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齐次树和完全图上具有合作机制的接触过程的极限定理

Limit theorems for contact processes with cooperative mechanisms on homogeneous trees and complete graphs

  • 摘要: 分别从底图为齐次树和完全图的情形给出了具有合作机制的接触过程的若干极限定理.首先得出了齐次树下具有合作机制的接触过程在给定点、给定时刻的极限方程的解;并利用微分方程不动点的思想,得到合作机制参数β的临界值;然后更换底图为完全图,研究完全图上具有合作机制的接触过程,得到了在给定时刻维数趋于无穷时的患病点密度;最后回顾了经典机制下的接触过程(即取β=0作为具有合作机制的接触过程的一种特例),并推导出了其患病点数目的极限函数.

     

    Abstract: Several limit theorems for contact processes with cooperation mechanisms were given from the case where the base maps are homogeneous trees and complete graphs. First, the solution of the limit equation of the contact process with the cooperation mechanism was obtained under the homogeneous tree at a given point and a given time. Next, by means of the idea of the fixed point of the differential equation, the critical value of the cooperation mechanism parameter β was obtained. Then, changing the base map to the complete map, the contact process with the cooperation mechanism was studied, and the density of diseased points at a given moment was obtained when the dimensionality tended to infinity. Finally, as a special case of the contact process of the mechanism, the contact process under the classic mechanism (that is β=0) was reviewed, and the limit function of the number of diseased points was derived.

     

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