Related papers: A counter-example to the probabilistic universal g…
We initiate the focused study of constant-cost randomized communication, with emphasis on its connection to graph representations. We observe that constant-cost randomized communication problems are equivalent to hereditary (i.e. closed…
We introduce a natural generalization of the Erd\H{o}s-R\'enyi random graph model in which random instances of a fixed motif are added independently. The binomial random motif graph $G(H,n,p)$ is the random (multi)graph obtained by adding…
We study common randomness generation problems where $n$ players aim to generate same sequences of random coin flips where some subsets of the players share an independent common coin which can be tossed multiple times, and there is a…
We study 3-random-like graphs, that is, sequences of graphs in which the densities of triangles and anti-triangles converge to 1/8. Since the random graph ${\mathcal G}_{n,1/2}$ is, in particular, 3-random-like, this can be viewed as a weak…
Answering a question of Claudet, we prove that the uniformly random graph $G\sim \mathbb G(n, 1/2)$ is $\Omega(\sqrt n)$-vertex-minor universal with high probability. That is, for some constant $\alpha\approx 0.911$, any graph on any…
A temporal random geometric graph is a random geometric graph in which all edges are endowed with a uniformly random time-stamp, representing the time of interaction between vertices. In such graphs, paths with increasing time stamps…
We consider the classic Moran process modeling the spread of genetic mutations, as extended to structured populations by Lieberman et al.\ (Nature, 2005). In this process, individuals are the vertices of a connected graph $G$. Initially,…
We investigate the threshold probability for connectivity of sparse graphs under weak assumptions. As a corollary this completely solve the problem for Cartesian powers of arbitrary graphs. In detail, let $G$ be a connected graph on $k$…
We consider random graphs with uniformly bounded edges on a Poisson point process conditioned to contain the origin. In particular we focus on the random connection model, the Boolean model and Miller-Abrahams random resistor network with…
We consider first order expressible properties of random perfect graphs. That is, we pick a graph $G_n$ uniformly at random from all (labelled) perfect graphs on $n$ vertices and consider the probability that it satisfies some graph…
An efficient implicit representation of an $n$-vertex graph $G$ in a family $\mathcal{F}$ of graphs assigns to each vertex of $G$ a binary code of length $O(\log n)$ so that the adjacency between every pair of vertices can be determined…
We present a simple proof for the universality of invariant and equivariant tensorized graph neural networks. Our approach considers a restricted intermediate hypothetical model named Graph Homomorphism Model to reach the universality…
We show that, under mild assumptions, the spectrum of a sum of independent random matrices is close to that of the Gaussian random matrix whose entries have the same mean and covariance. This nonasymptotic universality principle yields…
We prove a characterization of the structural conditions on matrices of sign-rank 3 and unit disk graphs (UDGs) which permit constant-cost public-coin randomized communication protocols. Therefore, under these conditions, these graphs also…
The unitary evolution maps in closed chaotic quantum graphs are known to have universal spectral correlations, as predicted by random matrix theory. In chaotic graphs with absorption the quantum maps become non-unitary. We show that their…
We show that for a sequence of random graphs Brouwer's conjecture holds true with probability tending to one as the number of vertices tends to infinity. Surprisingly, it was found that a similar statement holds true for weighted graphs…
For simple connected graphs with incommensurate bond lengths and with unitary symmetry we prove the Bohigas-Giannoni-Schmit conjecture in its most general form. Using supersymmetry and taking the limit of infinite graph size, we show that…
Random planar graphs have been the subject of much recent work. Many basic properties of the standard uniform random planar graph P_{n}, by which we mean a graph chosen uniformly at random from the set of all planar graphs with vertex set…
Graph-theoretic methods have seen wide use throughout the literature on multi-agent control and optimization. When communications are intermittent and unpredictable, such networks have been modeled using random communication graphs. When…
For a graph $G$ and a hereditary property $\mathcal{P}$, let $\text{ex}(G,\mathcal{P})$ denote the maximum number of edges of a subgraph of $G$ that belongs to $\mathcal{P}$. We prove that for every non-trivial hereditary property…