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.Keywords exchange · diffusion · multiset · hyperbag-graph · information retrieval · ranking This article is an extended version of [1] (pre-printed in arXiv:1809.00190v1): the text of the extended version is in blue, the text in black is the one of [1]. All the figures except Figure 2 have been either modified or added in this extended version to take into account the new developments. 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.

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. 7 Σ ρ R is the quotient set of Σ ρ by R 8 In a multiset repetitions of elements are allowed. For further details [10].

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-adjacencyk 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.