There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

다중혈관 관상동맥 환자에서 y-문합을 이용하여 양쪽 내흉동맥만을 사용한 우회술의 조기 성적

抗原变异可使得多种致病微生物易于逃避宿主免疫应答。表达在感染红细胞表面的恶性疟原虫红细胞表面蛋白1（PfPMP1）与感染红细胞、内皮细胞、树突状细胞以及胎盘的单个或多个受体作用，在黏附及免疫逃避中起关键的作用。每个单倍体基因组var基因家族编码约60种成员，通过启动转录不同的var基因变异体为抗原变异提供了分子基础。

疟原虫var基因转换速率变化导致抗原变异[英]／Paul H, Robert P, Christodoulou Z, et al//Proc Natl Acad Sci U S A

Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.

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The Structure and Function of Complex Networks

We propose and study a set of algorithms for discovering community structure in networks-natural divisions of network nodes into densely connected subgroups. Our algorithms all share two definitive features: first, they involve iterative removal of edges from the network to split it into communities, the edges removed being identified using any one of a number of possible "betweenness" measures, and second, these measures are, crucially, recalculated after each removal. We also propose a measure for the strength of the community structure found by our algorithms, which gives us an objective metric for choosing the number of communities into which a network should be divided. We demonstrate that our algorithms are highly effective at discovering community structure in both computer-generated and real-world network data, and show how they can be used to shed light on the sometimes dauntingly complex structure of networked systems.

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Finding and evaluating community structure in networks.

We discuss the occurrence of negative specific heat in a nonextensive system which has an equilibrium second-order phase transition.The specific heat is negative only in a transient regime before equilibration, in correspondence to long-lasting metastable states. For these states standard equilibrium Bolzmann-Gibbs thermodynamics does not apply and the system shows non-Gaussian velocity distributions, anomalous diffusion and correlation in phase space. Similar results have recently been found also in several other nonextensive systems, supporting the general validity of this scenario. These models seem also to support the conjecture that a nonexstensive statistical formalism, like the one proposed by Tsallis, should be applied in such cases. The theoretical scenario is not completely clear yet, but there are already many strong theoretical indications suggesting that, it can be wrong to consider the observation of an experimental negative specific heat as an unique and unambiguous signature of a standard equilibrium first-order phase transition.

Negative specific heat in out-of-equilibrium nonextensive systems

We propose a credit risk approach in which financial institutions, modelled as a portfolio of risky assets characterized by a probability of default and a correlation matrix, are the nodes of a network whose links are credit exposures that would be partially lost in case of neighbours' default. The systemic risk of the network is described in terms of the loss distribution over time obtained with a multi-period Montecarlo simulation process, during which the nodes can default, triggering a change in the probability of default in their neighbourhood as a contagion mechanism. In particular, we have considered the expected loss and introduced new measures of network stress called PDImpact and PDRank. They are expressed in monetary terms as the already known DebtRank and can be used to assess the importance of a node in the network. The model exhibits two regimes of 'weak' and 'strong' contagion, the latter characterized by the depletion of the loss distribution at intermediate losses in favour of fatter tails. Also, in systems with strong contagion, low average correlation between nodes corresponds to larger losses. This seems at odds with the diversification benefit obtained in standard credit risk models. Results suggest that the credit exposure network of the European global systemically important banks is in a weak contagion regime, but strong contagion could be approached in periods characterized by extreme volatility or in cases where the financial institutions are not adequately capitalized.

A hybrid approach to assess systemic risk in financial networks

We review some of the most recent results on the dynamics of the Hamiltonian Mean Field (HMF) model, a systems of N planar spins with ferromagnetic infiniterange interactions. We show, in particular, how some of the dynamical anomalies of the model can be interpreted and characterized in terms of the weak-ergodicity breaking proposed in frameworks of glassy systems. We also discuss the connections with the nonextensive thermodynamics proposed by Tsallis.

Metastability and anomalous behavior in the HMF Model: connections to nonextensive thermodynamics and glassy dynamics

By drawing on large-scale online data we construct and analyze the time-varying worldwide network of professional relationships among start-ups. The nodes of this network represent companies, while the links model the flow of employees and the associated transfer of know-how across companies. We use network centrality measures to assess, at an early stage, the likelihood of the long-term positive performance of a start-up, showing that the start-up network has predictive power and provides valuable recommendations doubling the current state of the art performance of venture funds. Our network-based approach not only offers an effective alternative to the labour-intensive screening processes of venture capital firms, but can also enable entrepreneurs and policy-makers to conduct a more objective assessment of the long-term potentials of innovation ecosystems and to target interventions accordingly.

Predicting success in the worldwide start-up network

Mapping the connectome through non-invasive imaging tools allows to study the wiring diagram of the brain in-vivo. The informative value of the resulting brain network depends on how much of the estimated connectivity is retained. We present a method, the efficiency cost optimization (ECO), to filter connectivity based on the optimization of the trade-off between the efficiency of a network and its wiring cost. As a result, we find an optimal connection density threshold that emphasizes the intrinsic properties of a given network, while preserving its sparsity. Such threshold can be determined a-priori, since the number of connections to retain only depends on the network size. We validate this result on connectomes, from micro- to macro-scales, obtained with different imaging modalities. Finally, we test the potential of ECO in discriminating brain states with respect to alternative filtering approaches. This method advances our ability to analyze and compare imaging connectomes in a fast and principled way.

Vito Latora

Papers

Negative specific heat in out-of-equilibrium nonextensive systems

A hybrid approach to assess systemic risk in financial networks

Metastability and anomalous behavior in the HMF Model: connections to nonextensive thermodynamics and glassy dynamics

Predicting success in the worldwide start-up network

Filtering information in imaging connectomics