Efficient network immunization under limited knowledge

Abstract

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 full assessment of the optimal nodes to immunize (or quarantine) in order to avoid epidemic spreading such as that of the current 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 an immunization with full knowledge. We develop an analytical framework for our method and determine the critical percolation 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)| and n as |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 non-zero when n increases from n = 1 to O(log N) (where N is the size of network). Thus, for SF networks, having knowledge of ≈log N nodes and immunizing them can dramatically 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.

Immunization of networks with limited knowledge and temporary immunity

“…Following the limited knowledge setting 10 , we assume n (1 ≤ n ≤ N) random nodes are observed at each time step t and the one with highest degree is immunized (i.e. removed) from the network.…”

“…The above process continues until a fraction of 1 − p nodes are immunized (regardless of whether it will be recovered) from G(V, E). Obviously, the case ρ = 0 suggests perfect immunity 10,15 and the other extremal case of ρ = 1 is trivial as no nodes will be removed and the network remains intact.…”

“…Moreover, assuming complete network data is uneconomical and even impractical for dealing with large-scale networks in many reallife situations. A recent work 10 has developed a percolationbased theoretical framework for targeted immunization under limited knowledge, where the degree information of a number n nodes is obtained at each step and the node with highest degree of them is immunized, i.e. removed.…”

“…Similar partiallyobserved networks have also been studied for problems like targeted immunization 11,12 , attack robustness 8,13 , and influence maximization 14 . These works, however, are from the perspective of blinding a portion of nodes through random or some specific sampling schemes, which is related but not equivalent to the limited knowledge notion studied in 10 . A mirror process with tunable limited knowledge is investigated very recently in 15 .…”

“…Here, we look into the interplay and influence of partially observed network and temporary immunity on networks motivated by the above consideration. The paper presents a concise analytical percolation theory for targeted immunization, where the degrees of n nodes are obtained at each step and we immunize the one with highest degree with a temporary immunity extending the framework of 10 . Both fixed immunity fading rate ρ and asymptotic complete loss rule ρ(t) are analyzed for networks with arbitrary degree distributions.…”