On the complexity of overlaying a hypergraph with a graph with bounded maximum degree

Let G and H be respectively a graph and a hypergraph defined on a same set of vertices, and let F be a graph. We say that G F-overlays a hyperedge S of H if the subgraph of G induced by S contains F as a spanning subgraph, and that G F-overlays H if it F-overlays every hyperedge of H. For a fixed graph F and a fixed integer k, the problem (∆ ≤ k)-F-Overlay consists in deciding whether there exists a graph with maximum degree at most k that F-overlays a given hypergraph H. In this paper, we prove that for any graph F which is neither complete nor anticomplete, there exists an integer np(F) such that (∆ ≤ k)-F-Overlay is NP-complete for all k ≥ np(F)

Problèmes de graphes motivés par des modèles basse et haute résolution de grands assemblages de protéines

To explain the biological function of a molecular assembly (MA), one has to know its structural description. It may be ascribed to two levels of resolution: low resolution (i.e. molecular interactions) and high resolution (i.e. relative position and orientation of each molecular subunit, called conformation). Our thesis aims to address the two problems from graph aspects.The first part focuses on low resolution problem. Assume that the composition (complexes) of a MA is known, we want to determine all interactions ofsubunits in the MA which satisfies some property. It can be modeled as a graph problem by representing a subunit as a vertex, then a subunit interaction is an edge, and a complex is an induced subgraph. In our work, we use the fact that a subunit has a bounded number of interactions. It leads to overlaying graph with bounded maximum degree. For a graph family F and a fixed integer k, given a hypergraph H = (V (H), E(H)) (whose edges are subsets of vertices) and an integer s, M AX (∆ ≤ k)-F -O VERLAY consists in deciding whether there exists a graph with degree at most k such that there are at least s hyperedges in which the subgraph induced by each hyperedge (complex) contains an element of F. When s = |E(H)|, it is called (∆ ≤ k)-F -O VERLAY . We present complexity dichotomy results (P vs. NP-complete) for MAX (∆ ≤ k)-F-OVERLAY and (∆ ≤ k)-F-OVERLAY depending on pairs (F, k).The second part presents our works motivated by high resolution problem. Assume that we are given a graph representing the interactions of subunits, a finite set of conformations for each subunit and a weight function assessing the quality of the contact between two subunits positioned in the assembly. Discrete Optimization of Multiple INteracting Objects (D OMINO ) aims to find conformations for the subunits maximizing a global utility function. We propose a new approach based on this problem in which the weight function is relaxed, CONFLICT COLORING . We present studies from both theoretical and experimental points of view. Regarding the theory, we provide a complexity dichotomy result and also algorithmic methods (approximation and fixed paramater tracktability). Regarding the experiments, we build instances of CONFLICT COLORING associated with Voronoi diagrams in the plane. The obtained statistics provide information on the dependencies of the existences of a solution, to parameters used in ourexperimental setup.Pour comprendre les fonctions biologiques d’un assemblage moléculaire (AM), il est utile d’en avoir une représentation structurale. Celle-ci peut avoir deux niveaux de résolution : basse résolution (i.e. interactions moléculaires) et haute résolution (i.e. position relative et orientation de chaque sous-unité, appelée conformation). Cette thèse s’intéresse à trouver de telles représentations à l’aide de graphes.Dans la première partie, nous cherchons des représentations basse résolution. Etant donné la composition des complexes d’un AM, notre but est de déterminer les interactions entre ses différentes sous-unités. Nous modélisons l’AM à l’aide d’un graphe : les sous-unités sont les sommets, les interactions entre elles sont les arêtes et un complexe est un sous-graphe induit. Utilisant le fait qu’une sous-unité n’a qu’un nombre limité d’interactions, nous arrivons au problème suivant. Pour un graphe F et un entier k fixés, étant donné un hypergraphe H et un entier s, MAX (∆ ≤ k)-F-OVERLAY consiste à décider s’il existe un graphe de degré au plus k tel qu’au moins s hyperarêtes de H induisent un sous-graphe contenant F (en tant que sous-graphe). La restriction au cas s = |E(H)| est appelée (∆ ≤ k)-F-OVERLAY . Nous donnons une dichotomie de complexité (P vs. NP-complet) pour MAX (∆ ≤ k)-F-OVERLAY et (∆ ≤ k)-F-OVERLAY en fonction du couple (F, k).Dans la seconde partie, nous nous attaquons à la haute résolution. Nous sont donnés un graphe représentant les interactions entre sous-unités, un ensemble de conformations possibles pour chaque sous-unité et une fonction de poids représentant la qualité de contact entre les conformations de deux sous-unités interagissant dans l’assemblage. Le problème Discrete Optimization of Multiple INteracting Objects (D OMINO ) consiste alors à trouver les conformations pour les sous-unités qui maximise une fonction d’utilité globale. Nous proposons une nouvelle approche à ce problème en relâchant la fonction de poids, ce qui mène au problème de graphe CONFLICT COLORING . Nous donnons tout d’abord des résultats de complexité et des algorithmes (d’approximation et à paramètre fixé). Nous menons ensuite des expérimentations sur des instances de CONFLICT COLORING associées à des diagrammes de Voronoi dans le plan. Les statistiques obtenues nous informent sur comment les parmètres de notre montage expérimental influe sur l’existence d’une solution

