Weak hypergraph regularity and linear hypergraphs

We consider conditions which allow the embedding of linear hypergraphs of fixed size. In particular, we prove that any k-uniform hypergraph H of positive uniform density contains all linear k-uniform hypergraphs of a given size. More precisely, we show that for all integers l >= k >= 2 and every d > 0 there exists Q > 0 for which the following holds: if His a sufficiently large k-uniform hypergraph with the property that the density of H induced on every vertex subset of size on is at least d, then H contains every linear k-uniform hypergraph F with l vertices. The main ingredient in the proof of this result is a counting lemma for linear hypergraphs, which establishes that the straightforward extension of graph epsilon-regularity to hypergraphs suffices for counting linear hypergraphs. We also consider some related problems. (C) 2009 Elsevier Inc. All rights reserved.

Propagation connectivity of random hypergraphs

We study the concept of propagation connectivity on random 3-uniform hypergraphs. This concept is inspired by a simple linear time algorithm for solving instances of certain constraint satisfaction problems. We derive upper and lower bounds for the propagation connectivity threshold, and point out some algorithmic implications.

Mathematical Morphology on Hypergraphs Using Vertex-Hyperedge Correspondence

The focus of this paper is to develop computationally efficient mathematical morphology operators on hypergraphs. To this aim we consider lattice structures on hypergraphs on which we build morphological operators. We develop a pair of dual adjunctions between the vertex set and the hyperedge set of a hypergraph , by defining a vertex-hyperedge correspondence. This allows us to recover the classical notion of a dilation/erosion of a subset of vertices and to extend it to subhypergraphs of . This paper also studies the concept of morphological adjunction on hypergraphs for which both the input and the output are hypergraphs

Morphological filtering on hypergraphs

The focus of this article is to develop computationally efficient mathematical morphology operators on hypergraphs. To this aim we consider lattice structures on hypergraphs on which we build morphological operators. We develop a pair of dual adjunctions between the vertex set and the hyper edge set of a hypergraph H, by defining a vertex-hyperedge correspondence. This allows us to recover the classical notion of a dilation/erosion of a subset of vertices and to extend it to subhypergraphs of H. Afterward, we propose several new openings, closings, granulometries and alternate sequential filters acting (i) on the subsets of the vertex and hyperedge set of H and (ii) on the subhypergraphs of a hypergraph

Small World Folksonomies: Clustering in Tri-Partite Hypergraphs

Many recent Web 2.0 resource sharing applications can be subsumed under the "folksonomy" moniker. Regardless of the type of resource shared, all of these share a common structure describing the assignment of tags to resources by users. In this report, we generalize the notions of clustering and characteristic path length which play a major role in the current research on networks, where they are used to describe the small-world effects on many observable network datasets. To that end, we show that the notion of clustering has two facets which are not equivalent in the generalized setting. The new measures are evaluated on two large-scale folksonomy datasets from resource sharing systems on the web.

Vertex splitting and connectivity augmentation in hypergraphs

We consider problems of splitting and connectivity augmentation in hypergraphs. In a hypergraph G = (V +s, E), to split two edges su, sv, is to replace them with a single edge uv. We are interested in doing this in such a way as to preserve a defined level of connectivity in V . The splitting technique is often used as a way of adding new edges into a graph or hypergraph, so as to augment the connectivity to some prescribed level. We begin by providing a short history of work done in this area. Then several preliminary results are given in a general form so that they may be used to tackle several problems. We then analyse the hypergraphs G = (V + s, E) for which there is no split preserving the local-edge-connectivity present in V. We provide two structural theorems, one of which implies a slight extension to Mader’s classical splitting theorem. We also provide a characterisation of the hypergraphs for which there is no such “good” split and a splitting result concerned with a specialisation of the local-connectivity function. We then use our splitting results to provide an upper bound on the smallest number of size-two edges we must add to any given hypergraph to ensure that in the resulting hypergraph we have λ(x, y) ≥ r(x, y) for all x, y in V, where r is an integer valued, symmetric requirement function on V*V. This is the so called “local-edge-connectivity augmentation problem” for hypergraphs. We also provide an extension to a Theorem of Szigeti, about augmenting to satisfy a requirement r, but using hyperedges. Next, in a result born of collaborative work with Zoltán Király from Budapest, we show that the local-connectivity augmentation problem is NP-complete for hypergraphs. Lastly we concern ourselves with an augmentation problem that includes a locational constraint. The premise is that we are given a hypergraph H = (V,E) with a bipartition P = {P1, P2} of V and asked to augment it with size-two edges, so that the result is k-edge-connected, and has no new edge contained in some P(i). We consider the splitting technique and describe the obstacles that prevent us forming “good” splits. From this we deduce results about which hypergraphs have a complete Pk-split. This leads to a minimax result on the optimal number of edges required and a polynomial algorithm to provide an optimal augmentation.

