Dynamic behavior analysis of an internet flow interaction model under cascading failures

Cascading failures in the internet have attracted recent attention due to their unpredictability and destructive consequences. Exploring the failure behavior patterns is necessary because they can provide effective intervention approaches to prevent huge network disasters. To analyze internet flow behaviors during cascading failures (chain reactions in router and link failures), we characterize the internet as two coupled networks, the router network and the flow network. Here the flow network is an abstract representation of data correlations obtained from the router network. We use this coupled network to build a cascading failure model for studying flow transmission and competition, which is reflected in bandwidth competition given by limited link capacity. We first study the dependency between routers and flows to explore the flow transmission efficiency when a failure event occurs. Moreover, we find that rerouting enables flow competition area (the number of flows with which one flow has a competitive relationship) to initially remain stable during a failure episode, but that it then quickly drops due to poor physical network connectivity. Additionally, in the early stage after the failure event, the degree of flow competition sharply increases because of the growing number of the flows and congestion. Subsequently, the flow competition decreases due to the failure of flow transmission.

Figure 1

An example of the internet and the corresponding flow network. (a) An example of the internet. Nodes represent routers, edges represent links. Flows numbered 1–5 are transmitted in this network; flow paths are shown with different colors. (b) The corresponding flow network. In the flow network, each node represents a packet flow, and the node number in panel (b) is in accord with the flow number in panel (a). Connections between flows represent the competitive relationships.

Figure 2

The coupled networks of the internet. The top layer is the flow network. The bottom layer is the router network. Connections between the two networks reveal the dependencies between flows and routers.

Figure 3

Dependency intensity when $\rho$ when $\alpha =1$. $t$ is the load redistribution time, revealing the failure process. For a BA coupled network and ER coupled network, the router number is $N_{\mathrm{Router}}=N_{\mathrm{BA}}=N_{\mathrm{ER}}=2000$; 5% of routers are initially attacked. $\beta =0.1$ to measure the average RTT of packets. 100 000 flows are randomly distributed with random priorities $\mathrm{\Phi }\in \{1,2,3,4,5\}$. The paths are assigned through the Dijkstra shortest path algorithm. The flow bandwidth demands obey a normal distribution $d\sim N(0.2,0.4)$. Simulation settings are the same below.

Figure 4

Competition area $g_F$ of flows. $g_F$ reflects the maximal flow competition area. The decreasing trend of during failures in BA flow networks is much more obvious than that in ER flow networks. $\alpha =1,2,4,8$ separately to adjust the link transmission capacity ($l$). $l_0=10$(Gps). Simulation settings are the same as in Fig. 3.

Figure 5

Competition degree $\gamma$ of flows. The flow competition degree in BA flow networks is more intense and concentrated. The difference of $\gamma$ and ${\gamma }_{\max }$ gets smaller as the failure continues. Simulation settings are the same as in Fig. 3.