A model of cascading failures in complex networks with an emergency recovery mechanism is proposed in this paper, and the cascading dynamics is investigated by running the proposed model on nearest-neighbor coupled network, Erdos-Renyi random graph network, Watts-Strogatz small-world network and Barabasi-Albert scale-free network respectively. New concepts in emergency recovery mechanism and the efficiency of networks are defined. And the effects of the parameters on the network efficiency and failure rate are investigated. Results demonstrate that the increase of the emergency recovery probability would reduce the network efficiency decreasing speed and the failure rate growing speed, and also improve the resilience of the network. And the greater the load capacity of the nodes in the network, the slower the speeds of network efficiency reducing and failure rate growing. Meanwhile, with the decrease of the overload node failure probability, the reducing speed of network efficiency and the growing speed of failure rate would reduce gradually. Furthermore, the changes of the network efficiency and failure rate during the process of cascading failures in different network topologies are analyzed. It is found that the rise of the heterogeneity of degree distribution increases the reducing speed of network efficiency and the growing speed of failure rate. All these results can help analyze the cascading dynamics in complex networks with an emergency recovery mechanism, and may provide a guidance for the controling of cascading failures and protecting against them in real-life complex networks.

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In this paper,we investigate diameter and average path length(APL) for Sierpinski pentagon based on its recursive construction and self-similar structure.We find that the diameter of Sierpinski pentagon is just the shortest path length between two nodes of generation 0. Deriving and solving the linear homogenous recurrence relation the diameter satisfies,we obtain rigorous solution for the diameter .We also obtain approximate solution for APL of Sierpinski pentagon, both diameter and APL grow approximately as a power-law function of network order N (t),with the exponent equals ln(1+ √ 3) ln(5) . Although the solution for APL is approximate,it is trusted because we have calculated all items of APL accurately except for the compensation( ∆t) of total distances between non-adjacent branches( Λ 1,3 t ) ,which is obtained approximately by least-squares curve fitting.The compensation( ∆t) is only a small part of total distances between non-adjacent branches( Λ 1,3 t ) and has little effect on APL.Further more,using the data obtained by iteration to test the fitting results,we find the relative error for ∆t is less than 10 −7 ,hence the approximate solution for average path length is almost accurate.

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