Multifractal analysis of weighted networks by a modified sandbox algorithm

Complex networks have attracted growing attention in many fields. As a generalization of fractal analysis, multifractal analysis (MFA) is a useful way to systematically describe the spatial heterogeneity of both theoretical and experimental fractal patterns. Some algorithms for MFA of unweighted complex networks have been proposed in the past a few years, including the sandbox (SB) algorithm recently employed by our group. In this paper, a modified SB algorithm (we call it SBw algorithm) is proposed for MFA of weighted networks.First, we use the SBw algorithm to study the multifractal property of two families of weighted fractal networks (WFNs): "Sierpinski" WFNs and "Cantor dust" WFNs. We also discuss how the fractal dimension and generalized fractal dimensions change with the edge-weights of the WFN. From the comparison between the theoretical and numerical fractal dimensions of these networks, we can find that the proposed SBw algorithm is efficient and feasible for MFA of weighted networks. Then, we apply the SBw algorithm to study multifractal properties of some real weighted networks ---collaboration networks. It is found that the multifractality exists in these weighted networks, and is affected by their edge-weights.Comment: 15 pages, 6 figures. Accepted for publication by Scientific Report

Multi-Echelon Data Envelopment Analysis Variable Returns to Scale Models for Performance Evaluation of Supply Chains

This paper develops a variable returns to scale multi-echelon data envelopment analysis (DEA) model to measure the efficiency of supply chain. The model is constructed at first with the assumption of serial sequence in a supply chain. The inputs of one stage become the output of the other stage in the multi-echelon structure. The traditional variable returns to scale model of DEA is modified to fit in the multi-echelon structure. The developed model helps to evaluate the supply network in a coordinated manner. It also provides helpful insights as how to improve the supply network performance