Network Infection Risk Analysis And Immunization Strategy Research Based On The Size Distribution Of Connected Components

Cutting off transmission routes of infectious diseases is an essential method in epi-demic prevention and control.Social activities need to maintain a certain level of connec-tivity.Hence,measures such as immunization,crowd isolation,community control,and traffic restrictions are usually adopted to divide crowd activities into connected areas of different sizes to prevent the spread of infectious diseases.Different epidemic prevention and control methods have disrupted network connectivity to varying degrees and affected the spreading dynamics carried by the contact network.This paper studies the quantifi-cation of network infection risk and the formulation of immunization strategies based on the size distribution of connected components in the network.The main research work and innovations are as follows:（1）Designed an indicator to quantify the risk of network infection:the gener-alized Herfindahl–Hirschman indexThe generalized Herfindahl index（GHI）was designed based on the size distribution of connected components for the network and the number of initial infection sources.This indicator comprehensively considered the limitation of the giant connected components and the spreading process occurring in all connected components,which measures the up-per bound of the expected infection’s prevalence（the fraction of infected nodes）in ran-dom outbreaks through combinatorial mathematical theoretical derivation.Meanwhile,relaxing a non-critical constraint,a simplified calculation method is proposed,signifi-cantly reducing computation time and suitable for large-scale networks.We theoretically proved the monotonicity of GHI and analyzed the influence of the difference between connected components on this indicator.Factors such as the number of connected com-ponents in the network and the number of initial infection sources on the risk of network infection were theoretically analyzed.The minimum number of partitions and the max-imum number of infection sources under the fixed number of connected components re-quired to reduce GHI were analyzed and deduced.（2）Analyze the infection risk of the network in the percolation process based on the generalized Herfindahl–Hirschman indexWe applied network percolation theory to numerically analyze GHI for the network in different percolation processes.Firstly,assuming that each node is initially infected with a certain probability,the number of infection sources satisfies the Bernoulli distri-bution.In this context,the GHI indicator for the network was derived in the form of a conditional probability formula.Meanwhile,the self-consistent equation of this indica-tor was established using the generating function method.The percolation curve of the network infection risk under random immunization and target immunization was theoret-ically analyzed.The theoretical results were compared with the experimental simulation results in several model networks.On one hand,the experimental results were consistent with the theoretical prediction results,verifying the correctness of the theoretical deriva-tion.On the other hand,different types of networks in different percolation processes were analyzed,which laid a theoretical foundation for formulating effective immuniza-tion strategies.（3）An effective network immunization strategy model is constructed based on the generalized Herfindahl–Hirschman indexAn immunization approach based on minimizing the GHI was developed to reduce the infection risk for individuals in the network.On one hand,the effect of this strat-egy is simulated and verified in synthetic and real networks.Experimental results show that this strategy can more effectively decrease the infection’s prevalence compared to high-degree adaptive,collective influence,and optimal dismantling strategies,especially in networks with community characteristics.On the other hand,the sensitivity of this strategy under different epidemiological models and parameters is analyzed.It shows that when the contagious disease parameters meet the infection rateβ∈[0.09,1]or the recovery rateμ∈[0,0.4],the effect of this strategy is better than those strategies.The sensitivity of this strategy to the deviation of prior information is analyzed,and it suggests that our strategy still maintains effective performance even if there is a certain deviation between the?I₀we obtained and the actual value I₀.The above experiments prove that the immunization strategy proposed based on the GHI indicator is suitable for most existing infectious diseases and can be better applied to realistic scenarios containing information noise.