arXiv subjects

Yuval Peled

Publications and source records attributed to Yuval Peled.

27 records · Page 2Linked to original sources

arXiv2018

On simple connectivity of random 2-complexes

The fundamental group of the $2$-dimensional Linial-Meshulam random simplicial complex $Y_2(n,p)$ was first studied by Babson, Hoffman and Kahle. They proved that the threshold probability for simple connectivity of $Y_2(n,p)$ is about $p\approx n^{-1/2}$. In this paper, we show that this threshold probability is at most $p\le (γn)^{-1/2}$, where $γ= 4^4/3^3$, and conjecture that this threshold is sharp. In fact, we show that $p=(γn)^{-1/2}$ is a sharp threshold probability for the stronger property that every cycle of length $3$ is the boundary of a subcomplex of $Y_2(n,p)$ that is homeomorphic to a disk. Our proof uses the Poisson paradigm, and relies on a classical result of Tutte on the enumeration of planar triangulations.

math.CO↗

arXiv2018

Enumeration and randomized constructions of hypertrees

Over thirty years ago, Kalai proved a beautiful $d$-dimensional analog of Cayley's formula for the number of $n$-vertex trees. He enumerated $d$-dimensional hypertrees weighted by the squared size of their $(d-1)$-dimensional homology group. This, however, does not answer the more basic problem of unweighted enumeration of $d$-hypertrees, which is our concern here. Our main result, Theorem 1.4, significantly improves the lower bound for the number of $d$-hypertrees. In addition, we study a random $1$-out model of $d$-complexes where every $(d-1)$-dimensional face selects a random $d$-face containing it, and show it has a negligible $d$-dimensional homology.

math.CO↗

arXiv2016

On the phase transition in random simplicial complexes

It is well-known that the $G(n,p)$ model of random graphs undergoes a dramatic change around $p=\frac 1n$. It is here that the random graph is, almost surely, no longer a forest, and here it first acquires a giant (i.e., order $Ω(n)$) connected component. Several years ago, Linial and Meshulam have introduced the $X_d(n,p)$ model, a probability space of $n$-vertex $d$-dimensional simplicial complexes, where $X_1(n,p)$ coincides with $G(n,p)$. Within this model we prove a natural $d$-dimensional analog of these graph theoretic phenomena. Specifically, we determine the exact threshold for the nonvanishing of the real $d$-th homology of complexes from $X_d(n,p)$. We also compute the real Betti numbers of $X_d(n,p)$ for $p=c/n$. Finally, we establish the emergence of giant shadow at this threshold. (For $d=1$ a giant shadow and a giant component are equivalent). Unlike the case for graphs, for $d\ge 2$ the emergence of the giant shadow is a first order phase transition.

math.PR↗

arXiv2016

Random simplicial complexes - around the phase transition

This article surveys some of the work done in recent years on random simplicial complexes. We mostly consider higher-dimensional analogs of the well known phase transition in $G(n, p)$ theory that occurs at $p = \frac 1n$ . Our main objective is to provide a more streamlined and unified perspective of some of the papers in this area.

math.CO↗

arXiv2016

Integral homology of random simplicial complexes

The random $2$-dimensional simplicial complex process starts with a complete graph on $n$ vertices, and in every step a new $2$-dimensional face, chosen uniformly at random, is added. We prove that with probability tending to $1$ as $n\to\infty$, the first homology group over $\mathbb Z$ vanishes at the very moment when all the edges are covered by triangular faces.

math.CO↗

arXiv2015

Extremal problems on shadows and hypercuts in simplicial complexes

Let $F$ be an $n$-vertex forest. We say that an edge $e\notin F$ is in the shadow of $F$ if $F\cup\{e\}$ contains a cycle. It is easy to see that if $F$ is "almost a tree", that is, it has $n-2$ edges, then at least $\lfloor\frac{n^2}{4}\rfloor$ edges are in its shadow and this is tight. Equivalently, the largest number of edges an $n$-vertex cut can have is $\lfloor\frac{n^2}{4}\rfloor$. These notions have natural analogs in higher $d$-dimensional simplicial complexes, graphs being the case $d=1$. The results in dimension $d>1$ turn out to be remarkably different from the case in graphs. In particular the corresponding bounds depend on the underlying field of coefficients. We find the (tight) analogous theorems for $d=2$. We construct $2$-dimensional "$\mathbb Q$-almost-hypertrees" (defined below) with an empty shadow. We also show that the shadow of an "$\mathbb F_2$-almost-hypertree" cannot be empty, and its least possible density is $Θ(\frac{1}{n})$. In addition we construct very large hyperforests with a shadow that is empty over every field. For $d\ge 4$ even, we construct $d$-dimensional $\mathbb{F} _2$-almost-hypertree whose shadow has density $o_n(1)$. Finally, we mention several intriguing open questions.

math.CO↗

arXiv2015

A random triadic process

Given a random 3-uniform hypergraph $H=H(n,p)$ on $n$ vertices where each triple independently appears with probability $p$, consider the following graph process. We start with the star $G_0$ on the same vertex set, containing all the edges incident to some vertex $v_0$, and repeatedly add an edge $xy$ if there is a vertex $z$ such that $xz$ and $zy$ are already in the graph and $xzy \in H$. We say that the process propagates if it reaches the complete graph before it terminates. In this paper we prove that the threshold probability for propagation is $p=\frac{1}{2\sqrt{n}}$. We conclude that $p=\frac{1}{2\sqrt{n}}$ is an upper bound for the threshold probability that a random 2-dimensional simplicial complex is simply connected.

math.CO↗

arXiv2013

On the 3-local profiles of graphs

For a graph G, let p_i(G), i=0,...,3 be the probability that three distinct random vertices span exactly i edges. We call (p_0(G),...,p_3(G)) the 3-local profile of G. We investigate the set ${\cal S}_3 \subset \mathbb R^4$ of all vectors (p_0,...,p_3) that are arbitrarily close to the 3-local profiles of arbitrarily large graphs. We give a full description of the projection of ${\cal S}_3$ to the (p_0, p_3) plane. The upper envelope of this planar domain is obtained from cliques on a fraction of the vertex set and complements of such graphs. The lower envelope is Goodman's inequality p_0+p_3\geq 1/4. We also give a full description of the triangle-free case, i.e., the intersection of ${\cal S}_3$ with the hyperplane p_3=0. This planar domain is characterized by an SDP constraint that is derived from Razborov's flag algebra theory.

math.CO↗

arXiv2013

On the densities of cliques and independent sets in graphs

Let r, s >= 2 be integers. Suppose that the number of blue r-cliques in a red/blue coloring of the edges of the complete graph K_n is known and fixed. What is the largest possible number of red s-cliques under this assumption? The well known Kruskal-Katona theorem answers this question for r=2 or s=2. Using the shifting technique from extremal set theory together with some analytical arguments, we resolve this problem in general and prove that in the extremal coloring either the blue edges or the red edges form a clique.

math.CO↗