论文标题
加权Erdos-Renyi图上局部密度依赖的Markov过程的平均场极限
Mean field limit of local density-dependent Markov processes on weighted Erdos-Renyi graphs
论文作者
论文摘要
我们研究了马尔可夫过程在大型加权Erdos-Renyi图上的渐近行为，其中顶点的过渡速率仅受邻居状态的影响以及边缘上的相应权重。我们发现，处于特定状态的顶点的比率将收敛到从平均场近似值获得的微分方程的解，如果图足够致密，即，平均程度至少属于$ n^{\ frac {\ frac {1} {1} {2} {2} {2}+ε} $。显示了瞬态制度中概率收敛的证明。
We study the asymptotic behaviour of Markov processes on large weighted Erdos-Renyi graphs where the transition rates of the vertices are only influenced by the state of their neighbours and the corresponding weight on the edges. We find the ratio of vertices being in a certain state will converge to the solution of a differential equation obtained from mean field approximation if the graph is dense enough, namely, the average degree is at least of order $N^{\frac{1}{2}+ε}$. Proof for convergence in probability in the transient regime is shown.
