From scale-free to Erdos-Rényi networks

We analyze a model that interpolates between scale-free and Erdos-Rényi networks. The model introduced generates a one-parameter family of networks and allows one to analyze the role of structural heterogeneity. Analytical calculations are compared with extensive numerical simulations in order to describe the transition between these two important classes of networks. Finally, an application of the proposed model to the study of the percolation transition is presented.

Figure 1

(Color online) Schematic representation of the general procedure for generating the networks. With probability $\alpha$ one of the $m$ links can be made with any of the nodes (and with the same uniform probability) that will take part in the final network. On the other hand, with probability $(1-\alpha )$ the link will be made only with those nodes that form the connected set at that time because the node will choose a preferential linking strategy.

Figure 2

(Color online) Size of the connected set $N\left(t\right)$ as a function of time. Solid lines correspond to the analytical results [Eq. (5)] and points are the Monte Carlo results of network construction [employing model A (Sec. 3A)]. The comparison is made for $\Omega ={10}^5$ and several values of $\alpha$. The parameters of the model are set to $A=m=m_0=1$.

Figure 3

(Color online) Model A. Monte Carlo simulation (points) versus mean field (lines) results for ${\widehat{k}}^{\mathrm{pa}}(t=\Omega )$ as a function of the birth time $t_0$ for different values of $\alpha$. The parameters of the model were $\Omega ={10}^5$ and $A=m=m_0=1$. The statistics of the Monte Carlo simulations were performed using ${10}^4$ networks for each value of $\alpha$.

Figure 4

(Color online) Monte Carlo results for the degree distribution $P\left(k\right)$ and rank-degree relation for several values of $\alpha$. (a), (c) The results for model A revealing a progressive increase of the tails decaying rate when $\alpha \to 1$. The results for model B [(b) and (d)] show how the decaying rate is not affected by $\alpha$. The networks were generated with the following parameters $\Omega ={10}^5$ and $m=m_0=3$ ($A=3$ for model A).

Figure 5

Average path length (a) and second moment of the degree distribution (b) as a function of $\alpha$. Both quantities are represented normalized by their respective values in the ER limit. The results clearly manifest the two different transitions of the models regarding the heterogeneity evolution along the interpolating path. The averaged networks had the following parameters $\Omega ={10}^4$ and $m=m_0=3$ ($A=3$ for model A).

Figure 6

(Color online) Ratio between the size of the giant connected component in networks generated using models A (upper) and B (bottom) and ER networks when a fraction $f$ of nodes is randomly deleted for different values of $\alpha$. Note that although the transition points for models A and B are located close to each other, the dependence with $\alpha$ is quite different. Network parameters are those of Fig. 5.