Hypergraphs with many Kneser colorings

For fixed positive integers r, k and E with 1 <= l < r and an r-uniform hypergraph H, let kappa(H, k, l) denote the number of k-colorings of the set of hyperedges of H for which any two hyperedges in the same color class intersect in at least l elements. Consider the function KC(n, r, k, l) = max(H epsilon Hn) kappa(H, k, l), where the maximum runs over the family H-n of all r-uniform hypergraphs on n vertices. In this paper, we determine the asymptotic behavior of the function KC(n, r, k, l) for every fixed r, k and l and describe the extremal hypergraphs. This variant of a problem of Erdos and Rothschild, who considered edge colorings of graphs without a monochromatic triangle, is related to the Erdos-Ko-Rado Theorem (Erdos et al., 1961 [8]) on intersecting systems of sets. (C) 2011 Elsevier Ltd. All rights reserved.

Weak quasi-randomness for uniform hypergraphs

We study quasi-random properties of k-uniform hypergraphs. Our central notion is uniform edge distribution with respect to large vertex sets. We will find several equivalent characterisations of this property and our work can be viewed as an extension of the well known Chung-Graham-Wilson theorem for quasi-random graphs. Moreover, let K(k) be the complete graph on k vertices and M(k) the line graph of the graph of the k-dimensional hypercube. We will show that the pair of graphs (K(k),M(k)) has the property that if the number of copies of both K(k) and M(k) in another graph G are as expected in the random graph of density d, then G is quasi-random (in the sense of the Chung-Graham-Wilson theorem) with density close to d. (C) 2011 Wiley Periodicals, Inc. Random Struct. Alg., 40, 1-38, 2012

Fast modularisation and aomic decomposition of ontologies using axiom dependency hypergraphs

In this paper we define the notion of an axiom dependency hypergraph, which explicitly represents how axioms are included into a module by the algorithm for computing locality-based modules. A locality-based module of an ontology corresponds to a set of connected nodes in the hypergraph, and atoms of an ontology to strongly connected components. Collapsing the strongly connected components into single nodes yields a condensed hypergraph that comprises a representation of the atomic decomposition of the ontology. To speed up the condensation of the hypergraph, we first reduce its size by collapsing the strongly connected components of its graph fragment employing a linear time graph algorithm. This approach helps to significantly reduce the time needed for computing the atomic decomposition of an ontology. We provide an experimental evaluation for computing the atomic decomposition of large biomedical ontologies. We also demonstrate a significant improvement in the time needed to extract locality-based modules from an axiom dependency hypergraph and its condensed version.

Strong Connectivity in Directed Hypergraphs and its Application to the Atomic Decomposition of Ontologies

