Related papers: Dirichlet draws are sparse with high probability
A majority coloring of a directed graph is a vertex-coloring in which every vertex has the same color as at most half of its out-neighbors. Kreutzer, Oum, Seymour, van der Zypen and Wood proved that every digraph has a majority 4-coloring…
Stochastic linear combinations of some random vectors are studied where the distribution of the random vectors and the joint distribution of their coefficients are Dirichlet. A method is provided for calculating the distribution of these…
We study the random loop model with crosses and bars on sparse random graphs. Our main objective is to prove the existence of macroscopic loops, in the sense that a loop visits a positive proportion of the vertices. We develop a…
The probability distribution for the number of top to bottom spanning clusters in Directed percolation in two and three dimensions appears to be universal and is of the form $P(n) \sim \exp(-\alpha n^2)$. We argue that $\alpha$ is a new…
We present a general method for obtaining the spectra of large graphs with short cycles using ideas from statistical mechanics of disordered systems. This approach leads to an algorithm that determines the spectra of graphs up to a high…
We provide generating functions for the popularity and the distribution of patterns of length at most three over the set of Dyck paths having a first return decomposition constrained by height.
Generalizing the concept of dense hypergraph, we say that a hypergraph is weakly dense, if no k in the half-open interval [2,sqrt(n)) is the degree of more than k^2 vertices. In our main result, we prove the famous Erdos-Faber-Lovasz…
Take a look around you -- in your family, your school or workplace, in the streets, and you see boys & girls in about equal proportion, and without any easily visible gender patterns in case of siblings. So, to the famous first order of…
Condorcet's paradox is a fundamental result in social choice theory which states that there exist elections in which, no matter which candidate wins, a majority of voters prefer a different candidate. In fact, even if we can select any $k$…
A sequence of graphs is FO-convergent if the probability of satisfaction of every first-order formula converges. A graph modeling is a graph, whose domain is a standard probability space, with the property that every definable set is Borel.…
We consider long-range percolation in dimension $d\geq 1$, where distinct sites $x$ and $y$ are connected with probability $p_{x,y}\in[0,1]$. Assuming that $p_{x,y}$ is translation invariant and that $p_{x,y}=\|x-y\|^{-s+o(1)}$ with $s>2d$,…
Sparse coding has been proposed as a theory of visual cortex and as an unsupervised algorithm for learning representations. We show empirically with the MNIST dataset that sparse codes can be very sensitive to image distortions, a behavior…
It is recognised now that a variety of real-life phenomena ranging from diffuson of cold atoms to motion of humans exhibit dispersal faster than normal diffusion. L\'evy walks is a model that excelled in describing such superdiffusive…
A distinguishing feature of certain intractable problems in prime number theory is the sparsity of the underlying sequence. Motivated by the general problem of finding primes in sparse polynomial sequences, we give an estimate for the…
This paper was removed due to an error in the proof (Claim 4.12 as stated is not true). The authors would like to thank Ilya Volkovich for pointing out a counterexample to this paper's main result in positive characteristic: If $F$ is a…
Dvorak (2013) gave a bound on the minimum size of a distance r dominating set in the terms of the maximum size of a distance 2r independent set and generalized coloring numbers, thus obtaining a constant factor approximation algorithm for…
We investigate the rank of random (symmetric) sparse matrices. Our main finding is that with high probability, any dependency that occurs in such a matrix is formed by a set of few rows that contains an overwhelming number of zeros. This…
Consider the set of Borel probability measures on $\mathbf{R}^k$ and endow it with the topology of weak convergence. We show that the subset of all probability measures which belong to the domain of attraction of some multivariate extreme…
Many problems of interest in computer science and information theory can be phrased in terms of a probability distribution over discrete variables associated to the vertices of a large (but finite) sparse graph. In recent years,…
The simple graphs $G=(V,E)$ that satisfy $|E'|\leq 2|V'|-l$ for any subgraph (and for $l=1,2,3$) are the $(2,l)$-sparse graphs. Those that also satisfy $|E|=2|V|-l$ are the $(2,l)$-tight graphs. These can be characterised by their…