arXiv subjects

Xavier Ouvrard

Publications and source records attributed to Xavier Ouvrard.

10 recordsLinked to original sources

arXiv2026

Probabilities

Probabilities is the English translation of the book Probabilités Tome 1 and Tome 2. The mathematic content is authored by Prof. Jean-Yves Ouvrard. The English version has been done by his eldest son Dr. Xavier Ouvrard. This probability theory book covers not only an introduction to this field, but also advanced concepts based on measure theory. The first part introduces the fundamentals of probability theory across 7 chapters, targeting bachelor level, including event algebras, random variables, independence, conditional probabilities, moments of discrete and continuous random variables, generating functions, and limit theorems. The second part contains 10 chapters and corresponds to master level. Following a brief introduction to measure theory, this part develops more advanced topics: probability measures and their complements, distributions and moments of random variables, modes of convergence, laws of large numbers, conditional expectation, Fourier transforms and characteristic functions, Gaussian random variables, convergence of measures, convergence in distribution, discrete-time stochastic processes, martingales, and Markov chains. The reader's work is greatly facilitated by the inclusion, in every chapter, of numerous exercises, all accompanied by detailed solutions that often provide substantial extensions to the theoretical material.

math.HO↗

arXiv2020

Hypergraphs: an introduction and review

Hypergraphs were introduced in 1973 by Bergé. This review aims at giving some hints on the main results that we can find in the literature, both on the mathematical side and on their practical usage. Particularly, different definitions of hypergraphs are compared, some unpublished work on the visualisation of large hypergraphs done by the author. This review does not pretend to be exhaustive.

cs.DM↗

arXiv2020

Tuning Ranking in Co-occurrence Networks with General Biased Exchange-based Diffusion on Hyper-bag-graphs

Co-occurence networks can be adequately modeled by hyper-bag-graphs (hb-graphs for short). A hb-graph is a family of multisets having same universe, called the vertex set. An efficient exchange-based diffusion scheme has been previously proposed that allows the ranking of both vertices and hb-edges. In this article, we extend this scheme to allow biases of different kinds and explore their effect on the different rankings obtained. The biases enhance the emphasize on some particular aspects of the network.

cs.SI↗

arXiv2019

Exchange-Based Diffusion in Hb-Graphs: Highlighting Complex Relationships

Most networks tend to show complex and multiple relationships between entities. Networks are usually modeled by graphs or hypergraphs; nonetheless a given entity can occur many times in a relationship: this brings the need to deal with multisets instead of sets or simple edges. Diffusion processes are useful to highlight interesting parts of a network: they usually start with a stroke at one vertex and diffuse throughout the network to reach a uniform distribution. Several iterations of the process are required prior to reaching a stable solution. We propose an alternative solution to highlighting the main components of a network using a diffusion process based on exchanges: it is an iterative two-phase step exchange process. This process allows to evaluate the importance not only of the vertices but also of the regrouping level. To model the diffusion process, we extend the concept of hypergraphs that are families of sets to families of multisets, that we call hb-graphs. This version is an extended version of arXiv:1809.00190v1: the overlaps with the v1 are in black, the new content is in blue. The contributions of this extended version are: the proofs of conservation and convergence of the extracted sequences of the diffusion process, as well as the illustration of the speed of convergence and comparison to classical and modified random walks; the algorithms of the exchange-based diffusion and the modified random walk; the application to a use case based on Arxiv publications. All the figures except one have been either modified or added in this extended version to take into account the new developments.

cs.DS↗

arXiv2019

The HyperBagGraph DataEdron: An Enriched Browsing Experience of Multimedia Datasets

Traditional verbatim browsers give back information in a linear way according to a ranking performed by a search engine that may not be optimal for the surfer. The latter may need to assess the pertinence of the information retrieved, particularly when s$\cdot$he wants to explore other facets of a multi-facetted information space. For instance, in a multimedia dataset different facets such as keywords, authors, publication category, organisations and figures can be of interest. The facet simultaneous visualisation can help to gain insights on the information retrieved and call for further searches. Facets are co-occurence networks, modeled by HyperBag-Graphs -- families of multisets -- and are in fact linked not only to the publication itself, but to any chosen reference. These references allow to navigate inside the dataset and perform visual queries. We explore here the case of scientific publications based on Arxiv searches.

