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A different model and mechanism for overload breakdown in growing networks has been considered in [28]. These authors focus on overloads caused by the growth of the network. Their model assigns the same capacity to every node in the network. In their analysis, when a node is overloaded, the links to that node are removed but the node itself is not removed and can be reconnected in the future. Their conclusion is that, to avoid overloads, the capacity must grow with the size of the network. Our model is different from the model in [28] as we assume the capacity to be node-dependent and the failed nodes to be permanently removed from the network. More importantly, we address the issues of intentional attack and random breakdown, and we study how the network collapses under overload failures induced by them. We assume that the time-scale for these events is much smaller than the time-scale in which the network grows
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note
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We choose to sort the links according to the initial distribution of ranges, instead of an updated distribution, because we want to address the relative importance of short-range and long-range links for the original network. In addition, in terms of attack efficiency, updating is time consuming
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Indeed, the range of a link can be regarded as the length of the second shortest path between the nodes that are connected to the link. Since we are considering the semi-random model, for which everything other than the connectivity distribution is random, the length of the second shortest path should also be correlated with the product of connectivities
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