Generalized Bose-Fermi Statistics and Structural Correlations in Weighted Networks

We derive a class of generalized statistics, unifying the Bose and Fermi ones, that describe any system where the first-occupation energies or probabilities are different from subsequent ones, as in the presence of thresholds, saturation, or aging. The statistics completely describe the structural correlations of weighted networks, which turn out to be stronger than expected and to determine significant topological biases. Our results show that the null behavior of weighted networks is different from what was previously believed, and that a systematic redefinition of weighted properties is necessary.

Figure 1

Ratio $⟨w_{ij}⟩/p_{ij}$ as a function of $y_i{y}_j$ for models 3 and 4. The values of $\overline{w}$, obtained for $y_i{y}_j=1$ (models 1 and 2), are highlighted as larger points.

Figure 2

Analytical results for random networks with strength sequence generated by the distribution $\rho (y)\propto y^{-\gamma }$ with (from top to bottom) $\gamma =1$, 2, 3, 4. All networks have the same link density and $N=10000$ vertices.