Related papers: On Sharp Thresholds in Random Geometric Graphs
The goal of this paper is to provide a general purpose result for the coupling of exploration processes of random graphs, both undirected and directed, with their local weak limits when this limit is a marked Galton-Watson process. This…
We introduce a new notion of resilience for constraint satisfaction problems, with the goal of more precisely determining the boundary between NP-hardness and the existence of efficient algorithms for resilient instances. In particular, we…
Pairwise Compatibility Graphs (PCGs) form a tree-metric graph class that originated in phylogeny and has since attracted sustained interest in graph theory. Several natural generalizations have been proposed in order to overcome the…
Asymptotic properties of a vector of length power functionals of random geometric graphs are investigated. More precisely, its asymptotic covariance matrix is studied as the intensity of the underlying homogeneous Poisson point process…
Kim defined a very general combinatorial abstraction of the diameter of polytopes called subset partition graphs to study how certain combinatorial properties of such graphs may be achieved in lower bound constructions. Using Lov\'asz'…
We address the following foundational question: what is the population, and sample, Frechet mean (or median) graph of an ensemble of inhomogeneous Erdos-Renyi random graphs? We prove that if we use the Hamming distance to compute distances…
We use the theory of graph limits to study several quasi-random properties, mainly dealing with various versions of hereditary subgraph counts. The main idea is to transfer the properties of (sequences of) graphs to properties of graphons,…
Multiple-choice load balancing has been a topic of intense study since the seminal paper of Azar, Broder, Karlin, and Upfal. Questions in this area can be phrased in terms of orientations of a graph, or more generally a k-uniform random…
For every $k \ge 1$, the $k$th cohomology group $H^k(X, \Q)$ of the random flag complex $X \sim X(n,p)$ passes through two phase transitions: one where it appears, and one where it vanishes. We describe the vanishing threshold and show that…
Suppose a $k$-uniform hypergraph $H$ that satisfies a certain regularity instance (that is, there is a partition of $H$ given by the hypergraph regularity lemma into a bounded number of quasirandom subhypergraphs of prescribed densities).…
We find sharp upper bounds on the order of the automorphism group of a hypersurface in complex projective space in every dimension and degree. In each case, we prove that the hypersurface realizing the upper bound is unique up to…
In this paper we study the one dimensional random geometric graph when the location of the nodes are independent and exponentially distributed. We derive exact results and the limit theorems for the connectivity and other properties…
Using methods and ideas from statistical mechanics, we propose a simple method for obtaining rigorous upper bounds for satisfiability transition in random boolean expressions composed of N variables and M clauses with K variables per…
Random geometric graphs consist of randomly distributed nodes (points), with pairs of nodes within a given mutual distance linked. In the usual model the distribution of nodes is uniform on a square, and in the limit of infinitely many…
In our previous paper, we presented the combinatorial theory for minimal isostatic pinned frameworks - Assur graphs - which arise in the analysis of mechanical linkages. In this paper we further explore the geometric properties of Assur…
Subgraph densities have been defined, and served as basic tools, both in the case of graphons (limits of dense graph sequences) and graphings (limits of bounded-degree graph sequences). While limit objects have been described for the…
There has been substantial interest in estimating the value of a graph parameter, i.e., of a real-valued function defined on the set of finite graphs, by querying a randomly sampled substructure whose size is independent of the size of the…
Random shapes arise naturally in many contexts. The topological and geometric structure of such objects is interesting for its own sake, and also for applications. In physics, for example, such objects arise naturally in quantum gravity, in…
The distribution $\mathsf{RGG}(n,\mathbb{S}^{d-1},p)$ is formed by sampling independent vectors $\{V_i\}_{i = 1}^n$ uniformly on $\mathbb{S}^{d-1}$ and placing an edge between pairs of vertices $i$ and $j$ for which $\langle V_i,V_j\rangle…
We establish an Expander Mixing Lemma for directed graphs in terms of the eigenvalues of an associated asymmetric transition probability matrix, extending the classical spectral inequality to the asymmetric setting. As an application, we…