Evolving Apollonian networks with small-world scale-free topologies

We propose two types of evolving networks: evolutionary Apollonian networks (EANs) and general deterministic Apollonian networks (GDANs), established by simple iteration algorithms. We investigate the two networks by both simulation and theoretical prediction. Analytical results show that both networks follow power-law degree distributions, with distribution exponents continuously tuned from 2 to 3. The accurate expression of clustering coefficient is also given for both networks. Moreover, the investigation of the average path length of EAN and the diameter of GDAN reveals that these two types of networks possess small-world feature. In addition, we study the collective synchronization behavior on some limitations of the EAN.

Figure 1

(Color online) A two-dimensional Apollonian packing of disks.

Figure 2

(Color online) Illustration of the two-dimensional deterministic Apollonian networks, showing the first two steps of iterative process.

Figure 3

(Color online) The cumulative degree distribution $P_{\mathrm{cum}}\left(k\right)$ at various $q$ values for dimension $d=2$. The circles (a), squares (b), stars (c), and triangles (d) denote the simulation results for networks with different evolutionary steps $t=1110$, $t=23$, $t=14$, and $t=12$, respectively. The four straight lines are the theoretical results of $\gamma (d,q)$ as provided by Eq. (6). All data are from the average of ten independent simulations.

Figure 4

(Color online) The clustering coefficient of the whole network as a function of $q$ and $d$. Results are averaged over ten network realizations for each datum.

Figure 5

(Color online) The dependence relation of $C$ on $d$ and $m$.

Figure 6

Semilogarithmic graph of the APL vs the network order $N$. In addition to $N$, APL depends on $d$ and $q$. Each data is obtained by ten independent network realizations. The lines are linear functions of $\ln N$.

Figure 7

(Color online) The eigenratio $R$ as a function of network order $N$. All quantities for random networks are averaged over 50 realizations.