The cascading vulnerability of the directed and weighted network

• The cascading failures based on directed and weighted network is studied.
• The ‘load-capacity’ model of the directed and weighted network is built.
• The ‘over-loading’ and the ‘short-loading’ cascading failure models based on the directed and weighted network were built.
• The power exponent ββ of ‘load-capacity’ function should be taken value (0, 1) for a good robustness of the power law network.
• The power exponent ββ of loading function should be taken value (0, 3) for a good robustness of the Poisson network.

The cascading failure can bring a huge loss for most real-world networks; but, we cannot uncover fully the mechanism and law of the cascading events occurrence. Most networks in which the cascading failure occurred are based on the various ‘flows’, such as power, oils, and information; moreover, the same link degree of the different nodes likely contain the different meanings, where some are large pivotal nodes and some are mini switching centers. Thus, these networks must be described by the directed and weighted network model. Besides, the ‘over-loading’ cascading failures were more analyzed and studied; but the cascading failures caused by ‘short-loading’ were less studied relatively. However, for some directed networks, such as power grids, oil pipe nets, gas pipe nets and information networks, the large-scale failures of network nodes substantially could be induced by ‘short-loading’ in a such similar way as ‘over-loading’. Based on the above reasons, in this paper, we first built the ‘load-capacity’ model of the directed and weighted network. Afterwards the ‘over-loading’ cascading failure model and the ‘short-loading’ cascading failure model based on the directed and weighted network were built. Meanwhile, applying the models to two typical real networks–Poisson distribution network and power law distribution network–intensive study and numerical analysis were carried out. Lastly, two classical networks simulation experiment results are provided. After the numerical and simulation analyses, we gained the following conclusions. For the power law network, the power exponent ββ of ‘load-capacity’ function should be taken value (0, 1) for a good robustness, and the minimum in-degree and out-degree should be increased respectively, meanwhile, the weight and the scaling exponents of the in-degree and the out-degree distributions should be increased synchronously in the interval (2, 3) for enhancing the resistibility of ‘over-loading’ and ‘short-loading’ failures. For the Poisson network, the power exponent ββ of loading function should be taken value (0, 3) for a good robustness, and the average weight and the average in-degree should be increased respectively restricting 2<β<32<β<3 for enhancing the resistibility of ‘over-loading’ and ‘short-loading’ failures.