Variable neighborhood search for extremal graphs. 5. Three ways to automate finding conjectures

AbstractThe AutoGraphiX system determines classes of extremal or near-extremal graphs with a variable neighborhood search heuristic. From these, conjectures may be deduced interactively. Three methods, a numerical, a geometric and an algebraic one are proposed to automate also this last step. This leads to automated deduction of previous conjectures, strengthening of a series of conjectures from Graffiti and obtention of several new conjectures, four of which are proved

Spectral Reconstruction and Isomorphism of graphs using variable neighbourhood search

The Euclidean distance between the eigenvalue sequences of graphs G and H, on the same number of vertices, is called the spectral distance between G and H. This notion is the basis of a heuristic algorithm for reconstructing a graph with prescribed spectrum. By using a graph Γ constructed from cospectral graphs G and H, we can ensure that G and H are isomorphic if and only if the spectral distance between Γ  and G+K2 is zero. This construction is exploited to design a heuristic algorithm for testing graph isomorphism. We present preliminary experimental results obtained by implementing these algorithms in conjunction with a meta-heuristic known as a variable neighbourhood search

Découverte par ordinateur en théorie des graphes

Le système AutoGraphiX -- Fonctions d'AutoGraphiX -- La recherche à voisinage variable -- Invariants disponibles dans AutoGraphiX -- Recherche automatique de conjectures -- Applications -- Étude de l'indice de Randié -- Étude de l'énergie d'un graphe -- Étude de l'index d'arbres avec contraintes de coloration -- Recherche d'arbres H-palindromiques -- Résultats et développemtns envisagés -- Conjectures de Graffiti réfutées -- Conjectures obtenues -- Recherche analytique de conjectures -- Définition automatiques des voisinages à utiliser -- Définition interactive d'invariants -- Énumération de familles de graphes -- Énumération de Benzenoides et Helicènes -- Génération ordonnée -- Énumération de polyhexes planaires simplement connectés -- Énumération de polyhexes simplement connectés -- Méthode par décomposition

Finding communities in networks in the strong and almost-strong sense

International audienceFinding communities, or clusters or modules, in networks can be done by optimizing an objective function defined globally and/or by specifying conditions which must be satisfied by all communities. Radicchi et al. [ Proc. Natl. Acad. Sci. USA 101 2658 (2004)] define a susbset of vertices of a network to be a community in the strong sense if each vertex of that subset has a larger inner degree than its outer degree. A partition in the strong sense has only strong communities. In this paper we first define an enumerative algorithm to list all partitions in the strong sense of a network of moderate size. The results of this algorithm are given for the Zachary karate club data set, which is solved by hand, as well as for several well-known real-world problems of the literature. Moreover, this algorithm is slightly modified in order to apply it to larger networks, keeping only partitions with the largest number of communities. It is shown that some of the partitions obtained are informative, although they often have only a few communities, while they fail to give any information in other cases having only one community. It appears that degree 2 vertices play a big role in forcing large inhomogeneous communities. Therefore, a weakening of the strong condition is proposed and explored: we define a partition in the almost-strong sense by substituting a nonstrict inequality to a strict one in the definition of strong community for all vertices of degree 2. Results, for the same set of problems as before, then give partitions with a larger number of communities and are more informative

Column generation algorithms for exact modularity maximization in networks

International audienceFinding modules, or clusters, in networks currently attracts much attention in several domains. The most studied criterion for doing so, due to Newman and Girvan [Phys. Rev. E 69, 026113 (2004)], is modularity maximization. Many heuristics have been proposed for maximizing modularity and yield rapidly near optimal solution or sometimes optimal ones but without a guarantee of optimality. There are few exact algorithms, prominent among which is a paper by Xu et al. [Eur. Phys. J. B 60, 231 (2007)]. Modularity maximization can also be expressed as a clique partitioning problem and the row generation algorithm of Grötschel and Wakabayashi [Math. Program. 45, 59 (1989)] applied. We propose to extend both of these algorithms using the powerful column generation methods for linear and non linear integer programming. Performance of the four resulting algorithms is compared on problems from the literature. Instances with up to 512 entities are solved exactly. Moreover, the computing time of previously solved problems are reduced substantially

