121 research outputs found
Strong Jumps and Lagrangians of Non-Uniform Hypergraphs
The hypergraph jump problem and the study of Lagrangians of uniform
hypergraphs are two classical areas of study in the extremal graph theory. In
this paper, we refine the concept of jumps to strong jumps and consider the
analogous problems over non-uniform hypergraphs. Strong jumps have rich
topological and algebraic structures. The non-strong-jump values are precisely
the densities of the hereditary properties, which include the Tur\'an densities
of families of hypergraphs as special cases. Our method uses a generalized
Lagrangian for non-uniform hypergraphs. We also classify all strong jump values
for -hypergraphs.Comment: 19 page
Biased landscapes for random Constraint Satisfaction Problems
The typical complexity of Constraint Satisfaction Problems (CSPs) can be
investigated by means of random ensembles of instances. The latter exhibit many
threshold phenomena besides their satisfiability phase transition, in
particular a clustering or dynamic phase transition (related to the tree
reconstruction problem) at which their typical solutions shatter into
disconnected components. In this paper we study the evolution of this
phenomenon under a bias that breaks the uniformity among solutions of one CSP
instance, concentrating on the bicoloring of k-uniform random hypergraphs. We
show that for small k the clustering transition can be delayed in this way to
higher density of constraints, and that this strategy has a positive impact on
the performances of Simulated Annealing algorithms. We characterize the modest
gain that can be expected in the large k limit from the simple implementation
of the biasing idea studied here. This paper contains also a contribution of a
more methodological nature, made of a review and extension of the methods to
determine numerically the discontinuous dynamic transition threshold.Comment: 32 pages, 16 figure
Multistability, intermittency, and hybrid transitions in social contagion models on hypergraphs
Although ubiquitous, interactions in groups of individuals are not yet thoroughly studied. Frequently, single groups are modeled as critical-mass dynamics, which is a widespread concept used not only by academics but also by politicians and the media. However, less explored questions are how a collection of groups will behave and how their intersection might change the dynamics. Here, we formulate this process as binary-state dynamics on hypergraphs. We showed that our model has a rich behavior beyond discontinuous transitions. Notably, we have multistability and intermittency. We demonstrated that this phenomenology could be associated with community structures, where we might have multistability or intermittency by controlling the number or size of bridges between communities. Furthermore, we provided evidence that the observed transitions are hybrid. Our findings open new paths for research, ranging from physics, on the formal calculation of quantities of interest, to social sciences, where new experiments can be designed
A hypergraph blow-up lemma
We obtain a hypergraph generalisation of the graph blow-up lemma proved by
Komlos, Sarkozy and Szemeredi, showing that hypergraphs with sufficient
regularity and no atypical vertices behave as if they were complete for the
purpose of embedding bounded degree hypergraphs.Comment: 102 pages, 1 figure, to appear in Random Structures and Algorithm
Topological and Geometric Methods with a View Towards Data Analysis
In geometry, various tools have been developed to explore the topology and other features
of a manifold from its geometrical structure. Among the two most powerful ones are the
analysis of the critical points of a function, or more generally, the closed orbits of a dynamical
system defined on the manifold, and the evaluation of curvature inequalities. When any
(nondegenerate) function has to have many critical points and with different indices, then the
topology must be rich, and when certain curvature inequalities hold throughout the manifold,
that constrains the topology. It has been observed that these principles also hold for metric
spaces more general than Riemannian manifolds, and for instance also for graphs. This
thesis represents a contribution to this program. We study the relation between the closed
orbits of a dynamical system and the topology of a manifold or a simplicial complex via the
approach of Floer. And we develop notions of Ricci curvature not only for graphs, but more
generally for, possibly directed, hypergraphs, and we draw structural consequences from
curvature inequalities. It includes methods that besides their theoretical importance can be
used as powerful tools for data analysis. This thesis has two main parts; in the first part we
have developed topological methods based on the dynamic of vector fields defined on smooth
as well as discrete structures. In the second
part, we concentrate on some curvature notions which already proved themselves as powerful
measures for determining the local (and global) structures of smooth objects. Our main
motivation here is to develop methods that are helpful for the analysis of complex networks.
