A multi-class approach for ranking graph nodes: models and experiments with incomplete data

After the phenomenal success of the PageRank algorithm, many researchers have extended the PageRank approach to ranking graphs with richer structures beside the simple linkage structure. In some scenarios we have to deal with multi-parameters data where each node has additional features and there are relationships between such features. This paper stems from the need of a systematic approach when dealing with multi-parameter data. We propose models and ranking algorithms which can be used with little adjustments for a large variety of networks (bibliographic data, patent data, twitter and social data, healthcare data). In this paper we focus on several aspects which have not been addressed in the literature: (1) we propose different models for ranking multi-parameters data and a class of numerical algorithms for efficiently computing the ranking score of such models, (2) by analyzing the stability and convergence properties of the numerical schemes we tune a fast and stable technique for the ranking problem, (3) we consider the issue of the robustness of our models when data are incomplete. The comparison of the rank on the incomplete data with the rank on the full structure shows that our models compute consistent rankings whose correlation is up to 60% when just 10% of the links of the attributes are maintained suggesting the suitability of our model also when the data are incomplete

Empirical Evaluation of Four Tensor Decomposition Algorithms

Higher-order tensor decompositions are analogous to the familiar Singular Value Decomposition (SVD), but they transcend the limitations of matrices (second-order tensors). SVD is a powerful tool that has achieved impressive results in information retrieval, collaborative filtering, computational linguistics, computational vision, and other fields. However, SVD is limited to two-dimensional arrays of data (two modes), and many potential applications have three or more modes, which require higher-order tensor decompositions. This paper evaluates four algorithms for higher-order tensor decomposition: Higher-Order Singular Value Decomposition (HO-SVD), Higher-Order Orthogonal Iteration (HOOI), Slice Projection (SP), and Multislice Projection (MP). We measure the time (elapsed run time), space (RAM and disk space requirements), and fit (tensor reconstruction accuracy) of the four algorithms, under a variety of conditions. We find that standard implementations of HO-SVD and HOOI do not scale up to larger tensors, due to increasing RAM requirements. We recommend HOOI for tensors that are small enough for the available RAM and MP for larger tensors

How we got Here? A Methodology to Study the Evolution of Economies

This paper proposes a methodology to analyze the evolution of the economic development of countries. Our approach is based upon the definition of temporal trajectories of countries in a common bidimensional space yielded by a High-Order Singular Value Decomposition (HOSVD). These trajectories are defined with respect to a pre-selected set of macroeconomic indicators and are appropriate for comparison purposes. To show the applicability of the proposed methodology we have used data from the World Bank concerning the economic and financial development of EU-27 over a 14-year span, that goes from 1995 to 2008. Based on this data we group the EU-27 state members according to their economic development, which is indicated by the position of their trajectories on the plane. We further perform individual analyses of the trajectories of Luxembourg, Germany and Portugal, aiming to both detect and interpret trends and changes in these economies. The results show that this methodology is of importance for economic studies, since it can help the design, monitoring and evaluation of specific economic policies, as well as provide an overview of the evolution of the studied economic phenomenon.European Union, HOSVD, International Comparisons, Temporal Trajectories

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

Une nouvelle approche de détection de communautés dans les réseaux multidimensionnels

L'analyse des graphes complexes, aussi appelés réseaux multidimensionnels ou réseaux multiplex, est l'un des nouveaux défis apparus en forage de données. Contrairement à la représentation classique de graphes où deux nœuds sont reliés par le biais d'une simple liaison, deux nœuds dans un réseau multidimensionnel se connectent par un ou plusieurs liens décrivant chacun une interaction spécifique dans une dimension particulière. Une des problématiques fondamentales étudiées dans ce domaine est la détection de communautés. Le but est de découvrir les sous-ensembles de nœuds densément connectés ou fortement interactifs, souvent, associés à des caractéristiques organisationnelles et fonctionnelles non connues à priori. Bien qu'elle ait fait l'objet de nombreuses études dans le contexte unidimensionnel, la détection de communautés dans les réseaux multidimensionnels demeure une question de recherche ouverte. C'est d'une part en raison des complexités inhérentes à ce type de réseaux et d'autre part, la conséquence de l'absence d'une définition universellement reconnue pour le concept de communauté multidimensionnelle. En dépit du nombre croissant de travaux abordant cette problématique, certains aspects demeurent peu ou pas abordés dans la littérature. En effet, les approches existantes souffrent d'au moins un des problèmes suivants : (1) La difficulté de fixer des valeurs propres aux paramètres d'entrée, (2) la sensibilité aux dimensions non pertinentes, et (3) l'incapacité de découvrir les sous-espaces de dimensions pertinentes associés aux communautés détectées. Afin de pallier les limites des approches existantes, nous présentons dans le cadre de ce mémoire une nouvelle approche de détection de communautés dans les réseaux multidimensionnels. Axée sur le principe de propagation d'étiquettes, l'approche développée vise l'identification automatique des structures denses dans les différents sous-espaces de dimensions, de même que leurs dimensions pertinentes via la maximisation d'une nouvelle fonction objective. L'efficacité de l'approche proposée est comparée à d'autres méthodes récentes par le biais d'une étude empirique détaillée sur différents réseaux synthétiques et réels. Les résultats obtenus démontrent la capacité de notre approche à identifier les communautés qui existent même dans des sous-espaces de faibles dimensions.\ud ______________________________________________________________________________ \ud MOTS-CLÉS DE L’AUTEUR : Détection de communautés, Réseaux multidimensionnels, Clustering

The structure and dynamics of multilayer networks

In the past years, network theory has successfully characterized the interaction among the constituents of a variety of complex systems, ranging from biological to technological, and social systems. However, up until recently, attention was almost exclusively given to networks in which all components were treated on equivalent footing, while neglecting all the extra information about the temporal- or context-related properties of the interactions under study. Only in the last years, taking advantage of the enhanced resolution in real data sets, network scientists have directed their interest to the multiplex character of real-world systems, and explicitly considered the time-varying and multilayer nature of networks. We offer here a comprehensive review on both structural and dynamical organization of graphs made of diverse relationships (layers) between its constituents, and cover several relevant issues, from a full redefinition of the basic structural measures, to understanding how the multilayer nature of the network affects processes and dynamics.Comment: In Press, Accepted Manuscript, Physics Reports 201