cs.SI↗

arXiv2018

Adjacency and Tensor Representation in General Hypergraphs.Part 2: Multisets, Hb-graphs and Related e-adjacency Tensors

HyperBagGraphs (hb-graphs as short) extend hypergraphs by allowing the hyperedges to be multisets. Multisets are composed of elements that have a multiplicity. When this multiplicity has positive integer values, it corresponds to non ordered lists of potentially duplicated elements. We define hb-graphs as family of multisets over a vertex set; natural hb-graphs correspond to hb-graphs that have multiplicity function with positive integer values. Extending the definition of e-adjacency to natural hb-graphs, we define different way of building an e-adjacency tensor, that we compare before having a final choice of the tensor. This hb-graph e-adjacency tensor is used with hypergraphs.

cs.DM↗

arXiv2018

On Adjacency and e-Adjacency in General Hypergraphs: Towards a New e-Adjacency Tensor

In graphs, the concept of adjacency is clearly defined: it is a pairwise relationship between vertices. Adjacency in hypergraphs has to integrate hyperedge multi-adicity: the concept of adjacency needs to be defined properly by introducing two new concepts: $k$-adjacency - $k$ vertices are in the same hyperedge - and e-adjacency - vertices of a given hyperedge are e-adjacent. In order to build a new e-adjacency tensor that is interpretable in terms of hypergraph uniformisation, we designed two processes: the first is a hypergraph uniformisation process (HUP) and the second is a polynomial homogeneisation process (PHP). The PHP allows the construction of the e-adjacency tensor while the HUP ensures that the PHP keeps interpretability. This tensor is symmetric and can be fully described by the number of hyperedges; its order is the range of the hypergraph, while extra dimensions allow to capture additional hypergraph structural information including the maximum level of $k$-adjacency of each hyperedge. Some results on spectral analysis are discussed.

math.CO↗

arXiv2018

Hypergraph Modeling and Visualisation of Complex Co-occurence Networks

Finding inherent or processed links within a dataset allows to discover potential knowledge. The main contribution of this article is to define a global framework that enables optimal knowledge discovery by visually rendering co-occurences (i.e. groups of linked data instances attached to a metadata reference) - either inherently present or processed - from a dataset as facets. Hypergraphs are well suited for modeling co-occurences since they support multi-adicity whereas graphs only support pairwise relationships. This article introduces an efficient navigation between different facets of an information space based on hypergraph modelisation and visualisation.

cs.SI↗

arXiv2018

Adjacency and Tensor Representation in General Hypergraphs Part 1: e-adjacency Tensor Uniformisation Using Homogeneous Polynomials

Adjacency between two vertices in graphs or hypergraphs is a pairwise relationship. It is redefined in this article as 2-adjacency. In general hypergraphs, hyperedges hold for $n$-adic relationship. To keep the $n$-adic relationship the concepts of $k$-adjacency and e-adjacency are defined. In graphs 2-adjacency and e-adjacency concepts match, just as $k$-adjacency and e-adjacency do for $k$-uniform hypergraphs. For general hypergraphs these concepts are different. This paper also contributes in a uniformization process of a general hypergraph to allow the definition of an e-adjacency tensor, viewed as a hypermatrix, reflecting the general hypergraph structure. This symmetric e-adjacency hypermatrix allows to capture not only the degree of the vertices and the cardinality of the hyperedges but also makes a full separation of the different layers of a hypergraph.

cs.DM↗

arXiv2017

Networks of Collaborations: Hypergraph Modeling and Visualisation

The acknowledged model for networks of collaborations is the hypergraph model. Nonetheless when it comes to be visualized hypergraphs are transformed into simple graphs. Very often, the transformation is made by clique expansion of the hyperedges resulting in a loss of information for the user and in artificially more complex graphs due to the high number of edges represented. The extra-node representation gives substantial improvement in the visualisation of hypergraphs and in the retrieval of information. This paper aims at showing qualitatively and quantitatively how the extra-node representation can improve the visualisation of hypergraphs without loss of information.

cs.SI↗