Large graphs are natural mathematical models for describing the structure of the data in a wide variety of fields, such as web mining, social networks, information retrieval, biological networks, etc. For all these applications, automatic tools are required to get a synthetic view of the graph and to reach a good understanding of the underlying problem. In particular, discovering groups of tightly connected vertices and understanding the relations between those groups is very important in practice. This paper shows how a kernel version of the batch Self Organizing Map can be used to achieve these goals via kernels derived from the Laplacian matrix of the graph, especially when it is used in conjunction with more classical methods based on the spectral analysis of the graph. The proposed method is used to explore the structure of a medieval social network modeled through a weighted graph that has been directly built from a large corpus of agrarian contracts.

An even (resp. odd) lollipop is the coalescence of a cycle of even (resp. odd) length and a path with pendant vertex as distinguished vertex. It is known that the odd lollipop is determined by its spectrum and the question is asked by W. Haemers, X. Liu and Y. Zhang for the even lollipop. A private communication of Behruz Tayfeh-Rezaie pointed out that an even lollipop with a cycle of length at least $6$ is determined by its spectrum but the result for lollipops with a cycle of length $4$ is still unknown. We give an unified proof for lollipops with a cycle of length not equal to $4$, generalize it for lollipops with a cycle of length $4$ and therefore answer the question. Our proof is essentially based on a method of counting closed walks.

A flag complex can be defined as a simplicial complex whose simplices correspond to complete subgraphs of its 1-skeleton taken as a graph. In this article, by introducing the notion of s-dismantlability, we shall define the s-homotopy type of a graph and show in particular that two finite graphs have the same s-homotopy type if, and only if, the two flag complexes determined by these graphs have the same simplicial simple-homotopy type (Theorem 2.10, part 1). This result is closely related to similar results established by Barmak and Minian ([2]) in the framework of posets and we give the relation between the two approaches (theorems 3.5 and 3.7). We conclude with a question about the relation between the s-homotopy and the graph homotopy defined in [5].

The different approaches developed to analyse the structure of complex networks have generated a large number of studies. In the field of social networks at least, studies mainly address the detection and analysis of communities. In this article, we challenge these approaches and focus on nodes that have meaningful local interactions able to identify the internal organization of communities or the way communities are assembled. We propose an algorithm, ItRich, to identify this type of nodes, based on the decomposition of a graph into successive, less and less dense, layers. Our method is tested on synthetic and real data sets and meshes well with other methods such as community detection or $k$-core decomposition.

Some structural characteristics of online discussions have been successfully modeled in the recent years. When parameters of these models are properly estimated, the models are able to generate synthetic discussions that are structurally similar to the real discussions. A common aspect of these models is that they consider that all users behave according to the same model. In this paper, we combine a growth-model with an Expectation-Maximization algorithm that finds different parameters for different latent groups of users. We use this method to find the different roles that coexist in the community. Moreover, we analyze whether we can predict users behaviors based on their roles. Indeed, we show that predictions are improved for some of the roles when compared with a simple growth model.

RésuméUn intervalle X d'un tournoi T est un ensemble de sommets de T tel que tout sommet extérieurà X domine ou est dominé par tous les sommets de X. Nous caractérisons les tournois dont tous les intervalles acycliques non vides sont des singletons et qui sont critiques pour cette propriété, c'est-à-dire que la suppression d'un sommet quelconque du tournoi donne naissanceà au moins un intervalle acyclique de plus de 2 sommets. Ces tournois sont exactement ceux construits comme la composition d'un tournoi quelconque avec des tournois circulants. Ce travail sur les intervalles acycliques aété motivé par la recherche de structures ordonnées dans des tournois pour lesquels aucun ordre médian ne s'impose naturellement. Pour citer cet article : J.F. Culus, B. Jouve, C. R. Acad. Sci. Paris, Ser. I 340 (2005). AbstractTournaments without acyclic interval. An interval X of a tournament T is a vertex subset of T such that any vertex not in X either dominates or is dominated by all of the vertices in X. We caracterize the tournaments such that the only non empty acyclic intervals are the singletons and which are critical for that property, that is whenever a vertex is removed at least one acyclic interval with more than 2 vertices is created. These tournaments are exactly those which are the composition of any tournament with circulant tournaments. That work on acyclic intervals was motivated by the study of tournaments for which no median order forced itself naturally.

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International audienceMedieval and modern tax documents (fieldbooks, “compoix”, cadasters...) give a rich spatial information. Whole territories are described plot per plot at different succesive periods. Nevertheless, historians don’t know how to relate different states of spatial information on the long term. Moreover, there are no plot plans before the 17th and 18th century. We want to overcome this methodological lock with the following proposition: modelling ancient plots described in tax documents using the topological proprieties of the graph theory. The translation of the spatial data into graphs should allow to set up a commun language between the historical documents, mapped and not mapped. The article talks about the fondamental method choices and the used process to transform the spatial information of ancient plans into graphs which will allow us not only to compare them together, but also to compare them with cadastral registers that have no plan. Finally, it is within a Geographic Information System that will be tested the pertinence of comparing possibilities while studying the plot changes according to the spatiotemporal operators of changement (creation, disappearance, stability, dilatation, contraction, fusion, fission, deformation).Les documents fiscaux médiévaux et modernes (terriers, compoix, cadastres…) offrent une information spatiale riche. Des territoires entiers sont décrits parcelle par parcelle à des époques successives. Les historiens ne parviennent toutefois jamais à corréler ces différents états de l’information spatiale sur la longue durée, d’autant qu’on ne dispose d’aucun plan parcellaire avant les xviie-xviiie siècles. C’est ce verrou méthodologique qu’il s’agit de dépasser en proposant, dans le cadre de l’ANR Modelespace, une modélisation des parcellaires anciens décrits dans les documents fiscaux en mettant en œuvre les propriétés topologiques de la théorie des graphes. La conversion des données spatiales en graphes doit permettre de mettre en place un langage commun entre les documents historiques. L’article traite des choix méthodologiques fondamentaux et de la démarche utilisée pour transformer l’information spatiale des plans anciens en graphes susceptibles d’être appariés non seulement entre eux, mais encore avec des graphes issus de matrices cadastrales sans plan. Au final, c’est au sein d’un Système d’Information Géographique que devra être testée la pertinence des possibilités de comparaisons en étudiant les recompositions parcellaires selon des opérateurs spatio-temporels de changement (création, disparition, stabilité, dilatation, contraction, fusion, fission, déformation)