In this paper, we consider the problem of assessing local clustering in complex networks. Various definitions for this measure have been proposed for the cases of networks having weighted edges, but less attention has been paid to both weighted and directed networks. We provide a new local clustering coefficient for this kind of networks, starting from those existing in the literature for the weighted and undirected case. Furthermore, we extract from our coefficient four specific components, in order to separately consider different link patterns of triangles. Empirical applications on several real networks from different frameworks and with different order are provided. The performance of our coefficient is also compared with that of existing coefficients in the literature.

The main contribution of the paper is to employ the financial market network as a useful tool to improve the portfolio selection process, where nodes indicate securities and edges capture the dependence structure of the system. Three different methods are proposed in order to extract the dependence structure between assets in a network context. Starting from this modified structure, we formulate and then we solve the asset allocation problem. We find that the portfolios obtained through a network-based approach are composed mainly of peripheral assets, which are poorly connected with the others. These portfolios, in the majority of cases, are characterized by an higher trade-off between performance and risk with respect to the traditional Global Minimum Variance (GMV) portfolio. Additionally, this methodology benefits of a graphical visualization of the selected portfolio directly over the graphic layout of the network, which helps in improving our understanding of the optimal strategy. IntroductionModern portfolio theory, which originated with Harry Markowitz's seminal paper in 1952 (see [23]) has stood the test of time and continues to be the intellectual foundation for real-world portfolio management. In this static framework, investors optimally allocate their wealth across a set of assets considering only the first and second moment of the returns' distribution. Despite the profound changes derived from this publication, the out-of-sample performance of Markowitz's prescriptions is often not as promising as expected. The poor performance of Markowitz's rule stems from the large estimation errors on the vector of expected returns (see [26]) and on the covariance matrix (see [17]) leading to the well-documented error-maximizing property discussed in [27]. The magnitude of this problem is evident when we acknowledge the modest improvements achieved by those models specifically designed to tackle the estimation risk (see [10]). Moreover, the evidence indicates that the simple yet effective equally-weighted portfolio rule has not been consistently out-performed by more sophisticated alternatives, as reported in [2,10]. The literature on portfolio selection has been extended on several directions. On one hand, some extensions modify the optimal problem, considering higher moments of returns' distribution (see [24] and the reference therein), exploiting alternative risk measures (see [6,19]) and as well as utility functions (see [37,13]), or proposing dynamic approaches (see among others [39,4]). On the other hand, there is now a vast literature on how to deal with the problem, that focuses on improved estimation procedures, considering that the optimal allocation is very sensitive to the estimation of moments and co-moments 1 .1 This sensitivity has generally been attributed to the tendency of the optimization to magnify the effects of estimation error. Michaud in [27] referred to "portfolio optimization "as "error maximization ". Efforts to improve parameters estimation procedure include among others the arXiv:18...

Node centrality is one of the most important and widely used concepts in the study of complex networks. Here, we extend the paradigm of node centrality in financial and economic networks to consider the changes of node ``importance"" produced not only by the variation of the topology of the system but also as a consequence of the external levels of risk to which the network as a whole is subjected. Starting from the ``Susceptible-Infected"" (SI) model of epidemics and its relation to the communicability functions of networks, we develop a series of risk-dependent centralities for nodes in (financial and economic) networks. We analyze here some of the most important mathematical properties of these risk-dependent centrality measures. In particular, we study the newly observed phenomenon of ranking interlacement, by means of which two entities may interlace their ranking positions in terms of risk in the network as a consequence of the change in the external conditions only, i.e., without any change in the topology. We test the risk-dependent centralities by studying two realworld systems: the network generated by collecting assets of the S\&P 100 and the corporate board network of the U.S. top companies, according to Forbes in 1999. We found that a high position in the ranking of the analyzed financial companies according to their risk-dependent centrality corresponds to companies more sensitive to the external market variations during the periods of crisis.

In this paper, we investigate the mesoscale structure of the World Trade Network. In this framework, a specific role is assumed by short- and long-range interactions, and hence by any suitably defined network-based distance between countries. Therefore, we identify clusters through a new procedure that exploits Estrada communicability distance and the vibrational communicability distance, which turn out to be particularly suitable for catching the inner structure of the economic network. The proposed methodology aims at finding the distance threshold that maximizes a specific quality function defined for general metric spaces. Main advantages regard the computational efficiency of the procedure as well as the possibility to inspect intercluster and intracluster properties of the resulting communities. The numerical analysis highlights peculiar relationships between countries and provides a rich set of information that can hardly be achieved within alternative clustering approaches.

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