Many empirical networks incorporate higher-order relations between elements and therefore
are naturally modeled as, possibly directed and/or weighted, hypergraphs, rather than merely
as graphs. In order to develop a systematic tool for the statistical analysis of such hypergraphs,
we propose a general definition of Ricci curvature on directed hypergraphs and explore the
consequences of that definition. The definition generalizes Ollivierâs definition for graphs.
It involves a carefully designed optimal transport problem between sets of vertices. We can
then characterize various classes of hypergraphs by their curvature. In the last chapter, we
show that our curvature notion is a powerful tool for determining complex local structures in
a variety of real and random networks modeled as (directed) hypergraphs. Furthermore, it
can nicely detect hyperloop structures; hyperloops are fundamental in some real networks
such as chemical reactions as catalysts in such reactions are faithfully modeled as vertices
of directed hyperloops. We see that the distribution of our curvature notion in real networks deviates
from random models
Clique Factors: Extremal and Probabilistic Perspectives
A K_r-factor in a graph G is a collection of vertex-disjoint copies of K_r covering the vertex set of G. In this thesis, we investigate these fundamental objects in three settings that lie at the intersection of extremal and probabilistic combinatorics.
Firstly, we explore pseudorandom graphs. An n-vertex graph is said to be (p,ÎČ)-bijumbled if for any vertex sets A, B â V (G), we have e( A, B) = p| A||B| ± ÎČâ|A||B|. We prove that for any 3 †r â N and c > 0 there exists an Δ > 0 such that any n-vertex (p, ÎČ)-bijumbled graph with n â rN, ÎŽ(G) â„ c p n and ÎČ â€ Î” p^{r â1} n, contains a K_r -factor. This implies a corresponding result for the stronger pseudorandom notion of (n, d, λ)-graphs. For the case of K_3-factors, this result resolves a conjecture of Krivelevich, Sudakov and SzabĂł from 2004 and it is tight due to a pseudorandom triangle-free construction of Alon. In fact, in this case even more is true: as a corollary to this result, we can conclude that the same condition of ÎČ = o( p^2n) actually guarantees that a (p, ÎČ)-bijumbled graph G contains every graph on n vertices with maximum degree at most 2.
Secondly, we explore the notion of robustness for K_3-factors. For a graph G and p â [0, 1], we denote by G_p the random sparsification of G obtained by keeping each edge of G independently, with probability p. We show that there exists a C > 0 such that if p â„ C (log n)^{1/3}n^{â2/3} and G is an n-vertex graph with n â 3N and ÎŽ(G) â„ 2n/3 , then with high probability G_p contains a K_3-factor. Both the minimum degree condition and the probability condition, up to the choice of C, are tight. Our result can be viewed as a common strengthening of the classical extremal theorem of CorrĂĄdi and Hajnal, corresponding to p = 1 in our result, and the famous probabilistic theorem of Johansson, Kahn and Vu establishing the threshold for the appearance of K_3-factors (and indeed all K_r -factors) in G (n, p), corresponding to G = K_n in our result. It also implies a first lower bound on the number of K_3-factors in graphs with minimum degree at least 2n/3, which gets close to the truth.
Lastly, we consider the setting of randomly perturbed graphs; a model introduced by Bohman, Frieze and Martin, where one starts with a dense graph and then adds random edges to it. Specifically, given any fixed 0 < α < 1 â 1/r we determine how many random edges one must add to an n-vertex graph G with ÎŽ(G) ℠α n to ensure that, with high probability, the resulting graph contains a K_r -factor. As one increases α we demonstrate that the number of random edges
required âjumpsâ at regular intervals, and within these intervals our result is best-possible. This work therefore bridges the gap between the seminal work of Johansson, Kahn and Vu mentioned above, which resolves the purely random case, i.e., α = 0, and that of Hajnal and SzemerĂ©di (and CorrĂĄdi and Hajnal for r = 3) showing that when α â„ 1 â 1/r the initial graph already hosts the
desired K_r -factor.Ein K_r -Faktor in einem Graphen G ist eine Sammlung von Knoten-disjunkten Kopien von K_r , die die Knotenmenge von G ĂŒberdecken. Wir untersuchen diese Objekte in drei Kontexten, die an der Schnittstelle zwischen extremaler und probabilistischer Kombinatorik liegen.
