Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020)

International audienceOriginating in arithmetics and logic, the theory of ordered sets is now a field of combinatorics that is intimately linked to graph theory, universal algebra and multiple-valued logic, and that has a wide range of classical applications such as formal calculus, classification, decision aid and social choice.This international conference “Algebras, graphs and ordered set” (ALGOS) brings together specialists in the theory of graphs, relational structures and ordered sets, topics that are omnipresent in artificial intelligence and in knowledge discovery, and with concrete applications in biomedical sciences, security, social networks and e-learning systems. One of the goals of this event is to provide a common ground for mathematicians and computer scientists to meet, to present their latest results, and to discuss original applications in related scientific fields. On this basis, we hope for fruitful exchanges that can motivate multidisciplinary projects.The first edition of ALgebras, Graphs and Ordered Sets (ALGOS 2020) has a particular motivation, namely, an opportunity to honour Maurice Pouzet on his 75th birthday! For this reason, we have particularly welcomed submissions in areas related to Maurice’s many scientific interests:• Lattices and ordered sets• Combinatorics and graph theory• Set theory and theory of relations• Universal algebra and multiple valued logic• Applications: formal calculus, knowledge discovery, biomedical sciences, decision aid and social choice, security, social networks, web semantics..

Properties of graphs with large girth

This thesis is devoted to the analysis of a class of iterative probabilistic algorithms in regular graphs, called locally greedy algorithms, which will provide bounds for graph functions in regular graphs with large girth. This class is useful because, by conveniently setting the parameters associated with it, we may derive algorithms for some well-known graph problems, such as algorithms to find a large independent set, a large induced forest, or even a small dominating set in an input graph G. The name ``locally greedy" comes from the fact that, in an algorithm of this class, the probability associated with the random selection of a vertex v is determined by the current state of the vertices within some fixed distance of v. Given r > 2 and an r-regular graph G, we determine the expected performance of a locally greedy algorithm in G, depending on the girth g of the input and on the degree r of its vertices. When the girth of the graph is sufficiently large, this analysis leads to new lower bounds on the independence number of G and on the maximum number of vertices in an induced forest in G, which, in both cases, improve the bounds previously known. It also implies bounds on the same functions in graphs with large girth and maximum degree r and in random regular graphs. As a matter of fact, the asymptotic lower bounds on the cardinality of a maximum induced forest in a random regular graph improve earlier bounds, while, for independent sets, our bounds coincide with asymptotic lower bounds first obtained by Wormald. Our result provides an alternative proof of these bounds which avoids sharp concentration arguments. The main contribution of this work lies in the method presented rather than in these particular new bounds. This method allows us, in some sense, to directly analyse prioritised algorithms in regular graphs, so that the class of locally greedy algorithms, or slight modifications thereof, may be applied to a wider range of problems in regular graphs with large girth

Some problems in combinatorial topology of flag complexes

In this work we study simplicial complexes associated to graphs and their homotopical and combinatorial properties. The main focus is on the family of flag complexes, which can be viewed as independence complexes and clique complexes of graphs. In the first part we study independence complexes of graphs using two cofibre sequences corresponding to vertex and edge removals. We give applications to the connectivity of independence complexes of chordal graphs and to extremal problems in topology and we answer open questions about the homotopy types of those spaces for particular families of graphs. We also study the independence complex as a space of configurations of particles in the so-called hard-core models on various lattices. We define, and investigate from an algorithmic perspective, a special family of combinatorially defined homology classes in independence complexes. This enables us to give algorithms as well as NP-hardness results for topological properties of some spaces. As a corollary we prove hardness of computing homology of simplicial complexes in general. We also view flag complexes as clique complexes of graphs. That leads to the study of various properties of Vietoris-Rips complexes of graphs. The last result is inspired by a problem in face enumeration. Using methods of extremal graph theory we classify flag triangulations of 3-manifolds with many edges. As a corollary we complete the classification of face vectors of flag simplicial homology 3-spheres

Une approche physique-statistique à diérents problèmes dans la théorie des réseaux

Statistical physics, originally developed to describe thermodynamic systems, has been playing for the last decades a central role in modelling an incredibly large and heterogeneous set of different phenomena taking forinstance place on social, economical or biological systems. Such a vast field of possible applications has been found also for networks, as a huge variety of systems can be described in terms of interconnected elements. After an introductory part introducing these themes as well as the role of abstract modelling in science, in this dissertation it will be discussed how a statistical physics approach can lead to new insights as regards three problems of interest in network theory: how some quantity can be optimally spread on a graph, how to explore it and how to reconstruct it frompartial information. Some final remarks on the importance such themes will likely preserve in the coming years conclude the work.La physique statistique, développée à l'origine pour décrire les systèmes thermodynamiques, a joué pendant les dernières décennies un rôle central dans la modélisation d'un ensemble incroyablement vaste et hétérogène de différents phénomènes qui ont lieu par exemple dans des systèmes sociaux, économiques ou biologiques.Un champ d'applications possibles aussi vaste a été trouvé aussi pour les réseaux, comme une grande variété de systèmes peut être décrite en termes d'éléments interconnectés. Après une partie introductive sur les thèmes abordés ainsi que sur le rôle de la modélisation abstraite dans la science, dans ce manuscrit seront décrites les nouvelles perspectives auxquelles on peut arriver en approchant d'une façon physico-statistique trois problèmes d'intérêt dans la théorie des réseaux: comment une certaine quantité peut se répandre de façon optimale sur un graphique, comment explorer un réseau et comment le reconstruire à partir d'un jeu d'informations partielles. Quelques remarques finales sur l'importance que ces thèmes préserveront dans les années à venir conclut le travail

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

Optimisation and information-theoretic principles in multiplex networks.

PhD ThesesThe multiplex network paradigm has proven very helpful in the study of many real-world complex systems, by allowing to retain full information about all the different possible kinds of relationships among the elements of a system. As a result, new non-trivial structural patterns have been found in diverse multi-dimensional networked systems, from transportation networks to the human brain. However, the analysis of multiplex structural and dynamical properties often requires more sophisticated algorithms and takes longer time to run compared to traditional single network methods. As a consequence, relying on a multiplex formulation should be the outcome of a trade-off between the level of information and the resources required to store it. In the first part of the thesis, we address the problem of quantifying and comparing the amount of information contained in multiplex networks. We propose an algorithmic informationtheoretic approach to evaluate the complexity of multiplex networks, by assessing to which extent a given multiplex representation of a system is more informative than a single-layer graph. Then, we demonstrate that the same measure is able to detect redundancy in a multiplex network and to obtain meaningful lower-dimensional representations of a system. We finally show that such method allows us to retain most of the structural complexity of the original system as well as the salient characteristics determining the behaviour of dynamical processes happening on it. In the second part of the thesis, we shift the focus to the modelling and analysis of some structural features of real-world multiplex systems throughout optimisation principles. We demonstrate that Pareto optimal principles provide remarkable tools not only to model real-world multiplex transportation systems but also to characterise the robustness of multiplex systems against targeted attacks in the context of optimal percolation