Crossovers in scale-free networks on geographical space

Complex networks are characterized by several topological properties: degree distribution, clustering coefficient, average shortest path length, etc. Using a simple model to generate scale-free networks embedded on geographical space, we analyze the relationship between topological properties of the network and attributes (fitness and location) of the vertices in the network. We find there are two crossovers for varying the scaling exponent of the fitness distribution.

Figure 1

(a) The degree distribution $P\left(k\right)$ obtained numerically for $m=3$, $d=2$, $N=10000$, and $\gamma =1.5$ (circles), $\gamma =2.5$ (squares), and $\gamma =3.5$ (triangles). These data are averaged over 100 realizations, and the bin is taken logarithmically to reduce noise. The solid curves stand for the theoretical predictions (18, 20). (b) The exponent values calculated using the maximum likelihood method are shown as a function of $\gamma$ for $m=3$, $d=2$, $N=10000$. (c) The ANND for the same parameters as in (a). The solid curves correspond to theoretical results (22, 23, 24). (d) The assortativity $r$ (degree correlation) obtained numerically is shown as a function of $\gamma$ for the same parameters as in (b).

Figure 2

The clustering coefficient as a function of $k$ for $m=3$, $N=10000$, and $\gamma =1.5$ (a), $\gamma =2.5$ (b), and $\gamma =3.5$ (c), where circles, squares, and triangles correspond to the numerical results for $d=1$, $d=2$, and $d=3$, respectively. In (a), these three plots coincide exactly. The solid curves correspond to the predictions from the numerical integrate of (25). In (d), the average clustering coefficient as a function of $\gamma$ for $m=3$, $N=10000$ and $d=1$ (circles), $d=2$ (squares), and $d=3$ (triangles).

Figure 3

(a) Log-linear plot of the average shortest path length $L$ vs the number of vertices $N$ for different values of $\gamma$ ($\gamma =4.0,3.5,3.0,2.5,2.0,1.5$ from top to bottom) and $d=2$. (b) The same data in the log-log plot.