Opinion disparity in hypergraphs with community structure

The division of a social group into subgroups with opposing opinions, which we refer to as opinion disparity, is a prevalent phenomenon in society. This phenomenon has been modeled by including mechanisms such as opinion homophily, bounded confidence interactions, and social reinforcement mechanisms. In this paper we study a complementary mechanism for the formation of opinion disparity based on higher-order interactions, i.e., simultaneous interactions between multiple agents. We present an extension of the planted partition model for uniform hypergraphs as a simple model of community structure and consider the hypergraph SIS model on a hypergraph with two communities where the binary ideology can spread via links (pairwise interactions) and triangles (three-way interactions). We approximate this contagion process with a mean-field model and find that for strong enough community structure, the two communities can hold very different average opinions. We determine the regimes of structural and infectious parameters for which this opinion disparity can exist and find that the existence of these disparities is much more sensitive to the triangle community structure than to the link community structure. We show that the existence and type of opinion disparities are extremely sensitive to differences in the sizes of the two communities.Comment: 14 pages, 8 figure

Recent developments in graph Ramsey theory

Given a graph H, the Ramsey number r(H) is the smallest natural number N such that any two-colouring of the edges of K_N contains a monochromatic copy of H. The existence of these numbers has been known since 1930 but their quantitative behaviour is still not well understood. Even so, there has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics. In this survey, we will describe some of this progress

The Power Of Locality In Network Algorithms

Over the last decade we have witnessed the rapid proliferation of large-scale complex networks, spanning many social, information and technological domains. While many of the tasks which users of such networks face are essentially global and involve the network as a whole, the size of these networks is huge and the information available to users is only local. In this dissertation we show that even when faced with stringent locality constraints, one can still effectively solve prominent algorithmic problems on such networks. In the first part of the dissertation we present a natural algorithmic framework designed to model the behaviour of an external agent trying to solve a network optimization problem with limited access to the network data. Our study focuses on local information algorithms --- sequential algorithms where the network topology is initially unknown and is revealed only within a local neighborhood of vertices that have been irrevocably added to the output set. We address both network coverage problems as well as network search problems. Our results include local information algorithms for coverage problems whose performance closely match the best possible even when information about network structure is unrestricted. We also demonstrate a sharp threshold on the level of visibility required: at a certain visibility level it is possible to design algorithms that nearly match the best approximation possible even with full access to the network structure, but with any less information it is impossible to achieve a reasonable approximation. For preferential attachment networks, we obtain polylogarithmic approximations to the problem of finding the smallest subgraph that connects a subset of nodes and the problem of finding the highest-degree nodes. This is achieved by addressing a decade-old open question of Bollobás and Riordan on locally finding the root in a preferential attachment process. In the second part of the dissertation we focus on designing highly time efficient local algorithms for central mining problems on complex networks that have been in the focus of the research community over a decade: finding a small set of influential nodes in the network, and fast ranking of nodes. Among our results is an essentially runtime-optimal local algorithm for the influence maximization problem in the standard independent cascades model of information diffusion and an essentially runtime-optimal local algorithm for the problem of returning all nodes with PageRank bigger than a given threshold. Our work demonstrates that locality is powerful enough to allow efficient solutions to many central algorithmic problems on complex networks

