马尔可夫链的连续极限及其在网络建模中的应用
Continuum Limits of Markov Chains with Application to Network Modeling
摘要：
本文研究了一类马氏链的连续极限。这种限制的调查是出于对非常大的网络建模的愿望。证明了在一定条件下，一列马氏链在某种意义下收敛于一个偏微分方程的解。基于这种收敛性，我们用偏微分方程来逼近马尔可夫链，对具有大量分量的网络进行建模。虽然传统的蒙特卡洛模拟非常大的网络实际上是不可行的，偏微分方程可以解决合理的计算开销使用成熟的数学工具。
英文摘要：
In this paper we investigate the continuum limits of a class of Markov chains. The investigation of such limits is motivated by the desire to model very large networks. We show that under some conditions, a sequence of Markov chains converges in some sense to the solution of a partial differential equation. Based on such convergence we approximate Markov chains modeling networks with a large number of components by partial differential equations. While traditional Monte Carlo simulation for very large networks is practically infeasible, partial differential equations can be solved with reasonable computational overhead using well-established mathematical tools.
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