Quantifying network heterogeneity.

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.