Problème de k-Séparateur

Let G be a vertex-weighted undirected graph. We aim to compute a minimum weight subset of vertices whose removal leads to a graph where the size of each connected component is less than or equal to a given positive number k. If k = 1 we get the classical vertex cover problem. Many formulations are proposed for the problem. The linear relaxations of these formulations are theoretically compared. A polyhedral study is proposed (valid inequalities, facets, separation algorithms). It is shown that the problem can be solved in polynomial time for many special cases including the path, the cycle and the tree cases and also for graphs not containing some special induced sub-graphs. Some (k + 1)-approximation algorithms are also exhibited. Most of the algorithms are implemented and compared. The k-separator problem has many applications. If vertex weights are equal to 1, the size of a minimum k-separator can be used to evaluate the robustness of a graph or a network. Another application consists in partitioning a graph/network into different sub-graphs with respect to different criteria. For example, in the context of social networks, many approaches are proposed to detect communities. By solving a minimum k-separator problem, we get different connected components that may represent communities. The k-separator vertices represent persons making connections between communities. The k-separator problem can then be seen as a special partitioning/clustering graph problemConsidérons un graphe G = (V,E,w) non orienté dont les sommets sont pondérés et un entier k. Le problème à étudier consiste à la construction des algorithmes afin de déterminer le nombre minimum de nœuds qu’il faut enlever au graphe G pour que toutes les composantes connexes restantes contiennent chacune au plus k-sommets. Ce problème nous l’appelons problème de k-Séparateur et on désigne par k-séparateur le sous-ensemble recherché. Il est une généralisation du Vertex Cover qui correspond au cas k = 1 (nombre minimum de sommets intersectant toutes les arêtes du graphe

