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and Arenas, A., Modeling structure and resilience of the dark network, Phys. and Barabási, A.-L., Error and attack tolerance of complex networks, Nature 406 378–382. J., Networks: An Introduction (Oxford University Press, Oxford, 2010). and Havlin, S., Complex Networks: Structure, Robustness and Function (Cambridge University Press, Cambridge, 2010). This unravels, e.g., a unique crossover phenomenon rooted in heterogeneous networks, which raises a caution that endeavor to promote network-level robustness could backfire when multi-hop tracing is involved. We test our theoretical results on synthetic homogeneous and heterogeneous networks, as well as on a selection of large-scale real-world networks. We develop analytical frameworks based upon generating function formalism and rate equation method, showing for instance continuous phase transition for G ( 2, 1 ) -core and discontinuous phase transition for G ( k, L ) -core with any other combination of k and L. The resulting subgraph is referred to as G ( k, L ) -core, extending the recently proposed G k -core and classical core of a network. Here we introduce the multi-hop generalized core percolation on complex networks, where nodes with degree less than k and their neighbors within L -hop distance are removed progressively from the network. Recent theoretical studies on network robustness have focused primarily on attacks by random selection and global vision, but numerous real-life networks suffer from proximity-based breakdown. This can be achieved by using Lawler’s method of loop erasure. Hence, it remains to transform γ into a γ ′ path with that property. Next, we identify subsets of with y j = y j ′, then the claim follows from the observation that k ≤ 2 # S ≤ 2 c ′ 1 # U ( i ) m. In other words, our results do not depend on the question whether the base stations are scattered at random in the Euclidean plane or are aligned according to a grid that is viewed from a random reference point. For instance, they can be applied to homogeneous Poisson point processes as well as randomly shifted lattices. Since we only assume stationarity and ergodicity, our results are valid under quite weak conditions on the spatial distribution of base stations. Here, r ≥ 0 is some scaling parameter controlling the intensity of base stations. We assume that they are of the form Y = r Y ( 1 ), where Y ( 1 ) is assumed to be a stationary and ergodic point process that is independent of X and has a finite and positive intensity λ ′.
The base stations constitute the second component. They are modeled by a homogeneous Poisson point process X in R d, d ≥ 2 with some intensity λ ∈ ( 0, ∞ ). The first component is formed by network users. It consists of two types of network components. Next, we provide a precise definition of the wireless spatial telecommunication network under consideration.
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