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

Abstract

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.

The HyperBagGraph DataEdron: An Enriched Browsing Experience of Datasets

“…We propose in this article another way to explore an information space by using hyper-bag-graphs (hb-graphs for short), an extension of hypergraphs to families of multisets. In [2], we show that hb-graphs enhance an exchange-based diffusion over co-occurence networks, providing a fine vertex and hbedge ranking, accounting for a hb-edge based weighting of vertices.…”

Exchange-based diffusion in Hb-Graphs

“…For the mathematical background, we give a minimal mathematical formalization. However, the interested reader can refer to [28] which contains all the necessary mathematics and to [26,27] for a full introduction on hb-graphs.…”

“…1. A more general figure of the principle of this iterative process is given in [24,28]. The iterative process conserves the overall value held by the vertices and the hb-edges, meaning that we have at…”

“…We want to address the following research question: "Can we find a network model and a diffusion process that not only rank vertices but also rank hb-edges in hb-graphs?". In [24], we have developed an iterative exchange approach in hb-graphs with two-phase steps that allows to extract information not only at the vertex level but also at the hb-edge level. In this article, which is an extended version of [24], we not only present the contributions of [24]-that included the introduction of the exchange-based diffusion process as a means to rank both vertices and hb-edges, and were formalizing the exchanges by using hb-graphs, and presenting a novel visualisation of co-occurrences network-, but also add new contributions.…”

“…In [24], we have validated our approach by using lab-generated hb-graphs. We continue here to use this approach, as mimicking real datasets by randomly generated ones is not only a warranty for reproducibility, but also for robustness of the results obtained.…”