Leontief Meets Markov: Sectoral Vulnerabilities Through Circular Connectivity

Economists have been aware of the mapping between an Input-Output (I-O, hereinafter) table and the adjacency matrix of a weighted digraph for several decades (Solow, Econometrica 20(1):29–46, 1952). An I-O table may be interpreted as a network in which edges measure money flows to purchase inputs that go into production, whilst vertices represent economic industries. However, only recently the language and concepts of complex networks (Newman 2010) have been more intensively applied to the study of interindustry relations (McNerney et al. Physica A Stat Mech Appl, 392(24):6427–6441, 2013). The aim of this paper is to study sectoral vulnerabilities in I-O networks, by connecting the formal structure of a closed I-O model (Leontief, Rev Econ Stat, 19(3):109–132, 1937) to the constituent elements of an ergodic, regular Markov chain (Kemeny and Snell 1976) and its chance process specification as a random walk on a graph. We provide an economic interpretation to a local, sector-specific vulnerability index based on mean first passage times, computed by means of the Moore-Penrose inverse of the asymmetric graph Laplacian (Boley et al. Linear Algebra Appl, 435(2):224–242, 2011). Traversing from the most central to the most peripheral sector of the economy in 60 countries between 2005 and 2015, we uncover cross-country salient roles for certain industries, pervasive features of structural change and (dis)similarities between national economies, in terms of their sectoral vulnerabilities.

Transposed Markov Matrix as a New Decision Tool of How to Choose among Competing Investment Options in Academic Medicine

Medical institutions face the challenge of promoting excellence in a variety of competing focus areas, such as grants, publications, income, research, faculty, variety, patient care and teaching. A transposed Markov chain is used analyse the interactions between the various focus areas and their transition towards steady-state. In contradistinction with a regular Markov chain, in the transposed chain used for the present analysis, the sum ofinputs(rather thanoutputs) of each individual state is 100%, whereas the outputs are left to assume any possible value. The mathematics of calculating the steady state conditions of a transposed Markov matrix are similar to those of a regular Markov matrix. The analysis shows that a focus area more dependent on other areas is also more likely to lose its investment, whereas largely self-reliant areas will generate the largest return. Full strength of all academic focus areas can be achieved only by investments in all areas. In academic systems with one or several exclusively self-reliant focus areas, only investment in these particular areas will invigorate the system, as all other investments are bound to dissipate over time. The newly developed decision tool of a transposed Markov matrix could be helpful in stochastic modelling of medical phenomena.

A note on cumulative sums of markovian variables

Consider a positive regular Markov chain X0, X1, X2,… with s(s finite) number of states E1, E2,… E8, and a transition probability matrix P = (pij) where = , and an initial probability distribution given by the vector p0. Let {Zr} be a sequence of random variables such that and consider the sum SN = Z1+Z2+ … ZN. It can easily be shown that (cf. Bartlett [1] p. 37), where λ1(t), λ2(t)…λ1(t) are the latent roots of P(t) ≡ (pijethij) and si(t) and t′i(t) are the column and row vectors corresponding to λi(t), and so constructed as to give t′i(t)Si(t) = 1 and t′i(t), si(o) = si where t′i(t) and si are the corresponding column and row vectors, considering the matrix .