We study the resilience of complex networks against attacks in which nodes are targeted intelligently, but where disabling a node has a cost to the attacker which depends on its degree. Attackers have to meet these costs with limited resources, which constrains their actions. A network's integrity is quantified in terms of the efficacy of the process that it supports. We calculate how the optimal attack strategy and the most attack-resistant network degree statistics depend on the node removal cost function and the attack resources. The resilience of networks against intelligent attacks is found to depend strongly on the node removal cost function faced by the attacker. In particular, if node removal costs increase sufficiently fast with the node degree, power law networks are found to be more resilient than Poissonian ones, even against optimized intelligent attacks. For cost functions increasing quadratically in the node degrees, intelligent attackers cannot damage the network more than random damages would.

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Recent observations of two-photon absorption within the configuration 4f7 of Gd3+ in LaFs have indicated an anomalously high line strength for the transition 8S7,2+ 6P7,2 compared with .+ 6 P~,~ and *S7;2 -+ 6P3i2. It is shown that the standard expression for the two-photon intensity is unusually small for all three transitions and that a higher-order correction, involving matrix elements of the spin-orbit interaction within the intermediate configuration 4f65d, predominates. The principal term is scalar, implying (in the absence of J mixing by the crystal field) isotropic contributions to the various components of the AJ = 0 transition. The line strengths that have been observed can be accounted for reasonably well without having to invoke variations in the radial integral (4f lr15d) or inadequacies in the closure approximation. Expressions for the transition matrix elements are given for the general configuration iN.