نمونه متن انگلیسی مقاله
Despite degree distributions give some insights about how heterogeneous a network is, they fail in giving a unique quantitative characterization of network heterogeneity. This is particularly the case when several different distributions fit for the same network, when the number of data points is very scarce due to network size, or when we have to compare two networks with completely different degree distributions. Here we propose a unique characterization of network heterogeneity based on the difference of functions of node degrees for all pairs of linked nodes. We show that this heterogeneity index can be expressed as a quadratic form of the Laplacian matrix of the network, which allows a spectral representation of network heterogeneity. We give bounds for this index, which is equal to zero for any regular network and equal to one only for star graphs. Using it we study random networks showing that those generated by the Erdös-Rényi algorithm have zero heterogeneity, and those generated by the preferential attachment method of Barabási and Albert display only 11% of the heterogeneity of a star graph. We finally study 52 real-world networks and we found that they display a large variety of heterogeneities. We also show that a classification system based on degree distributions does not reflect the heterogeneity properties of real-world networks.
Complex networks are the structural skeleton of biological, ecological, technological, and socioeconomic systems [1,2]. They are formed by a set of nodes V representing the entities of these systems and a set of links E representing relationships between pairs of nodes [3,4]. Despite the disparate nature of the systems represented by these networks they share several universal topological properties, such as small worldness , scale freeness 6, the existence of network motifs 7, and self-similarity characteristics . A great deal of attention has been paid to the scale-free property shared by many real-world networks, which contrasts with the regularity observed in random network models like the one proposed by Erdös and Rényi (ER) .
We have defined here an index that accounts for the heterogeneity of a network by using the sum of differences of some function of the node degrees for linked pairs of nodes. This index is then expressed as a quadratic form of the Laplacian matrix of the network, also allowing a spectral representation on the basis of Laplacian eigenvalues and eigenvectors.