Fault-tolerant gates on hypergraph product codes

L’un des défis les plus passionnants auquel nous sommes confrontés aujourd’hui est la perspective de la construction d’un ordinateur quantique de grande échelle. L’information quantique est fragile et les implémentations de circuits quantiques sont imparfaites et su- jettes aux erreurs. Pour réaliser un tel ordinateur, nous devons construire des circuits quan- tiques tolérants aux fautes capables d’opérer dans le monde réel. Comme il sera expliqué plus loin, les circuits quantiques tolérant aux fautes nécessitent plus de ressources que leurs équivalents idéaux, sans bruit. De manière générale, le but de mes recherches est de minimiser les ressources nécessaires à la construction d’un circuit quantique fiable. Les codes de correction d’erreur quantiques protègent l’information des erreurs en l’encodant de manière redondante dans plusieurs qubits. Bien que la redondance requière un plus grand nombre de qubits, ces qubits supplé- mentaires jouent un rôle de protection: cette redondance sert de garantie. Si certains qubits sont endommagés en raison d’un circuit défectueux, nous pourrons toujours récupérer l’informations. Préparer et maintenir des qubits pendant des durées suffisamment longues pour effectuer un calcul s’est révélé être une tâche expérimentale difficile. Il existe un écart important entre le nombre de qubits que nous pouvons contrôler en laboratoire et le nombre requis pour implementer des algorithmes dans lesquels les ordinateurs quantiques ont le dessus sur ceux classiques. Par conséquent, si nous voulons contourner ce problème et réaliser des circuits quantiques à tolérance aux fautes, nous devons rendre nos constructions aussi efficaces que possible. Nous devons minimiser le surcoût, défini comme le nombre de qubits physiques nécessaires pour construire un qubit logique. Dans un article important, Gottesman a montré que, si certains types de codes de correction d’erreur quantique existaient, cela pourrait alors conduire à la construction de circuits quantiques tolérants aux fautes avec un surcoût favorable. Ces codes sont appelés codes éparses. La proposition de Gottesman décrivait des techniques pour exécuter des opérations logiques sur des codes éparses quantiques arbitraires. Cette proposition était limitée à certains égards, car elle ne permettait d’exécuter qu’un nombre constant de portes logiques par unité de temps. Dans cette thèse, nous travaillons avec une classe spécifique de codes éparses quantiques appelés codes de produits d’hypergraphes. Nous montrons comment effectuer des opérations sur ces codes en utilisant une technique appelée déformation du code. Notre technique généralise les codages basés sur les défauts topologiques dans les codes de surface aux codes de produits d’hypergraphes. Nous généralisons la notion de perforation et montrons qu’elle peut être exprimée naturellement dans les codes de produits d’hypergraphes. Comme cela sera expliqué en détail, les défauts de perforation ont eux-mêmes une portée limitée. Pour réaliser une classe de portes plus large, nous intro- duisons un nouveau défaut appelé trou de ver basé sur les perforations. À titre d’exemple, nous illustrons le fonctionnement de ce défaut dans le contexte du code de surface. Ce défaut a quelques caractéristiques clés. Premièrement, il préserve la propriété éparses du code au cours de la déformation, contrairement à une approche naïve qui ne garantie pas cette propriété. Deuxièmement, il généralise de manière simple les codes de produits d’hypergraphes. Il s’agit du premier cadre suffisamment riche pour décrire les portes tolérantes aux fautes de cette classe de codes. Enfin, nous contournons une limitation de l’approche de Gottesman qui ne permettait d’effectuer qu’un certain nombre de portes logiques à un moment donné. Notre proposition permet d’opérer sur tous les qubits encodés à tout moment.One of the most exciting challenges that faces us today is the prospect of building a scalable quantum computer. Implementations of quantum circuits are imperfect and prone to error. In order to realize a scalable quantum computer, we need to construct fault-tolerant quantum circuits capable of working in the real world. As will be explained further below, fault-tolerant quantum circuits require more resources than their ideal, noise-free counterparts. Broadly, the aim of my research is to minimize the resources required to