Generalised thresholding of hidden variable network models with scale-free property

Abstract

The hidden variable formalism (based on the assumption of some intrinsic node parameters) turned out to be a remarkably efficient and powerful approach in describing and analyzing the topology of complex networks. Owing to one of its most advantageous property - namely proven to be able to reproduce a wide range of different degree distribution forms - it has become a standard tool for generating networks having the scale-free property. One of the most intensively studied version of this model is based on a thresholding mechanism of the exponentially distributed hidden variables associated to the nodes (intrinsic vertex weights), which give rise to the emergence of a scale-free network where the degree distribution p(k) ~ k^-γ is decaying with an exponent of γ = 2. Here we propose a generalization and modification of this model by extending the set of connection probabilities and hidden variable distributions that lead to the aforementioned degree distribution, and analyze the conditions leading to the above behavior analytically. In addition, we propose a relaxation of the hard threshold in the connection probabilities, which opens up the possibility for obtaining sparse scale free networks with arbitrary scaling exponent.

Figure 1

Complementary cumulative distribution of the node degrees F(k) in a network of size N = 20000, obtained from simulations for the Weibull fitness distribution in Table 1, at a scale parameter c = 2, and a linking function defined in (13), shown on logarithmic scale. The solid line is decreasing as k⁻¹, which is corresponding to the decay characteristics of F(k) in SF networks with γ = 2.

Figure 2

The complementary cumulative distribution of the node degrees F(k) for four different networks, generated with the same ρ(x) and f(x, y), but with different β parameters. The fitness distribution was chosen to be ρ(x) = 3x²exp(−x³) and corresponding linking function f(x, y) is given in (31). In panel (a) we used β = 0.5, and the decay characteristics of the resulting F(k) seem to be close to that of SF networks with $\gamma \simeq 3.12$ (shown by the solid line). The parameter β was increased to β = 0.7 in panel (b), where the decay of F(k) suggests a γ value of $\gamma \simeq 2.47$. In panels (c,d) we increased β further to β = 1.0 and β = 5.0, reducing the γ exponent to γ = 2.13 and γ = 2.05 respectively. In panel (a,b) $\mathrm{\Delta }=\frac{1}{\beta }\ln N$, while in panel (c,d) we used Δ = lnN for obtaining sparse networks.

Figure 3

Scaling exponent γ of the degree distribution as a function of the effective temperature 1/β in the model with soft thresholding. The data points correspond to simulation results on networks of size N = 20000, where the fitness distribution was chosen to be ρ(x) = 3x²exp(−x³) and the linking function f(x, y) was given by (31). The dashed line shows the (approximate) analytic results given by (35) and (38).

Figure 4

(a) Clustering coefficient as a function of node degrees $\overline{C}(k)$ for four different networks each of them containing N = 20000 nodes. All networks were generated by using the same exponential fitness distribution but with different connection functions f_Δ(x + y) = f_Δ(z) corresponding to specific forms of (5) displayed by different colours and indicated in panel (b).