Related papers: Density theorems for bipartite graphs and related …
Empirical networks are often globally sparse, with a small average number of connections per node, when compared to the total size of the network. However, this sparsity tends not to be homogeneous, and networks can also be locally dense,…
One of the central questions in Ramsey theory asks how small can be the size of the largest clique and independent set in a graph on $N$ vertices. By the celebrated result of Erd\H{o}s from 1947, the random graph on $N$ vertices with edge…
Finding dense bipartite subgraphs and detecting the relations among them is an important problem for affiliation networks that arise in a range of domains, such as social network analysis, word-document clustering, the science of science,…
We analyze the computational complexity of the following computational problems called Bounded-Density Edge Deletion and Bounded-Density Vertex Deletion: Given a graph $G$, a budget $k$ and a target density $\tau_\rho$, are there $k$ edges…
We determine to within a constant factor the threshold for the property that two random k-uniform hypergraphs with edge probability p have an edge-disjoint packing into the same vertex set. More generally, we allow the hypergraphs to have…
In this paper we raise the question of how to compress sparse graphs. By introducing the idea of redundancy, we find a way to measure the overlap of neighbors between nodes in networks. We exploit symmetry and information by making use of…
Graph clustering is an important algorithmic technique for analysing massive graphs, and has been widely applied in many research fields of data science. While the objective of most graph clustering algorithms is to find a vertex set of low…
We study a well known noisy model of the graph isomorphism problem. In this model, the goal is to perfectly recover the vertex correspondence between two edge-correlated Erd\H{o}s-R\'{e}nyi random graphs, with an initial seed set of…
It appeared recently that the classical random graph model used to represent real-world complex networks does not capture their main properties. Since then, various attempts have been made to provide accurate models. We study here a model…
In a recent breakthrough, Kelley and Meka (FOCS 2023) obtained a strong upper bound on the density of sets of integers without nontrivial three-term arithmetic progressions. In this work, we extend their result, establishing similar bounds…
The use of tools from analysis to approach problems in graph theory has become an active area of research. Usually such methods are applied to problems involving dense graphs and hypergraphs; here we give the an extension of such methods to…
Dense subgraph discovery aims to find a dense component in edge-weighted graphs. This is a fundamental graph-mining task with a variety of applications and thus has received much attention recently. Although most existing methods assume…
The problem of finding the densest subgraph in a given graph has several applications in graph mining, particularly in areas like social network analysis, protein and gene analyses etc. Depending on the application, finding dense subgraphs…
We develop a new technique that allows us to show in a unified way that many well-known combinatorial theorems, including Tur\'an's theorem, Szemer\'edi's theorem and Ramsey's theorem, hold almost surely inside sparse random sets. For…
Many real-world networks can be modeled as graphs. Finding dense subgraphs is a key problem in graph mining with applications in diverse domains. In this paper, we consider two variants of the densest subgraph problem where multiple graph…
We introduce the Density Formula for (topological) drawings of graphs in the plane or on the sphere, which relates the number of edges, vertices, crossings, and sizes of cells in the drawing. We demonstrate its capability by providing…
In this manuscript a unified framework for conducting inference on complex aggregated data in high dimensional settings is proposed. The data are assumed to be a collection of multiple non-Gaussian realizations with underlying undirected…
We propose a statistical model for graphs with a core-periphery structure. To do this we define a precise notion of what it means for a graph to have this structure, based on the sparsity properties of the subgraphs of core and periphery…
Let d = (d1, d2, ..., dn) be a vector of non-negative integers with even sum. We prove some basic facts about the structure of a random graph with degree sequence d, including the probability of a given subgraph or induced subgraph.…
We study subgraphs that appear in large Ramsey graphs for a given graph $F$. The recent girth Ramsey theorem of the first two authors asserts that there are Ramsey graphs such that all small subgraphs are `forests of copies of $F$'…