Efficient network immunization under limited knowledge

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Targeted immunization of centralized nodes in large-scale networks has attracted significant attention.However,in real-world scenarios,knowledge and observations of the network may be limited,thereby precluding a fall assessment of the optimal nodes to immunize(or quarantine)in order to avoid epidemic spreading such as that of the current coronavirus disease(COVID-19)epidemic.Here,we study a novel immunization strategy where only n nodes are observed at a time and the most central among these n nodes is immunized.This process can globally immunize a network.We find that even for small n(≈10)there is significant improvement in the immunization(quarantine),which is very close to the levels of immunization with full knowledge.We develop an analytical framework for our method and determine the critical perco-lation threshold pc and the size of the giant component P∞ for networks with arbitrary degree distributions P(k).In the limit of n → ∞ we recover prior work on targeted immunization,whereas for n = 1 we recover the known case of random immunization.Between these two extremes,we observe that,as n increases,pc increases quickly towards its optimal value under targeted immunization with complete information.In particular,we find a new general scaling relationship between |pc(∞)-pc(n)| andnas |pc(∞)-pc(n)|~n-1exp(-αn).For scale-free(SF)networks,where P(k)~k-γ,2<γ<3,we find that pc has a transition from zero to nonzero when n increases from n = 1 to O(log N)(where N is the size of the network).Thus,for SF networks,having knowledge of ≈log N nodes and immunizing the most optimal among them can dra-matically reduce epidemic spreading.We also demonstrate our limited knowledge immunization strategy on several real-world networks and confirm that in these real networks,pc increases significantly even for small n.
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