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Simi Haber

Publications and source records attributed to Simi Haber.

15 recordsLinked to original sources
arXiv2026

On the most reliable graphs with fixed redundancy

The all-terminal reliability of a graph $G$ is the probability that $G$ remains connected when each edge fails independently with probability $p$. For fixed $n$ and $m$, the uniformly most reliable problem asks which graph with $n$ vertices and $m$ edges maximizes reliability for all $p \in [0,1]$. Although such graphs do not always exist, optimal graphs in the regime $p \to 0$ always do and are determined by the structure of their minimal cut sets. We establish a structural characterization of graphs that are most reliable near $p=0$. Our results partially resolve a conjecture of Bourel et al., showing that, under suitable conditions, regular graphs with maximal girth are optimal. Extending this analysis to graphs with fixed redundancy $r=m-(n-1)$ and sufficiently large $n$, we show that the most reliable graphs are obtained by subdividing the most reliable cubic graphs with $2(r-1)$ vertices. The general conjecture remains open. Unlike previous results, which resolved only small redundancy cases or very dense regimes, our approach yields a substantial extension of the known range. We determine the unique cubic candidates for uniformly most reliable graphs for all redundancy levels $m-n \le 19$, and prove the non-existence of uniformly most reliable graphs for several infinite families with fixed redundancy and asymptotically large $n$. These results significantly enlarge both the candidate class and the range of provable non-existence.

math.CO↗
arXiv2023

Unsupervised Estimation of Ensemble Accuracy

Ensemble learning combines several individual models to obtain a better generalization performance. In this work we present a practical method for estimating the joint power of several classifiers. It differs from existing approaches which focus on "diversity" measures by not relying on labels. This makes it both accurate and practical in the modern setting of unsupervised learning with huge datasets. The heart of the method is a combinatorial bound on the number of mistakes the ensemble is likely to make. The bound can be efficiently approximated in time linear in the number of samples. We relate the bound to actual misclassifications, hence its usefulness as a predictor of performance. We demonstrate the method on popular large-scale face recognition datasets which provide a useful playground for fine-grain classification tasks using noisy data over many classes.

cs.AI↗
arXiv2022

Ensemble dependence of the critical behavior of a system with long-range interaction and quenched randomness

We propose a hybrid model governed by the Blume-Emery-Griffiths (BEG) Hamiltonian with a mean-field-like interaction, where the spins are randomly quenched such that some of them are "pure" Ising and the others admit the BEG set of states. It is found, by varying the concentration of the Ising spins, that the model displays different phase portraits in concentration-temperature parameter space, within the canonical and the microcanonical ensembles. Phenomenological indications that these portraits are rich and rather unusual are provided.

cond-mat.stat-mech↗
arXiv2021

Efficient covering of convex domains by congruent discs

In this paper, we consider the problem of covering a plane region with unit discs. We present an improved upper bound and the first nontrivial lower bound on the number of discs needed for such a covering, depending on the area and perimeter of the region. We provide algorithms for efficient covering of convex polygonal regions using unit discs. We show that the computational complexity of the algorithms is pseudo-polynomial in the size of the input and the output. We also show that these algorithms provide a constant factor approximation of the optimal covering of the region.

cs.CG↗
arXiv2021

Changeover phenomenon in randomly colored Potts models

A hybrid Potts model where a random concentration $p$ of the spins assume $q_0$ states and a random concentration $1-p$ of the spins assume $q>q_0$ states is introduced. It is known that when the system is homogeneous, with an integer spin number $q_0$ or $q$, it undergoes a second or a first order transition, respectively. It is argued that there is a concentration $p^\ast$ such that the transition nature of the model is changed at $p^\ast$. This idea is demonstrated analytically and by simulations for two different types of interaction: the usual square lattice nearest neighboring and mean field all-to-all. Exact expressions for the second order critical line in concentration-temperature parameter space of the mean field model together with some other related critical properties, are derived.

cond-mat.stat-mech↗
arXiv2020

Isomorphism for Random $k$-Uniform Hypergraphs

We study the isomorphism problem for random hypergraphs. We show that it is solvable in polynomial time for the binomial random $k$-uniform hypergraph $H_{n,p;k}$, for a wide range of $p$. We also show that it is solvable w.h.p. for random $r$-regular, $k$-uniform hypergraphs $H_{n,r;k},r=O(1)$.

math.CO↗
arXiv2019

Optimal shattering of complex networks

We consider optimal attacks or immunization schemes on different models of random graphs. We derive bounds for the minimum number of nodes needed to be removed from a network such that all remaining components are fragments of negligible size. We obtain bounds for different regimes of random regular graphs, Erd\H{o}s-R\'enyi random graphs, and scale free networks, some of which are tight. We show that the performance of attacks by degree is bounded away from optimality. Finally we present a polynomial time attack algorithm and prove its optimal performance in certain cases.

