Phase transition and hysteresis in scale-free network traffic

We model information traffic on scale-free networks by introducing the node queue length $L$ proportional to the node degree and its delivering ability $C$ proportional to $L$. The simulation gives the overall capacity of the traffic system, which is quantified by a phase transition from free flow to congestion. It is found that the maximal capacity of the system results from the case of the local routing coefficient $\varphi$ slightly larger than zero, and we provide an analysis for the optimal value of $\varphi$. In addition, we report for the first time the fundamental diagram of flow against density, in which hysteresis is found, and thus we can classify the traffic flow with four states: free flow, saturated flow, bistable, and jammed.

Figure 1

(Color online) Overall capacity of a network with $N=1000$, $m0=m=5$, $\alpha =1$ (a), $\alpha =2$ (b), and $\beta =0.2$. The capacity is characterized by the critical value of $R_c$ for different $\varphi$. In (a), $\alpha =1$, ${\varphi }_{\mathit{\text{optimal}}}=0.3$, and $R_c^{\mathit{max}}=18.7$. In (b), $\alpha =2$, ${\varphi }_{\mathit{\text{optimal}}}=0.1$, and $R_c^{\mathit{max}}=42.2$. In both cases, the maximum of $R_c$ corresponds to a $\varphi$ slightly greater than zero marked by a dashed line. The data are obtained by averaging $R_c$ over ten network realizations.

Figure 2

(Color online) Average travel time for a network with $N=1000$, $m_0=m=5$, $\alpha =2$, and $\beta =0.2$. (a) Average travel time $⟨T⟩$ versus $\varphi$ for $R=1$, 2, and 5. The data are truncated because the system jams when $\varphi$ is either too large or too small. (b) The variation of $⟨T⟩$ versus $R$ when $\varphi$ is fixed. The data are also truncated when the system jams.

Figure 3

(Color online) Fundamental diagram for a $N=1000$ network with $m0=m=5$, $\alpha =1$, $\beta =0.2$, and different $\varphi$. The data are averaged over ten typical simulations on one realization of network. In each chart, the solid square line shows the flux variation when adding packets to the system (increase density), while the empty circle line shows the flux variation when drawing out the packet from the system (decrease density). The sudden transition density values are 0.26 and 0.23 $(\varphi =0.0)$, 0.40 and 0.34 $(\varphi =0.3)$, 0.26 and 0.15 $(\varphi =-0.5)$, 0.15 and 0.13 $(\varphi =-0.7)$. For different realizations of the network, the fundamental charts are similar, but with a small difference in the transition values. The arrows in the charts of $\varphi =0.3$ and $-0.5$ show the hysteresis as a guide for the eyes.

Figure 4

(Color online) (a) Fundamental diagram for a $N=5000$ network with $m0=m=5$, $\alpha =1$, $\beta =0.2$, and $\varphi =0.1$. (b) The averaged number of jammed nodes $⟨N_{jv}⟩$. The symbols for an increasing or decreasing density are the same as in Fig. 3. One can see that the two sudden change points 0.40 and 0.14 in both charts are equal. The arrows are showing the hysteresis as a guide for the eyes.