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

Upper and lower bounds for the mixed degree-Kirchhoff index

We introduce the mixed degree-Kirchhoff index, a new molecular descriptor defined by R^(G) = ?i<j(di/dj+dj/di)Rij, where di is the degree of the vertex i and Rij is the effective resistance between vertices i and j. We give general upper and lower bounds for bR(G) and show that, unlike other related descriptors, it attains its largest asymptotic value (order n4), among barbell graphs, for the highly asymmetric lollipop graph. We also give more refined lower (order n2) and upper (order n3) bounds for c-cyclic graphs in the cases 0 ? c ? 6. For this latter purpose we use a close relationship between our new mixed degree-Kirchhoff index and the inverse degree, prior bounds we found for the inverse degree of c-cyclic graphs, and suitable expressions for the largest and smallest effective resistances of c-cyclic graphs

New Upper and Lower Bounds for the Additive Degree-Kirchhoff Index

Given a simple connected graph on N vertices with size | E | and degree sequence 1 2 ... N d d d , the aim of this paper is to exhibit new upper and lower bounds for the additive degree- Kirchhoff index in closed forms, not containing effective resistances but a few invariants (N,| E | and the degrees i d ) and applicable in general contexts. In our arguments we follow a dual approach: along with a traditional toolbox of inequalities we also use a relatively newer method in Mathematical Chemistry, based on the majorization and Schur-convex functions. Some theoretical and numerical examples are provided, comparing the bounds obtained here and those previously known in the literature. (doi: 10.5562/cca2282

Kirchhoffian indices for weighted digraphs

The resistance indices, namely the Kirchhoff index and its generalisations, have undergone intense critical scrutiny in recent years. Based on random walks, we derive three Kirchhoffian indices for strongly connected and weighted digraphs. These indices are expressed in terms of (i) hitting times and (ii) the trace and eigenvalues of suitable matrices associated to the graph, namely the asymmetric Laplacian, the diagonally scaled Laplacian and their MoorePenrose inverses. The appropriateness of the generalised Kirchhoff index as a measure of network robustness is discussed, providing an alternative interpretation which is supported by an empirical application to the World Trade Network

A majorization method for localizing graph topological indices

This paper presents a unified approach for localizing some relevant graph topological indices via majorization techniques. Through this method, old and new bounds are derived and numerical examples are provided, showing how former results in the literature could be improved.Comment: 11 page

Higher order assortativity in complex networks

Assortativity was first introduced by Newman and has been extensively studied and applied to many real world networked systems since then. Assortativity is a graph metrics and describes the tendency of high degree nodes to be directly connected to high degree nodes and low degree nodes to low degree nodes. It can be interpreted as a first order measure of the connection between nodes, i.e. the first autocorrelation of the degree-degree vector. Even though assortativity has been used so extensively, to the author's knowledge, no attempt has been made to extend it theoretically. This is the scope of our paper. We will introduce higher order assortativity by extending the Newman index based on a suitable choice of the matrix driving the connections. Higher order assortativity will be defined for paths, shortest paths, random walks of a given time length, connecting any couple of nodes. The Newman assortativity is achieved for each of these measures when the matrix is the adjacency matrix, or, in other words, the correlation is of order 1. Our higher order assortativity indexes can be used for describing a variety of real networks, help discriminating networks having the same Newman index and may reveal new topological network features.Comment: 24 pages, 16 figure

Bounds for the global cyclicity index of a general network via weighted majorization

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