physics.soc-ph↗
arXiv2018

Unusual changeover in the transition nature of local-interaction Potts models

A combinatorial approach is used to study the critical behavior of a $q$-state Potts model with a round-the-face interaction. Using this approach it is shown that the model exhibits a first order transition for $q>3$. A second order transition is numerically detected for $q=2$. Based on these findings, it is deduced that for some two-dimensional ferromagnetic Potts models with completely local interaction, there is a changeover in the transition order at a critical integer $q_c\leq 3$. This stands in contrast to the standard two-spin interaction Potts model where the maximal integer value for which the transition is continuous is $q_c=4$. A lower bound on the first order critical temperature is additionally derived.

cond-mat.stat-mech↗
arXiv2017

Ferromagnetic Potts models with multisite interaction

We study the $q$ states Potts model with four site interaction on the square lattice. Based on the asymptotic behaviour of lattice animals, it is argued that when $q\leq 4$ the system exhibits a second-order phase transition, and when $q > 4$ the transition is first order. The $q=4$ model is borderline. We find ${1}/{\ln q}$ to be an upper bound on $T_c$, the exact critical temperature. Using a low-temperature expansion, we show that $1/(\theta\ln q)$, where $\theta>1$ is a $q$-dependent geometrical term, is an improved upper bound on $T_c$. In fact, our findings support $T_c=1/(\theta\ln q)$. This expression is used to estimate the finite correlation length in first-order transition systems. These results can be extended to other lattices. Our theoretical predictions are confirmed numerically by an extensive study of the four-site interaction model using the Wang-Landau entropic sampling method for $q=3,4,5$. In particular, the $q=4$ model shows an ambiguous finite-size pseudocritical behaviour.

cond-mat.stat-mech↗
arXiv2015

Random graphs and Lindstrom quantifiers for natural graph properties

We study zero-one laws for random graphs. We focus on the following question that was asked by many: Given a graph property P, is there a language of graphs able to express P while obeying the zero-one law? Our results show that on the one hand there is a (regular) language able to express connectivity and k-colorability for any constant k and still obey the zero-one law. On the other hand we show that in any (semiregular) language strong enough to express Hamiltonicity one can interpret arithmetic and thus the zero-one law fails miserably. This answers a question of Blass and Harary.

math.LO↗
arXiv2012

An almost linear time algorithm for finding Hamilton cycles in sparse random graphs with minimum degree at least three

We describe an algorithm for finding Hamilton cycles in random graphs. Our model is the random graph $G=\gc$. In this model $G$ is drawn uniformly from graphs with vertex set $[n]$, $m$ edges and minimum degree at least three. We focus on the case where $m=cn$ for constant $c$. If $c$ is sufficiently large then our algorithm runs in $O(n^{1+o(1)})$ time and succeeds w.h.p.

math.CO↗
arXiv2011

On the game chromatic number of sparse random graphs

Given a graph G and an integer k, two players take turns coloring the vertices of G one by one using k colors so that neighboring vertices get different colors. The first player wins iff at the end of the game all the vertices of $G$ are colored. The game chromatic number \chi_g(G) is the minimum k for which the first player has a winning strategy. The paper \cite{BFS} began the analysis of the asymptotic behavior of this parameter for a random graph G_{n,p}. This paper provides some further analysis for graphs with constant average degree i.e. np=O(1) and for random regular graphs.

math.CO↗
arXiv2010

The number of F-matchings in almost every tree is a zero residue

For graphs F and G an F-matching in G is a subgraph of G consisting of pairwise vertex disjoint copies of F. The number of F-matchings in G is denoted by s(F,G). We show that for every fixed positive integer m and every fixed tree F, the probability that s(F,T_n) = 0 mod m, where T_n is a random labeled tree with n vertices, tends to one exponentially fast as n grows to infinity. A similar result is proven for induced F-matchings. This generalizes a recent result of Wagner who showed that the number of independent sets in a random labeled tree is almost surely a zero residue.

math.CO↗
arXiv2005

The isoperimetric constant of the random graph process

The isoperimetric constant of a graph $G$ on $n$ vertices, $i(G)$, is the minimum of $\frac{|\partial S|}{|S|}$, taken over all nonempty subsets $S\subset V(G)$ of size at most $n/2$, where $\partial S$ denotes the set of edges with precisely one end in $S$. A random graph process on $n$ vertices, $\widetilde{G}(t)$, is a sequence of $\binom{n}{2}$ graphs, where $\widetilde{G}(0)$ is the edgeless graph on $n$ vertices, and $\widetilde{G}(t)$ is the result of adding an edge to $\widetilde{G}(t-1)$, uniformly distributed over all the missing edges. We show that in almost every graph process $i(\widetilde{G}(t))$ equals the minimal degree of $\widetilde{G}(t)$ as long as the minimal degree is $o(\log n)$. Furthermore, we show that this result is essentially best possible, by demonstrating that along the period in which the minimum degree is typically $Θ(\log n)$, the ratio between the isoperimetric constant and the minimum degree falls from 1 to 1/2, its final value.

math.PR↗