Traffic dynamics based on local routing protocol on a scale-free network

We propose a packet routing strategy with a tunable parameter $\alpha$ based on the local structural information of a scale-free network. As free traffic flow on the communication networks is key to their normal and efficient functioning, we focus on the network capacity that can be measured by the critical point of phase transition from free flow to congestion. Simulations show that the maximal capacity corresponds to $\alpha =-1$ in the case of identical nodes' delivering ability. To explain this, we investigate the number of packets of each node depending on its degree in the free flow state and observe the power law behavior. Other dynamic properties including average packets traveling time and traffic load are also studied. Inspiringly, our results indicate that some fundamental relationships exist between the dynamics of synchronization and traffic on the scale-free networks.

Figure 1

(Color online) The order parameter $\eta$ versus $R$ for BA network with different free parameter $\alpha$. Other parameters are networks size $N=1000$ and $C=10$.

Figure 2

The critical $R_c$ versus $\alpha$ with network size $N=1000$ and constant node capacity $C=10$. The maximum of $R_c$ corresponds to $\alpha =-1$ marked by a dotted line.

Figure 3

(Color online) Average number of packets of nodes $n\left(k\right)$ depending on degree $k$ with PIA for different $\alpha$, the network size $N=1000$. In the inset, the line is the theoretical prediction, the red circles and dark squires are the values measured from the simulation with PIA for $N=1000$ and without PIA for $N=5000$, respectively.

Figure 4

(Color online) The distribution of packets on each node for different chosen parameter $\alpha$ with (a) $\alpha >0$ and (b) $\alpha <0$. In (a) $P\left(n\right)$ follows power-law distribution, while in (b) $P\left(n\right)$ approximately exhibits a Poisson distribution.

Figure 5

(Color online) Packets traveling time distribution for different $\alpha$ all in the steady states. The simulation last for $50000$ time steps.

Figure 6

(Color online) The evolution of $N_p$ for $R>R_c$. Here, ${\alpha }_c$ takes $-1.5$ corresponding to the critical point $R_c=39$.

Figure 7

(Color online) The ratio between $\mathrm{\Delta }{N}_p$ and time step interval $\mathrm{\Delta }t$ versus $R$ (a) and versus $R-R_c$ the rescaling of $R$ (b) for different $\alpha$. In (b) three curves collapse to a single line with the slope $\approx 0.7$ marked by a dashed line.

Figure 8

(Color online) The variance of $R_c$ with the increasing of $m$.

Figure 9

The critical generating rate $R_c$ versus $\alpha$ in the case of node capacity proportional to its degree $C_i=k_i$. Here, $k_{\min }$ is set to be 3 and network size $N=1000$. The maximum of $R_c$ corresponds to $\alpha =0$ marked by a dotted line.