Condensation and topological phase transitions in a dynamical network model with rewiring of the links

Growing network models with both heterogeneity of the nodes and topological constraints can give rise to a rich phase structure. We present a simple model based on preferential attachment with rewiring of the links. Rewiring probabilities are modulated by the negative fitness of the nodes and by the constraint for the network to be a simple graph. At low temperatures and high rewiring rates, this constraint induces a Bose-Einstein condensation of paths of length 2, i.e., a new phase transition with an extended condensate of links. The phase space of the model includes further transitions in the size of the connected component and the degeneracy of the network.

Figure 1

Border of the critical region for condensation, according to Eq. (16), for different values of $\kappa$.

Figure 2

Log-log plot of $k_{\max }/2mt$ as a function of $t$ in a model with $m=2$, $r=2$, $g\left(\varepsilon \right)=3{\varepsilon }^2$ for different temperatures below and above the critical temperature $T_c\sim 0.5$, averaged over 100 runs. For comparison, the dashed line corresponds to $k_{\max }/2mt=0.025/ln\left(t\right)$.

Figure 3

Plot of $p_2$ as a function of $r$ for different times from ${10}^3$ to ${10}^6$ in a model with $m=2$, $T=0.125$, $g\left(\varepsilon \right)=3{\varepsilon }^2$, averaged over 100 runs.

Figure 4

Plot of $\left|\mu \right|=ln\left[\right(r-1)C/2r]/\beta$ as a function of $r$ for different times from ${10}^3$ to ${10}^6$ in a model with $m=2$, $T=0.125$, $g\left(\varepsilon \right)=3{\varepsilon }^2$, averaged over 100 runs. The vertical line corresponds to the critical value of $r$ for the phase transition. The border between the Fermi-like and the Bose-like regimes is at $r=1$.

Figure 5

Plot of data collapse for the finite-size scaling of $p_2\left(t\right)$ across the transition at $r_c\simeq 1.014$ in a model with $m=2$, $T=0.125$, $g\left(\varepsilon \right)=3{\varepsilon }^2$, averaged over 100 runs. The scaling parameters are $\alpha \simeq -1/12$ and $\beta \simeq -1/36$.

Figure 6

Plot of $P_2$ at $t={10}^6$ in a model with $m=2$, $g\left(\varepsilon \right)=3{\varepsilon }^2$, averaged over 100 runs.

Figure 7

Time-scaling exponent of the paths of length 2 ${\nu }_{P_2}$ in the r-T plane, computed between $t={10}^4$ and $t={10}^6$ in a model with $m=2$, $g\left(\varepsilon \right)=3{\varepsilon }^2$, averaged over 100 runs. The transition corresponds to ${\nu }_{P_2}=1$. The line represents the transition line predicted by Eq. (16).

Figure 8

Plot of $p_2$ as a function of $r$ in a model with $m=2$, $T=0$ for different times from ${10}^3$ to ${10}^6$, averaged over 100 runs.

Figure 9

Plot of $s$ as a function of $r$ in a model with $m=2$, $T=0$ for different times from ${10}^3$ to ${10}^6$, averaged over 100 runs. The bottom line is the predicted asymptotic value of $s$.

Figure 10

Plot of $s\left(t\right)$ for different values of $r$ in a model with $m=2$, $T=0$. The points represent the average over 100 runs of simulations while the continuous lines are the analytical predictions.

Figure 11

Time-scaling exponent of the number of connected nodes ${\nu }_S$ in the r-T plane, computed between $t={10}^4$ and $t={10}^6$ in a model with $m=2$, $g\left(\varepsilon \right)=3{\varepsilon }^2$, averaged over 100 runs. The transition corresponds to ${\nu }_S<1$.

Figure 12

Plot of data collapse for the finite-size scaling of $s\left(t\right)$ across the transition at $r_c\simeq 1.014$ in a model with $m=2$, $T=0.125$, $g\left(\varepsilon \right)=3{\varepsilon }^2$, averaged over 100 runs. The scaling parameters are $\alpha \simeq -1/12$ and $\beta \simeq -1/36$.

Figure 13

Time-scaling exponent of the clustering coefficient ${\nu }_C$ in the r-T plane, computed between $t={10}^4$ and $t={10}^6$ in a model with $m=2$, $g\left(\varepsilon \right)=3{\varepsilon }^2$, averaged over 100 runs. The transition would correspond to ${\nu }_C=0$.

Figure 14

Time-scaling exponent of the degeneracy ${\nu }_K$ in the r-T plane, computed between $t={10}^4$ and $t={10}^6$ in a model with $m=2$, $g\left(\varepsilon \right)=3{\varepsilon }^2$, averaged over 100 runs. The transition corresponds to ${\nu }_K=1/2$.