Modeling cascading failures in complex networks based on radiate circle

This article introduces a new approach for defining the initial load of a node in a network and investigate how to allocate the initial load so as to maximize the network robustness against cascading failures and minimize the protection cost. Motivated by the radiate circle of a node in a network, we define the initial load of node i in a network to be Li=(ki)α0(∑i1∈Γiki1)α1(∑i2∈Γi1ki2)α2…(∑in∈Γin−1kin)αn with ki and Γi being the degree of node i and the set of its neighboring nodes, respectively, where α0,α1,α2,…, and αn are tunable parameters, governing the strength of the node initial load, and generally n=12D, of which D represents the diameter of a network. According to the definition of the initial load of a node and the local preferential redistribution mechanism of the load of a failed node, we construct a cascading model. We then present exact analytical solutions for the critical threshold βc as a metric of the network robustness, where there should evidently exist some crossover behavior of the system from large scale breakdown to no breakdown. Surprisingly, both analytically and numerically, we find that, when α0+α1+α2+⋯+αn=1, all networks with no degree–degree correlation can reach the strongest robust level against cascading failures and the network robustness has a positive with the average degree of a network. Our findings highlight how to allocate the initial load and construct the network so as to obtain the strongest robustness.