construct a reliable quantum circuit. Quantum error correcting codes protect information from errors by encoding our information redundantly into qubits. Although the number of qubits that we require increases, this redundancy serves as a buffer – in the event that some qubits are damaged because of a faulty circuit, we will still be able to recover our information. Preparing and maintaining qubits for durations long enough to perform a computation has proved to be a challenging experimental task. There is a large gap between the number of qubits we can control in the lab and the number required to implement algorithms where quantum computers have the upper hand over classical ones. Therefore, if we want to circumvent this bottleneck, we need to make fault-tolerant quantum circuits as efficient as possible. To be precise, we need to minimize the overhead, defined as the number of physical qubits required to construct a logical qubit. In an important paper, Gottesman showed that if certain kinds of quantum error correcting codes were to exist, then this could lead to constructions of fault-tolerant quantum circuits with favorable overhead. These codes are called quantum Low-Density Parity-Check (LDPC) codes. Gottesman’s proposal described techniques to perform gates on generic quantum LDPC codes. This proposal limited the number of logical gates that could be performed at any given time. In this thesis, we work with a specific class of quantum LDPC codes called hypergraph product codes. We demonstrate how to perform gates on these codes using a technique called code deformation. Our technique generalizes defect-based encodings in the surface code to hypergraph product codes. We generalize puncture defects and show that they can be expressed naturally in hypergraph product codes. As will be explained in detail, puncture defects are themselves limited in scope; they only permit a limited set of gates. To perform a larger class of gates, we introduce a novel defect called a wormhole that is based on punctures. As an example, we illustrate how this defect works in the context of the surface code. This defect has a few key features. First, it preserves the LDPC property of the code over the course of code deformation. At the outset, this property was not guaranteed. Second, it generalizes in a straightforward way to hypergraph product codes. This is the first framework that is rich enough to describe fault-tolerant gates on this class of codes. Finally, we circumvent a limitation in Gottesman’s approach which only allowed a limited number of logical gates at any given time. Our proposal allows to access the entire code block at any given time

Multilayer Networks

In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time, and include other types of complications. Such systems include multiple subsystems and layers of connectivity, and it is important to take such "multilayer" features into account to try to improve our understanding of complex systems. Consequently, it is necessary to generalize "traditional" network theory by developing (and validating) a framework and associated tools to study multilayer systems in a comprehensive fashion. The origins of such efforts date back several decades and arose in multiple disciplines, and now the study of multilayer networks has become one of the most important directions in network science. In this paper, we discuss the history of multilayer networks (and related concepts) and review the exploding body of work on such networks. To unify the disparate terminology in the large body of recent work, we discuss a general framework for multilayer networks, construct a dictionary of terminology to relate the numerous existing concepts to each other, and provide a thorough discussion that compares, contrasts, and translates between related notions such as multilayer networks, multiplex networks, interdependent networks, networks of networks, and many others. We also survey and discuss existing data sets that can be represented as multilayer networks. We review attempts to generalize single-layer-network diagnostics to multilayer networks. We also discuss the rapidly expanding research on multilayer-network models and notions like community structure, connected components, tensor decompositions, and various types of dynamical processes on multilayer networks. We conclude with a summary and an outlook.Comment: Working paper; 59 pages, 8 figure