Los hipergrafos dirigidos se han empleado en problemas relacionados con lógica proposicional, bases de datos relacionales, linguística computacional y aprendizaje automático. Los hipergrafos dirigidos han sido también utilizados como alternativa a los grafos (bipartitos) dirigidos para facilitar el estudio de las interacciones entre componentes de sistemas complejos que no pueden ser fácilmente modelados usando exclusivamente relaciones binarias. En este contexto, este tipo de representación es conocida como hiper-redes. Un hipergrafo dirigido es una generalización de un grafo dirigido especialmente adecuado para la representación de relaciones de muchos a muchos. Mientras que una arista en un grafo dirigido define una relación entre dos de sus nodos, una hiperarista en un hipergrafo dirigido define una relación entre dos conjuntos de sus nodos. La conexión fuerte es una relación de equivalencia que divide el conjunto de nodos de un hipergrafo dirigido en particiones y cada partición define una clase de equivalencia conocida como componente fuertemente conexo. El estudio de los componentes fuertemente conexos de un hipergrafo dirigido puede ayudar a conseguir una mejor comprensión de la estructura de este tipo de hipergrafos cuando su tamaño es considerable. En el caso de grafo dirigidos, existen algoritmos muy eficientes para el cálculo de los componentes fuertemente conexos en grafos de gran tamaño. Gracias a estos algoritmos, se ha podido averiguar que la estructura de la WWW tiene forma de “pajarita”, donde más del 70% del los nodos están distribuidos en tres grandes conjuntos y uno de ellos es un componente fuertemente conexo. Este tipo de estructura ha sido también observada en redes complejas en otras áreas como la biología. Estudios de naturaleza similar no han podido ser realizados en hipergrafos dirigidos porque no existe algoritmos capaces de calcular los componentes fuertemente conexos de este tipo de hipergrafos. En esta tesis doctoral, hemos investigado como calcular los componentes fuertemente conexos de un hipergrafo dirigido. En concreto, hemos desarrollado dos algoritmos para este problema y hemos determinado que son correctos y cuál es su complejidad computacional. Ambos algoritmos han sido evaluados empíricamente para comparar sus tiempos de ejecución. Para la evaluación, hemos producido una selección de hipergrafos dirigidos generados de forma aleatoria inspirados en modelos muy conocidos de grafos aleatorios como Erdos-Renyi, Newman-Watts-Strogatz and Barabasi-Albert. Varias optimizaciones para ambos algoritmos han sido implementadas y analizadas en la tesis. En concreto, colapsar los componentes fuertemente conexos del grafo dirigido que se puede construir eliminando ciertas hiperaristas complejas del hipergrafo dirigido original, mejora notablemente los tiempos de ejecucion de los algoritmos para varios de los hipergrafos utilizados en la evaluación. Aparte de los ejemplos de aplicación mencionados anteriormente, los hipergrafos dirigidos han sido también empleados en el área de representación de conocimiento. En concreto, este tipo de hipergrafos se han usado para el cálculo de módulos de ontologías. Una ontología puede ser definida como un conjunto de axiomas que especifican formalmente un conjunto de símbolos y sus relaciones, mientras que un modulo puede ser entendido como un subconjunto de axiomas de la ontología que recoge todo el conocimiento que almacena la ontología sobre un conjunto especifico de símbolos y sus relaciones. En la tesis nos hemos centrado solamente en módulos que han sido calculados usando la técnica de localidad sintáctica. Debido a que las ontologías pueden ser muy grandes, el cálculo de módulos puede facilitar las tareas de re-utilización y mantenimiento de dichas ontologías. Sin embargo, analizar todos los posibles módulos de una ontología es, en general, muy costoso porque el numero de módulos crece de forma exponencial con respecto al número de símbolos y de axiomas de la ontología. Afortunadamente, los axiomas de una ontología pueden ser divididos en particiones conocidas como átomos. Cada átomo representa un conjunto máximo de axiomas que siempre aparecen juntos en un modulo. La decomposición atómica de una ontología es definida como un grafo dirigido de tal forma que cada nodo del grafo corresponde con un átomo y cada arista define una dependencia entre una pareja de átomos. En esta tesis introducimos el concepto de“axiom dependency hypergraph” que generaliza el concepto de descomposición atómica de una ontología. Un modulo en una ontología correspondería con un componente conexo en este tipo de hipergrafos y un átomo de una ontología con un componente fuertemente conexo. Hemos adaptado la implementación de nuestros algoritmos para que funcionen también con axiom dependency hypergraphs y poder de esa forma calcular los átomos de una ontología. Para demostrar la viabilidad de esta idea, hemos incorporado nuestros algoritmos en una aplicación que hemos desarrollado para la extracción de módulos y la descomposición atómica de ontologías. A la aplicación la hemos llamado HyS y hemos estudiado sus tiempos de ejecución usando