Graph Theory and Economic Models: from Small to Large Size Applications

doi:10.1016/j.endm.2007.01.065

Electronic Notes in Discrete Mathematics

Volume 28, 1 March 2007, Pages 469-476

https://doi.org/10.1016/j.endm.2007.01.065 Get rights and content

Abstract

This empirical study explores the structure of macroeconomic models using major concepts and algorithms of the graph theory. Different sizes of applications with dynamic effects are considered. We will firstly examine the matching problem when assigning the equations to the variables. We'll also propose a simple method for improving the regular circular embedding of graphs on the basis of one of the longest circuit and adequate permutations. The determination of the maximal list of edge-disjoint circuits also produces an useful insight into the structure. A typology of the interdependent variables is proposed using the all- pairs shortest paths matrix. This classification is based on both the emissions of nodes towards the rest of the directed graph and the perturbations that the rest of the graph exerts on these nodes. The computations have been done using the softwares MATHEMATICA^® 5.1, LINDO 6.1 and our own programs in Fortran 77L and C++.

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Citation Excerpt :
Since the emergence of graph signal processing (GSP), the theory and applications of GSP have been extensively studied [1–12]. Graphs can be used to capture many various phenomena in nature where data and objects are invariably inter-connected such as human brain [2], ecological, financial [13], and social [14] networks. GSP has found a wide range of applications including wavelet/filter bank design [15–18], learning graphs from observed data [19–21], graph signal restoration [22,23], image/point cloud processing [24], and deep learning on graphs [25], to name but a few.
In the graph signal processing literature, most methods were developed based on the assumption of Gaussian noise since it can lead to computationally efficient and mathematically tractable solutions. Unfortunately, the Gaussian distribution cannot capture the sharp spikes and tail heaviness of the signal noise in various natural phenomena. The α-stable distribution is a generalization of the Gaussian distribution and is a more appropriate model for such impulsive heavy-tailed noise. In this paper, we consider the problem of adaptive estimation and sparse sampling for signals defined over graphs in the presence of impulsive α-stable noise. To tackle the problems associated with α-stable noise, the graph signal estimation problem is formulated as a minimum dispersion (MD)-based optimization. A novel adaptive least mean pth power (LMP) algorithm is proposed for robust estimation of band-limited graph signals from partial observations in α-stable noise environments. The mean square performance of the proposed LMP algorithm is theoretically analyzed. To handle the case of band-limited graph signals with unknown and time-varying bandwidth and spectral contents, the sparse nature of the Graph Fourier transform of the signal is exploited to develop an adaptive graph sampling technique. Specifically, the proposed sparse sampling technique identifies the spectral signal support via sparse online estimation building on the iterative shrinkage-thresholding algorithm. Numerical simulation studies are presented to corroborate the performance advantages of the proposed adaptive estimation and sparse sampling algorithms.
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Graph Theory and Economic Models: from Small to Large Size Applications

Abstract

Graphs and Algorithms

Semi-reduced Forms of Econometric Models