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Risk-dependent centrality in economic and financial networks

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 submitted. 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 real-world systems: the network generated by collecting assets of the S\&P 100 and the corporate board network of the US 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.