una selección de ontologías muy conocidas del área biomédica, la mayoría disponibles en el portal de Internet NCBO. Los resultados de la evaluación muestran que los tiempos de ejecución de HyS son mucho mejores que las aplicaciones más rápidas conocidas. ABSTRACT Directed hypergraphs are an intuitive modelling formalism that have been used in problems related to propositional logic, relational databases, computational linguistic and machine learning. Directed hypergraphs are also presented as an alternative to directed (bipartite) graphs to facilitate the study of the interactions between components of complex systems that cannot naturally be modelled as binary relations. In this context, they are known as hyper-networks. A directed hypergraph is a generalization of a directed graph suitable for representing many-to-many relationships. While an edge in a directed graph defines a relation between two nodes of the graph, a hyperedge in a directed hypergraph defines a relation between two sets of nodes. Strong-connectivity is an equivalence relation that induces a partition of the set of nodes of a directed hypergraph into strongly-connected components. These components can be collapsed into single nodes. As result, the size of the original hypergraph can significantly be reduced if the strongly-connected components have many nodes. This approach might contribute to better understand how the nodes of a hypergraph are connected, in particular when the hypergraphs are large. In the case of directed graphs, there are efficient algorithms that can be used to compute the strongly-connected components of large graphs. For instance, it has been shown that the macroscopic structure of the World Wide Web can be represented as a “bow-tie” diagram where more than 70% of the nodes are distributed into three large sets and one of these sets is a large strongly-connected component. This particular structure has been also observed in complex networks in other fields such as, e.g., biology. Similar studies cannot be conducted in a directed hypergraph because there does not exist any algorithm for computing the strongly-connected components of the hypergraph. In this thesis, we investigate ways to compute the strongly-connected components of directed hypergraphs. We present two new algorithms and we show their correctness and computational complexity. One of these algorithms is inspired by Tarjan’s algorithm for directed graphs. The second algorithm follows a simple approach to compute the stronglyconnected components. This approach is based on the fact that two nodes of a graph that are strongly-connected can also reach the same nodes. In other words, the connected component of each node is the same. Both algorithms are empirically evaluated to compare their performances. To this end, we have produced a selection of random directed hypergraphs inspired by existent and well-known random graphs models like Erd˝os-Renyi and Newman-Watts-Strogatz. Besides the application examples that we mentioned earlier, directed hypergraphs have also been employed in the field of knowledge representation. In particular, they have been used to compute the modules of an ontology. An ontology is defined as a collection of axioms that provides a formal specification of a set of terms and their relationships; and a module is a subset of an ontology that completely captures the meaning of certain terms as defined in the ontology. In particular, we focus on the modules computed using the notion of syntactic locality. As ontologies can be very large, the computation of modules facilitates the reuse and maintenance of these ontologies. Analysing all modules of an ontology, however, is in general not feasible as the number of modules grows exponentially in the number of terms and axioms of the ontology. Nevertheless, the modules can succinctly be represented using the Atomic Decomposition of an ontology. Using this representation, an ontology can be partitioned into atoms, which are maximal sets of axioms that co-occur in every module. The Atomic Decomposition is then defined as a directed graph such that each node correspond to an atom and each edge represents a dependency relation between two atoms. In this thesis, we introduce the notion of an axiom dependency hypergraph which is a generalization of the atomic decomposition of an ontology. A module in the ontology corresponds to a connected component in the hypergraph, and the atoms of the ontology to the strongly-connected components. We apply our algorithms for directed hypergraphs to axiom dependency hypergraphs and in this manner, we compute the atoms of an ontology. To demonstrate the viability of this approach, we have implemented the algorithms in the application HyS which computes the modules of ontologies and calculate their atomic decomposition. In the thesis, we provide an experimental evaluation of HyS with a selection of large and prominent biomedical ontologies, most of which are available in the NCBO Bioportal. HyS outperforms state-of-the-art implementations in the tasks of extracting modules and computing the atomic decomposition of these ontologies.

