Correlations in weighted networks

We develop a statistical theory to characterize correlations in weighted networks. We define the appropriate metrics quantifying correlations and show that strictly uncorrelated weighted networks do not exist due to the presence of structural constraints. We also introduce an algorithm for generating maximally random weighted networks with arbitrary $P(k,s)$ to be used as null models. The application of our measures to real networks reveals the importance of weights in a correct understanding and modeling of these heterogeneous systems.

Figure 1

(Color online) Correlation measures for a random WCN generated with the algorithm defined in the text with $P\left(k\right)\sim k^{-2.5}$ and $\overline{s}\left(k\right)\propto k^{1.5}$.

Figure 2

(Color online) Correlation measures for real networks. From top to bottom, the U.S. airport network (USAN) for the year 2005, the world trade web (WTW) for the year 2000, and the scientific collaboration network (SCN).