Zuerst untersuchen wir Pseudozufallsgraphen. Ein Graph heiĂt (p,ÎČ)-bijumbled, wenn fĂŒr beliebige Knotenmengen A, B â V (G) gilt e( A, B) = p| A||B| ± ÎČâ|A||B|. Wir beweisen, dass es fĂŒr jedes 3 †r â N und c > 0 ein Δ > 0 gibt, so dass jeder n-Knoten (p, ÎČ)-bijumbled Graph mit n â rN, ÎŽ(G) â„ c p n und ÎČ â€ Î” p^{r â1} n, einen K_r -Faktor enthĂ€lt. Dies impliziert ein entsprechendes Ergebnis fĂŒr den stĂ€rkeren Pseudozufallsbegriff von (n, d, λ)-Graphen. Im Fall von K_3-Faktoren, löst dieses Ergebnis eine Vermutung von Krivelevich, Sudakov und SzabĂł aus
dem Jahr 2004 und ist durch eine pseudozufĂ€llige K_3-freie Konstruktion von Alon bestmöglich. TatsĂ€chlich ist in diesem Fall noch mehr wahr: als Korollar dieses Ergebnisses können wir schlieĂen, dass die gleiche Bedingung von ÎČ = o( p^2n) garantiert, dass ein (p, ÎČ)-bijumbled Graph G jeden Graphen mit maximalem Grad 2 enthĂ€lt.
Zweitens untersuchen wir den Begriff der Robustheit fĂŒr K_3-Faktoren. FĂŒr einen Graphen G und p â [0, 1] bezeichnen wir mit G_p die zufĂ€llige Sparsifizierung von G, die man erhĂ€lt, indem man jede Kante von G unabhĂ€ngig von den anderen Kanten mit einer Wahrscheinlichkeit p behĂ€lt. Wir zeigen, dass, wenn p â„ C (log n)^{1/3}n^{â2/3} und G ein n-Knoten-Graph mit n â 3N und ÎŽ(G) â„ 2n/3 ist, G_pmit hoher Wahrscheinlichkeit (mhW) einen K_3-Faktor enthĂ€lt. Sowohl die Bedingung des minimalen Grades als auch die Wahrscheinlichkeitsbedingung sind bestmöglich. Unser Ergebnis ist eine VerstĂ€rkung des klassischen extremalen Satzes von CorrĂĄdi und Hajnal, entsprechend p = 1 in unserem Ergebnis, und des berĂŒhmten probabilistischen Satzes von Johansson, Kahn und Vu, der den Schwellenwert fĂŒr das Auftreten eines K_3-Faktors (und aller K_r -Faktoren) in G (n, p) festlegt, entsprechend G = K_n in unserem Ergebnis. Es impliziert auch eine erste untere Schranke fĂŒr die Anzahl der K_3-Faktoren in Graphen mit einem minimalen Grad von mindestens 2n/3, die der Wahrheit nahe kommt.
SchlieĂlich betrachten wir die Situation von zufĂ€llig gestörten Graphen; ein Modell, bei dem man mit einem dichten Graphen beginnt und dann zufĂ€llige Kanten hinzufĂŒgt. Wir bestimmen, bei gegebenem 0 < α < 1 â 1/r, wie viele zufĂ€llige Kanten man zu einem n-Knoten-Graphen G mit ÎŽ(G) ℠α n hinzufĂŒgen muss, um sicherzustellen, dass der resultierende Graph mhW einen K_r -Faktor enthĂ€lt. Wir zeigen, dass, wenn man α erhöht, die Anzahl der benötigten Zufallskanten in regelmĂ€Ăigen AbstĂ€nden âspringt", und innerhalb dieser AbstĂ€nde unser Ergebnis bestmöglich ist. Diese Arbeit schlieĂt somit die LĂŒcke zwischen der oben erwĂ€hnten bahnbrechenden Arbeit von Johansson, Kahn und Vu, die den rein zufĂ€lligen Fall, d.h. α = 0, löst, und der Arbeit von Hajnal und SzemerĂ©di (und CorrĂĄdi und Hajnal fĂŒr r = 3), die zeigt, dass der ursprĂŒngliche Graph bereits den gewĂŒnschten K_r -Faktor enthĂ€lt, wenn α â„ 1 â 1/r ist
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