Computing hypergraph width measures exactly

Hypergraph width measures are a class of hypergraph invariants important in studying the complexity of constraint satisfaction problems (CSPs). We present a general exact exponential algorithm for a large variety of these measures. A connection between these and tree decompositions is established. This enables us to almost seamlessly adapt the combinatorial and algorithmic results known for tree decompositions of graphs to the case of hypergraphs and obtain fast exact algorithms. As a consequence, we provide algorithms which, given a hypergraph H on n vertices and m hyperedges, compute the generalized hypertree-width of H in time O*(2n) and compute the fractional hypertree-width of H in time O(1.734601n.m).1

On some Density Theorems in Number Theory and Group Theory

Gowers, dans son article sur les matrices quasi-aléatoires, étudie la question, posée par Babai et Sos, de l'existence d'une constante $c>0$ telle que tout groupe fini possède un sous-ensemble sans produit de taille supérieure ou égale a $c|G|$. En prouvant que, pour tout nombre premier $p$ assez grand, le groupe $PSL_2(\mathbb{F}_p)$ (d'ordre noté $n$) ne posséde aucun sous-ensemble sans produit de taille $c n^{8/9}$, il y répond par la négative. Nous allons considérer le probléme dans le cas des groupes compacts finis, et plus particuliérement des groupes profinis $SL_k(\mathbb{Z}_p)$ et $Sp_{2k}(\mathbb{Z}_p)$. La premiére partie de cette thése est dédiée à l'obtention de bornes inférieures et supérieures exponentielles pour la mesure suprémale des ensembles sans produit. La preuve nécessite d'établir préalablement une borne inférieure sur la dimension des représentations non-triviales des groupes finis $SL_k(\mathbb{Z}/(p^n\mathbb{Z}))$ et $Sp_{2k}(\mathbb{Z}/(p^n\mathbb{Z}))$. Notre théoréme prolonge le travail de Landazuri et Seitz, qui considérent le degré minimal des représentations pour les groupes de Chevalley sur les corps finis, tout en offrant une preuve plus simple que la leur. La seconde partie de la thése à trait à la théorie algébrique des nombres. Un polynome monogéne $f$ est un polynome unitaire irréductible à coefficients entiers qui endengre un corps de nombres monogéne. Pour un nombre premier $q$ donné, nous allons montrer, en utilisant le théoréme de densité de Tchebotariov, que la densité des nombres premiers $p$ tels que $t^q -p$ soit monogéne est supérieure ou égale à $(q-1)/q$. Nous allons également démontrer que, quand $q=3$, la densité des nombres premiers $p$ tels que $\mathbb{Q}(\sqrt[3]{p})$ soit non monogéne est supérieure ou égale à $1/9$.

Droites sur les hypergraphes

Le Théorème de Sylvester-Gallai affirme que dans un ensemble fini S de points dans le plan, où les points ne sont pas tous sur une même droite, il y a une droite qui passe par exactement deux points de S. Chvátal [14] a étendu la notion de droites aux espaces métriques arbitraires et a fait une conjecture généralisant le Théorème de Sylvester-Gallai. Chen [10] a démontré cette conjecture qui s’appelle maintenant le Théorème de Sylvester-Chvátal. En 1943, Erdos [18] a remarqué un corollaire pour le Théorème de Sylvester-Gallai affirmant que, dans un ensemble fini V de points dans le plan, où les points ne sont pas tous sur une droite, le nombre de droites qui passent par au moins deux points de V est au moins |V |. De Bruijn et Erdos [7] ont généralisé ce corollaire, en utilisant une définition généralisée de droite (voir Chapitre 2) et ont prouvé que tout ensemble de n points, où les points ne sont pas tous sur une même droite, détermine au moins n droites distinctes. Dans le présent mémoire, nous allons étudier les théorèmes mentionnés ci-dessus. Nous allons aussi considérer le Théorème de De Bruijn-Erdos dans le cadre des hypergraphes et des espaces métriques.

Network properties of folksonomies

Social resource sharing systems like YouTube and del.icio.us have acquired a large number of users within the last few years. They provide rich resources for data analysis, information retrieval, and knowledge discovery applications. A first step towards this end is to gain better insights into content and structure of these systems. In this paper, we will analyse the main network characteristics of two of the systems. We consider their underlying data structures – socalled folksonomies – as tri-partite hypergraphs, and adapt classical network measures like characteristic path length and clustering coefficient to them. Subsequently, we introduce a network of tag co-occurrence and investigate some of its statistical properties, focusing on correlations in node connectivity and pointing out features that reflect emergent semantics within the folksonomy. We show that simple statistical indicators unambiguously spot non-social behavior such as spam.