Generalized Markov stability of network communities

We address the problem of community detection in networks by introducing a general definition of Markov stability, based on the difference between the probability fluxes of a Markov chain on the network at different time scales. The specific implementation of the quality function and the resulting optimal community structure thus become dependent both on the type of Markov process and on the specific Markov times considered. For instance, if we use a natural Markov chain dynamics and discount its stationary distribution -- that is, we take as reference process the dynamics at infinite time -- we obtain the standard formulation of the Markov stability. Notably, the possibility to use finite-time transition probabilities to define the reference process naturally allows detecting communities at different resolutions, without the need to consider a continuous-time Markov chain in the small time limit. The main advantage of our general formulation of Markov stability based on dynamical flows is that we work with lumped Markov chains on network partitions, having the same stationary distribution of the original process. In this way the form of the quality function becomes invariant under partitioning, leading to a self-consistent definition of community structures at different aggregation scales

Non-mean-field Critical Exponent in a Mean-field Model : Dynamics versus Statistical Mechanics

The mean-field theory tells that the classical critical exponent of susceptibility is the twice of that of magnetization. However, the linear response theory based on the Vlasov equation, which is naturally introduced by the mean-field nature, makes the former exponent half of the latter for families of quasistationary states having second order phase transitions in the Hamiltonian mean-field model and its variances. We clarify that this strange exponent is due to existence of Casimir invariants which trap the system in a quasistationary state for a time scale diverging with the system size. The theoretical prediction is numerically confirmed by $N$-body simulations for the equilibrium states and a family of quasistationary states.Comment: 6 pages, 3 figure

The scientific influence of nations on global scientific and technological development

Determining how scientific achievements influence the subsequent process of knowledge creation is a fundamental step in order to build a unified ecosystem for studying the dynamics of innovation and competitiveness. Relying separately on data about scientific production on one side, through bibliometric indicators, and about technological advancements on the other side, through patents statistics, gives only a limited insight on the key interplay between science and technology which, as a matter of fact, move forward together within the innovation space. In this paper, using citation data of both research papers and patents, we quantify the direct influence of the scientific outputs of nations on further advancements in science and on the introduction of new technologies. Our analysis highlights the presence of geo-cultural clusters of nations with similar innovation system features, and unveils the heterogeneous coupled dynamics of scientific and technological advancements. This study represents a step forward in the buildup of an inclusive framework for knowledge creation and innovation

Universal Database for Economic Complexity

We present an integrated database suitable for the investigations of the Economic development of countries by using the Economic Fitness and Complexity framework. Firstly, we implement machine learning techniques to reconstruct the database of Trade of Services and we integrate it with the database of the Trade of the physical Goods, generating a complete view of the International Trade and denoted the Universal database. Using this data, we derive a statistically significant network of interaction of the Economic activities, where preferred paths of development and clusters of High-Tech industries naturally emerge. Finally, we compute the Economic Fitness, an algorithmic assessment of the competitiveness of countries, removing the unexpected misbehaviour of Economies under-represented by the sole consideration of the Trade of the physical Goods

Linear response theory for long-range interacting systems in quasistationary states

Long-range interacting systems, while relaxing to equilibrium, often get trapped in long-lived quasistationary states which have lifetimes that diverge with the system size. In this work, we address the question of how a long-range system in a quasistationary state (QSS) responds to an external perturbation. We consider a long-range system that evolves under deterministic Hamilton dynamics. The perturbation is taken to couple to the canonical coordinates of the individual constituents. Our study is based on analyzing the Vlasov equation for the single-particle phase space distribution. The QSS represents stable stationary solution of the Vlasov equation in the absence of the external perturbation. In the presence of small perturbation, we linearize the perturbed Vlasov equation about the QSS to obtain a formal expression for the response observed in a single-particle dynamical quantity. For a QSS that is homogeneous in the coordinate, we obtain an explicit formula for the response. We apply our analysis to a paradigmatic model, the Hamiltonian mean-field model, that involves particles moving on a circle under Hamilton dynamics. Our prediction for the response of three representative QSSs in this model (the water-bag QSS, the Fermi-Dirac QSS, and the Gaussian QSS) is found to be in good agreement with $N$-particle simulations for large $N$. We also show the long-time relaxation of the water-bag QSS to the Boltzmann-Gibbs equilibrium state.Comment: 13 pages, 4 figures; v2: typos fixed; v3: small changes, close to the published versio

Geography of science: competitiveness and inequality

We characterize the temporal dynamics of Scientific Fitness, as defined by the Economic Fitness and Complexity (EFC) framework, and R&D expenditures at the geographic scale of nations. Our analysis highlights common patterns across similar research systems, and shows how develop-ing nations (China in particular) are quickly catching up with the developed world. This paints the picture of a general growth of scientific and technical capabilities of nations induced by the spreading of information typical of the scientific environment. Shifting the focus of the analysis to the regional level, we find that even developed nations display a considerable level of inequal-ity in the Scientific Fitness of their internal regions. Further, we assess comparatively how the competitiveness of each geographic region is distributed over the spectrum of research sectors. Overall, the Scientific Fitness represents the first high quality estimation of the scientific strength of nations and regions, opening new policy-making applications for better allocating resources, filling inequality gaps and ultimately promoting innovation

The scientific impact of nations on scientific and technological development

Determining how scientific achievements influence the subsequent process of knowledge creation is a fundamental step in order to build a unified ecosystem for studying the dynamics of innovation and competitiveness. Yet, relying separately on data about scientific production on one side, through bibliometric indicators, and about technological advancements on the other side, through patents statistics, gives only a limited insight on the key interplay between science and technology which, as a matter of fact, move forward together within the innovation space. In this paper, using citation data of both scientific papers and patents, we quantify the direct impact of the scientific outputs of nations on further advancements in science and on the introduction of new technologies. Our analysis highlights the presence of geo-cultural clusters of nations with similar innovation system features, and unveils the heterogeneous coupled dynamics of scientific and technological success. This study represents a first step in the buildup of a comprehensive framework for knowledge creation and innovation

Ranking species in complex ecosystems through nestedness maximization

Identifying the rank of species in a social or ecological network is a difficult task, since the rank of each species is invariably determined by complex interactions stipulated with other species. Simply put, the rank of a species is a function of the ranks of all other species through the adjacency matrix of the network. A common system of ranking is to order species in such a way that their neighbours form maximally nested sets, a problem called nested maximization problem (NMP). Here we show that the NMP can be formulated as an instance of the Quadratic Assignment Problem, one of the most important combinatorial optimization problem widely studied in computer science, economics, and operations research. We tackle the problem by Statistical Physics techniques: we derive a set of self-consistent nonlinear equations whose fixed point represents the optimal rankings of species in an arbitrary bipartite mutualistic network, which generalize the Fitness-Complexity equations widely used in the field of economic complexity. Furthermore, we present an efficient algorithm to solve the NMP that outperforms state-of-the-art network-based metrics and genetic algorithms. Eventually, our theoretical framework may be easily generalized to study the relationship between ranking and network structure beyond pairwise interactions, e.g. in higher-order networks.Comment: 28 pages; 2 figure