Algorithms for network modularity maximization

National audienceNetworks are often used to represent complex systems arising in a variety of fields. Social networks model interactions among people. Telecommunication networks model communications betwen them, such as in the WorldWide Web. Transportation networks model movements of goods and passengers. Biological networks model interactions between organisms, such as in food networks. A network (or graph) G = (V,E) is composed of a set of vertices, representing the entities of the system under study, and a set of edges joining pairs of vertices and representing a relation holding for such pairs. Identifying communities, or clusters, in complex networks is a topic of particular interest and is currently a very active research domain

An algorithm for cost optimization of PMU and communication infrastructure in WAMS

International audiencePower system state estimation relies increasingly on PMU measurements to ef-fectively control and monitor growing and stressed transmission networks whichare also affected by transient and dynamic events. High PMU cost has moti-vated optimal PMU placement solutions but recent works have shown the effectof communication infrastructure cost in PMU configuration. In this paper,we present a new method for the design of Wide Area Measurement Systems.A topological analysis algorithm based on the Variable Neighbourhood Searchheuristic is proposed and tested in several networks, including the common IEEEtest networks and the 5804-bus Brazilian transmission system. Our results showthe flexibility, effectiveness, and scalability of the proposed methodology whencompared with recent research presented in the literature

Analyse automatique des données scripturales prétraitées par des outils de visualization

RÉSUMÉ: Plusieurs méthodes pour analyser le processus d'écriture ont été utilisés afin de comprendre les stratégies des scripteurs. L'outil principal pour analyser le processus d'écriture est le fichier log, qui contient de façon exhaustive et détaillée l'ensemble des opérations effectuées par le scripteur lors de la rédaction d'un texte. Les données qui y sont emmagasinées sont de quantité considérable et lorsqu'elles ne sont pas préalablement traitées, elles sont hostiles à être analysées par l'humain. Parmi les outils d'analyse utilisés, les représentations du processus d'écriture permettent l'agrégation des données grâce à un pré-traitement. Les structures sous-jacentes des données ainsi représentées sont généralement plus propices à l'analyse que les données brutes. Cet article vise à démontrer différentes méthodes d'analyse automatique pouvant être appliquées à ces structures afin de trouver ou confirmer des structures et tendances à travers les données. ABSTRACT: Several methods to analyze the writing process were used in order to understand the strategies of the writers. The main tool to analyze the writing process is the log file which contains all the operations performed by the writer when writing a text, in a comprehensive and detailed way. The data stored in it is of considerable amount and when not previously treated, it is not made to be analyzed by humans. Among the analytical tools used, the representations of the writing process allow aggregation of data through a pre-treatment. The underlying data structures as shown by these tools are generally conducive to analyzing the raw data afterwards. This article aims to demonstrate various automatic analysis methods that can be applied to these structures to find or confirm the structures and trends through data

Optimizing C-RAN Backhaul Topologies: A Resilience-Oriented Approach Using Graph Invariants

ABSTRACT: At the verge of the launch of the first commercial fifth generation (5G) system, trends in wireless and optical networks are proceeding toward increasingly dense deployments, supporting resilient interconnection for applications that carry higher and higher capacity and tighter latency requirements. These developments put increasing pressure on network backhaul and drive the need for a re-examination of traditional backhaul topologies. Challenges of impending networks cannot be tackled by star and ring approaches due to their lack of intrinsic survivability and resilience properties, respectively. In support of this re-examination, we propose a backhaul topology design method that formulates the topology optimization as a graph optimization problem by capturing both the objective and constraints of optimization in graph invariants. Our graph theoretic approach leverages well studied mathematical techniques to provide a more systematic alternative to traditional approaches to backhaul design. Specifically, herein, we optimize over some known graph invariants, such as maximum node degree, topology diameter, average distance, and edge betweenness, as well as over a new invariant called node Wiener impact, to achieve baseline backhaul topologies that match the needs for resilient future wireless and optical networks