Mathematical analysis of k-path Laplacian operators on simple graphs

A set of links and nodes are the fundamental units or components used to represent complex networks. Over the last few decades, network studies have expanded and matured, increasingly making use of complex mathematical tools. Complex networks play a significant role in the propagation of processes, which include for example the case of epidemic spreading, the diffusion process, synchronisation or the consensus process. Such dynamic processes are critically important in achieving understanding of the behaviour of complex systems at different levels of complexity - examples might be the brain and modern man-made infrastructures. Although part of the study of the diffusion of information in the dynamic processes, it is generally supposed that interactions in networks originate only from a node, spreading to its nearest neighbours, there also exist long-range interactions (LRI), which can be transmitted from a node to others not directly connected. The focus of this study is on dynamic processes on networks where nodes interact with not only their nearest neighbours but also through certain LRIs. The generalised k-path Laplacian operators (LOs) Lk, which account for the hop of a diffusive particle to its non-nearest neighbours in a graph, control this diffusive process, describing hops of nodes vi at distance k; here the distance is measured as the length of the shortest path between two nodes. In this way the introduction of the k-path LOs can facilitate conducting more precise studies of network dynamics in different applications. This thesis aims to study a generalised diffusion equation employing the transformed generalised k-path LOs for a locally finite infinite graph. This generalised diffusion equation promotes both normal and super diffusive processes on infinite graphs. Furthermore, this thesis develops a new theoretical mathematical framework for describing superdiffusion processes that use a transform of the k-path LOs defined on infinite graphs. The choice of the transform appeared to be vitally important as the probability of a long jump should be great enough. As described by other researchers the fractional diffusion equation (FDE) formed the mathematical framework employed to describe this anomalous diffusion. In this regard,it is taken that the diffusive particle is not just hopping to its nearest node but also to any other node of the network with a probability that scales according to the distance between the two places. Initially, we extend the k-path LOs above to consider a connected and locally finite infinite network with a bounded degree and investigate a number of the properties of these operators, such as their self-adjointness and boundedness. Then, three different transformations of the k-path LOs, i.e. the Laplace, Factorial and Mellin transformations as well as their properties, are studied.In addition, in order to show a number of applications of these operators and the transformed ones, the transformed k-path LOs are used to obtain a generalised diffusion process for one-dimensional and two-dimensional infinite graphs.First, the infinite path graph is studied, where it is possible to prove that when the Laplacian- and factorial-transformed operators are used in the generalised diffusion equation, the diffusive processes observed are always normal, independent of the transform parameters. It is then proven analytically that when the k-path LOs are transformed via a Mellin transform and plugged into the diffusion equation, the result is a super diffusive process for certain values of the exponent in the transform. Secondly, we generalise the results on the superdiffusive behaviour generated by transforming k-path LOs from one-dimensional graphs to 2-dimensional ones. Our attention focuses on the Abstract Cauchy problem in an infinite square lattice. A generalised diffusion equation on a square lattice corresponding to Mellin transforms of the k-path Laplacian is investigated. Similar to the one-dimensional case also for the graph embedded in two-dimensional space,we could observe superdiffusive behaviour for the Mellin transformed k-path Laplacian. In comparison to the one-dimensional case, the conclusion reached is that the asymptotic behaviour of the solution of the Cauchy problem is much subtler.A set of links and nodes are the fundamental units or components used to represent complex networks. Over the last few decades, network studies have expanded and matured, increasingly making use of complex mathematical tools. Complex networks play a significant role in the propagation of processes, which include for example the case of epidemic spreading, the diffusion process, synchronisation or the consensus process. Such dynamic processes are critically important in achieving understanding of the behaviour of complex systems at different levels of complexity - examples might be the brain and modern man-made infrastructures. Although part of the study of the diffusion of information in the dynamic processes, it is generally supposed that interactions in networks originate only from a node, spreading to its nearest neighbours, there also exist long-range interactions (LRI), which can be transmitted from a node to others not directly connected. The focus of this study is on dynamic processes on networks where nodes interact with not only their nearest neighbours but also through certain LRIs. The generalised k-path Laplacian operators (LOs) Lk, which account for the hop of a diffusive particle to its non-nearest neighbours in a graph, control this diffusive process, describing hops of nodes vi at distance k; here the distance is measured as the length of the shortest path between two nodes. In this way the introduction of the k-path LOs can facilitate conducting more precise studies of network dynamics in different applications. This thesis aims to study a generalised diffusion equation employing the transformed generalised k-path LOs for a locally finite infinite graph. This generalised diffusion equation promotes both normal and super diffusive processes on infinite graphs. Furthermore, this thesis develops a new theoretical mathematical framework for describing superdiffusion processes that use a transform of the k-path LOs defined on infinite graphs. The choice of the transform appeared to be vitally important as the probability of a long jump should be great enough. As described by other researchers the fractional diffusion equation (FDE) formed the mathematical framework employed to describe this anomalous diffusion. In this regard,it is taken that the diffusive particle is not just hopping to its nearest node but also to any other node of the network with a probability that scales according to the distance between the two places. Initially, we extend the k-path LOs above to consider a connected and locally finite infinite network with a bounded degree and investigate a number of the properties of these operators, such as their self-adjointness and boundedness. Then, three different transformations of the k-path LOs, i.e. the Laplace, Factorial and Mellin transformations as well as their properties, are studied.In addition, in order to show a number of applications of these operators and the transformed ones, the transformed k-path LOs are used to obtain a generalised diffusion process for one-dimensional and two-dimensional infinite graphs.First, the infinite path graph is studied, where it is possible to prove that when the Laplacian- and factorial-transformed operators are used in the generalised diffusion equation, the diffusive processes observed are always normal, independent of the transform parameters. It is then proven analytically that when the k-path LOs are transformed via a Mellin transform and plugged into the diffusion equation, the result is a super diffusive process for certain values of the exponent in the transform. Secondly, we generalise the results on the superdiffusive behaviour generated by transforming k-path LOs from one-dimensional graphs to 2-dimensional ones. Our attention focuses on the Abstract Cauchy problem in an infinite square lattice. A generalised diffusion equation on a square lattice corresponding to Mellin transforms of the k-path Laplacian is investigated. Similar to the one-dimensional case also for the graph embedded in two-dimensional space,we could observe superdiffusive behaviour for the Mellin transformed k-path Laplacian. In comparison to the one-dimensional case, the conclusion reached is that the asymptotic behaviour of the solution of the Cauchy problem is much subtler

A Connectedness Constraint for Learning Sparse Graphs

Graphs are naturally sparse objects that are used to study many problems involving networks, for example, distributed learning and graph signal processing. In some cases, the graph is not given, but must be learned from the problem and available data. Often it is desirable to learn sparse graphs. However, making a graph highly sparse can split the graph into several disconnected components, leading to several separate networks. The main difficulty is that connectedness is often treated as a combinatorial property, making it hard to enforce in e.g. convex optimization problems. In this article, we show how connectedness of undirected graphs can be formulated as an analytical property and can be enforced as a convex constraint. We especially show how the constraint relates to the distributed consensus problem and graph Laplacian learning. Using simulated and real data, we perform experiments to learn sparse and connected graphs from data.Comment: 5 pages, presented at the European Signal Processing Conference (EUSIPCO) 201

From local averaging to emergent global behaviors: the fundamental role of network interconnections

Distributed averaging is one of the simplest and most studied network dynamics. Its applications range from cooperative inference in sensor networks, to robot formation, to opinion dynamics. A number of fundamental results and examples scattered through the literature are gathered here and originally presented, emphasizing the deep interplay between the network interconnection structure and the emergent global behavior.Comment: 10 page