Risk-Dependent Centrality in Economic and Financial Networks

Abstract

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.

Matrix functions in network analysis

“…The communicability function (6) has proved very useful in a number of applications. Some recent examples include: 3 The study of the spread of contagion in financial networks [16,140]; 4 The study of fracture networks in rocks [138]; 5 The study of influence of papers in collaboration and citation networks [153]; 6 The modeling of concept learning as weighted networks [108]; 7 The characterization of robustness in infrastructural and transportation networks [8,9,117]; 8 The analysis of urban traffic flow [1]; 9 The community detection problem [52,67,149]; 10 The identification of sparse and dense subgraphs of large graphs [80].…”

“…In other words, the steady state is just "thermal equilibrium." The rate of convergence of the solution vector x(t) reflects structural properties of the graph and is governed by the algebraic connectivity of G. In more detail, let (16) be the eigendecomposition of the Laplacian of the (connected) graph G. Recalling that the eigenvalues of L satisfy 0 = 1 < 2 ≤ • • • ≤ n and that the eigenvector associated with the simple zero eigenvalue, normalized in the 2 -norm, is…”

“…where Λ is the diagonal matrix with entries given by 1 = 0, 2 , … , n (cf. (16)). In this definition, the principal (positive) branch of the th root i is chosen.…”

From Big Data to Econophysics and Its Use to Explain Complex Phenomena

“…Highlighted among studies relating complex networks and the contagion effect are Gai and Kapadia (2010), Glasserman and Young (2015), or Acemoglu et al (2015), who analyze the risk of financial networks and propose that financial contagion shows a form of transition phase, and to a certain extent, strong interconnections serve as a shock propagation mechanism, leading to the fragility of a given network. It is crucial to analyze the changing relevance of the nodes, according to Bartesaghi et al (2020), which in the context of crises and their potential financial implications, could also be an interesting issue for analysis using big data.…”

New results and open problems on subgraph centrality

“…In some cases, one or more parameters are added to the purely graph-theoretic structure, possibly as a way to model some external influence on the network. For example, one may want to take into account the existence of tensions between members of a social network [10], or of heightened risk conditions for an economic or financial network [3].…”