Universal robustness characteristic of weighted networks against cascading failure

We investigate the cascading failure on weighted complex networks by adopting a local weighted flow redistribution rule, where the weight of an edge is $(k_i{k}_j{)}^{\theta }$ with $k_i$ and $k_j$ being the degrees of the nodes connected by the edge. Assume that a failed edge leads only to a redistribution of the flow passing through it to its neighboring edges. We found that the weighted complex network reaches the strongest robustness level when the weight parameter $\theta =1$, where the robustness is quantified by a transition from normal state to collapse. We determined that this is a universal phenomenon for all typical network models, such as small-world and scale-free networks. We then confirm by theoretical predictions this universal robustness characteristic observed in simulations. We furthermore explore the statistical characteristics of the avalanche size of a network, thus obtaining a power-law avalanche size distribution together with a tunable exponent by varying $\theta$. Our findings have great generality for characterizing cascading-failure-induced disasters in nature.

Figure 1

Illustration of the LWFRR triggered by an edge-cut-based attack. Edge $ij$ is broken and the flow along it is redistributed to the neighboring edges connecting to both ends of $ij$. Among these neighbor edges, the one with higher flow will receive higher shared flow from the broken edge.

Figure 2

(Color online) Normalized avalanche size $S_N$ as a function of threshold $T$ for (a) different values of $\theta$ on both BA and NW networks (network size 5000) and (b) different network sizes for BA networks. (c) Both the $S_N$ of edges and nodes for BA networks. The phase transition points are the same. The average degree of BA networks is $⟨k⟩=4$. The coordination number of NW networks is 2 and the parameter $p=1.0$. Here, if all edges of a node is cut, the node fails. The avalanche size of node is the number of failed nodes normalized by the total number of nodes in the network.

Figure 3

(Color online) Robustness $T_c$ as a function of $\theta$ for different average degrees $⟨k⟩$ on BA networks and different $p$ on NW networks. $p=1$ corresponds to a random network. $N=5000$, and each data point is averaged over ten different network realizations. For the BA model, $⟨k⟩=2m$, where $m$ is the number of edges attached to the existing nodes from the new node and $k_{\min }=m$. For the NW model, $⟨k⟩=2(n_0+p)$, where $n_0$ is the coordinate number and $p$ is the probability for each node to receive one edge.

Figure 4

(Color online) A comparison of analytical results and theoretical predictions on $T_c$ vs $k_{\min }$ ($\theta =0$ and $\theta =1$) for BA networks and vs. $⟨k⟩$ $(\theta =1)$ for NW networks. Data points are by simulations and curves are by theory.

Figure 5

(Color online) Distributions of the avalanche sizes for both (a) BA and (b) NW networks for different $\theta$. The inset of (a) shows the exponent $\eta$ of the power-law distribution, $P\left(s\right)\sim s^{-\eta }$, depending on $\theta$. $N=5000$ and $\delta =0.01$. The avalanche size is calculated as follows. At each time step, after the cascading process stops, record the number of broken edges as the avalanche size at this time step. Then recover all broken edges and set their flows to zero. At the next time step, add flow $\delta$ to randomly selected edges and repeat the cascading process. After the time step has lasted ${10}^8$, we obtain the avalanche distribution $P\left(s\right)$ of the sizes recorded at each time step. If no cascading failures occur, $s=0$.