a graph $G$, we define its density $0 \leq \sigma(S) \leq 1$ as the ratio of the number of edges of $G$ spanned by the vertices of $S$ to ${|S| \choose 2}$. We show that, given a graph $G$ with $n$ vertices and…
In this paper, we present a novel way to summarize the structure of large graphs, based on non-parametric estimation of edge density in directed multigraphs. Following coclustering approach, we use a clustering of the vertices, with a…
Small subgraph counts can be used as summary statistics for large random graphs. We use the Stein-Chen method to derive Poisson approximations for the distribution of the number of subgraphs in the stochastic block model which are…
We explicitly quantify the empirically observed phenomenon that estimation under a stochastic block model (SBM) is hard if the model contains classes that are similar. More precisely, we consider estimation of certain functionals of random…
We propose a new hierarchy of semidefinite programming relaxations for inference problems. As test cases, we consider the problem of community detection in block models. The vertices are partitioned into $k$ communities, and a graph is…
The stochastic block model (SBM) is a random graph model with different group of vertices connecting differently. It is widely employed as a canonical model to study clustering and community detection, and provides a fertile ground to study…
The minimal number of nodes required to multilaterate a network endowed with geodesic distance (i.e., to uniquely identify all nodes based on shortest path distances to the selected nodes) is called its metric dimension. This quantity is…
We analyze the bootstrap percolation process on the stochastic block model (SBM), a natural extension of the Erd\"{o}s--R\'{e}nyi random graph that allows representing the "community structure" observed in many real systems. In the SBM,…
Motivated by the prevalent data science applications of processing large-scale graph data such as social networks and biological networks, this paper investigates lossless compression of data in the form of a labeled graph. Particularly, we…
Finding dense subgraphs of a large graph is a standard problem in graph mining that has been studied extensively both for its theoretical richness and its many practical applications. In this paper we introduce a new family of dense…
Community detection in graphs often relies on ad hoc algorithms with no clear specification about the node partition they define as the best, which leads to uninterpretable communities. Stochastic block models (SBM) offer a framework to…
Dense subgraph discovery methods are routinely used in a variety of applications including the identification of a team of skilled individuals for collaboration from a social network. However, when the network's node set is associated with…
The densest $k$-subgraph problem is the problem of finding a $k$-vertex subgraph of a graph with the maximum number of edges. In order to solve large instances of the densest $k$-subgraph problem, we introduce two algorithms that are based…
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…
Graph spectral techniques for measuring graph similarity, or for learning the cluster number, require kernel smoothing. The choice of kernel function and bandwidth are typically chosen in an ad-hoc manner and heavily affect the resulting…
Inference for the stochastic blockmodel is currently of burgeoning interest in the statistical community, as well as in various application domains as diverse as social networks, citation networks, brain connectivity networks…
The distribution of the eigenvalues of a Hermitian matrix (or of a Hermitian matrix pencil) reveals important features of the underlying problem, whether a Hamiltonian system in physics, or a social network in behavioral sciences. However,…
The problem of finding dense components of a graph is a widely explored area in data analysis, with diverse applications in fields and branches of study including community mining, spam detection, computer security and bioinformatics. This…
In this paper we prove the strong consistency of several methods based on the spectral clustering techniques that are widely used to study the community detection problem in stochastic block models (SBMs). We show that under some weak…
Random graphs have been widely used in statistics, for example in network analysis and graphical models. In some applications, the data may contain an inherent hierarchical ordering among its vertices, which prevents directed edges between…