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Clustering and closure coefficients are among the most widely applied indicators in the description of the topological structure of a network. Many distinct definitions have been proposed over time, particularly in the case of weighted networks, where the choice of the weight attributed to the triangles is a crucial aspect. In the present work, in the framework of weighted directed multilayer networks, we extend the classical clustering and closure coefficients through the introduction of the clumping coefficient, which generalizes them to incomplete triangles of any type. We then organize the class of these coefficients in a systematic taxonomy in the more general context of weighted directed multilayer networks. Such cohesion coefficients have also been adapted to the different scales that characterize a multilayer network, in order to grasp their structure from different perspectives. We also show how the tensor formalism allows incorporating the new definitions, as well as all those existing in the literature, in a single unified writing, in such a way that a suitable choice of the involved adjacency tensors allows obtaining each of them. Finally, through some applications to simulated networks, we show the effectiveness of the proposed coefficients in capturing different peculiarities of the network structure on different scales.

In this paper, we provide a methodology suitable to identify relevant insurance companies in a systemic risk framework. To this end, we propose a complex network approach where insurers are linked to form a global interconnected system. In particular, we extend the current literature proposing an approach based on a directed weighted multilayer network. The reciprocal influence between insurers is indeed considered in each period and in subsequent periods, calibrating arc weights on the basis of specific risk measures. Hub and authority scores are then used to assess the prominence of a company in spreading and receiving risk from the others.

This study shows that interconnections in the global banking network decreased due to the global financial crisis and subsequent regulatory initiatives. It also proves the effectiveness of the regulations of global systemically important banks in inducing these banks to contain their systemic nature

Several definitions of clustering coefficient for weighted networks have been proposed in literature, 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 already 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.