5 research outputs found
On the editing distance of graphs
An edge-operation on a graph is defined to be either the deletion of an
existing edge or the addition of a nonexisting edge. Given a family of graphs
, the editing distance from to is the smallest
number of edge-operations needed to modify into a graph from .
In this paper, we fix a graph and consider , the set of
all graphs on vertices that have no induced copy of . We provide bounds
for the maximum over all -vertex graphs of the editing distance from
to , using an invariant we call the {\it binary chromatic
number} of the graph . We give asymptotically tight bounds for that distance
when is self-complementary and exact results for several small graphs
Absolutely avoidable order-size pairs in hypergraphs
For fixed integer , we call a pair of integers, ,
, if there is , such
that for any pair of integers with and there is an -uniform hypergraph on vertices and edges
that contains no induced sub-hypergraph on vertices and edges. Some
pairs are clearly not absolutely avoidable, for example is not
absolutely avoidable since any sufficiently sparse hypergraph on at least
vertices contains independent sets on vertices. Here we show that for any
and , either the pair or
the pair is absolutely avoidable. Next,
following the definition of Erd\H{o}s, F\"uredi, Rothschild and S\'os, we
define the of a pair as . We show that for most pairs satisfy , and that for , there
exists no pair of density 1
MĂ©trologie des graphes de terrain, application Ă la construction de ressources lexicales et Ă la recherche d'information
This thesis is organized in two parts : the first part focuses on measures of similarity (or proximity) between vertices of a graph, the second part on clustering methods for bipartite graph. A new measure of similarity between vertices, based on short time random walks, is introduced. The main advantage of the method is that it is insensitive to the density of the graph. A broad state of the art of similarities between vertices is then proposed, as well as experimental comparisons of these measures. This is followed by the proposal of a robust method for comparing graphs sharing the same set of vertices. This measure is shown to be applicable to the comparison and merging of synonymy networks. Finally an application for the enrichment of lexical resources is presented. It consists in providing candidate synonyms on the basis of already existing links. In the second part, a parallel between formal concept analysis and clustering of bipartite graph is established. This parallel leads to the particular case where a partition of one of the vertex groups can be determined whereas there is no corresponding partition on the other group of vertices. A simple method that addresses this problem is proposed and evaluated. Finally, a system of automatic classification of search results (Kodex) is presented. This system is an application of previously seen clustering methods. An evaluation on a collection of two million web pages shows the benefits of the approach and also helps to understand some differences between clustering methods.Cette thĂšse s'organise en deux parties : une premiĂšre partie s'intĂ©resse aux mesures de similaritĂ© (ou de proximitĂ©) dĂ©finies entre les sommets d'un graphe, une seconde aux mĂ©thodes de clustering de graphe biparti. Une nouvelle mesure de similaritĂ© entre sommets basĂ©e sur des marches alĂ©atoires en temps courts est introduite. Cette mĂ©thode a l'avantage, en particulier, d'ĂȘtre insensible Ă la densitĂ© du graphe. Il est ensuite proposĂ© un large Ă©tat de l'art des similaritĂ©s entre sommets, ainsi qu'une comparaison expĂ©rimentale de ces diffĂ©rentes mesures. Cette premiĂšre partie se poursuit par la proposition d'une mĂ©thode robuste de comparaison de graphes partageant le mĂȘme ensemble de sommets. Cette mĂ©thode est mise en application pour comparer et fusionner des graphes de synonymie. Enfin une application d'aide Ă la construction de ressources lexicales est prĂ©sentĂ©e. Elle consiste Ă proposer de nouvelles relations de synonymie Ă partir de l'ensemble des relations de synonymie dĂ©jĂ existantes. Dans une seconde partie, un parallĂšle entre l'analyse formelle de concepts et le clustering de graphe biparti est Ă©tabli. Ce parallĂšle conduit Ă l'Ă©tude d'un cas particulier pour lequel une partition d'un des groupes de sommets d'un graphe biparti peut-ĂȘtre dĂ©terminĂ©e alors qu'il n'existe pas de partitionnement correspondant sur l'autre type de sommets. Une mĂ©thode simple qui rĂ©pond Ă ce problĂšme est proposĂ©e et Ă©valuĂ©e. Enfin Kodex, un systĂšme de classification automatique des rĂ©sultats d'une recherche d'information est prĂ©sentĂ©. Ce systĂšme est une application en RI des mĂ©thodes de clustering vues prĂ©cĂ©demment. Une Ă©valuation sur une collection de deux millions de pages web montre les avantages de l'approche et permet en outre de mieux comprendre certaines diffĂ©rences entre mĂ©thodes de clustering