In this note, we study discrete time majority dynamics over an inhomogeneous random graph G obtained by including each edge e in the complete graph \(K_n\) independently with probability \(p_n(e)\). Each vertex is independently assigned an initial state \(+1\) (with probability \(p_+\)) or \(-1\) (with probability \(1-p_+\)), updated at each time step following the majority of its neighbors’ states. Under some regularity and density conditions of the edge probability sequence, if \(p_+\) is smaller than a threshold, then G will display a unanimous state \(-1\) asymptotically almost surely, meaning that the probability of reaching consensus tends to one as \(n\rightarrow \infty \). The consensus reaching process has a clear difference in terms of the initial state assignment probability: In a dense random graph \(p_+\) can be near a half, while in a sparse random graph \(p_+\) has to be vanishing. The size of a dynamic monopoly in G is also discussed.

在本注释中，我们研究了不均匀随机图G上的离散时间多数动力学，该随机图G是通过以概率\（p_n（e）\）独立地将每个边e包括在完整图\（K_n \）中而获得的。每个顶点被独立分配一个初始状态\（+ 1 \）（概率为\（p _ + \））或\（-1 \）（概率为（（1-p _ + \））），并在随后的每个时间步进行更新邻国的大多数国家。在边缘概率序列的某些规律性和密度条件下，如果\（p _ + \）小于阈值，则G将显示一致状态\（-1 \）渐近地几乎可以肯定地表示，这意味着达成共识的可能性趋于为\（n \ rightarrow \ infty \）。就初始状态分配概率而言，共识达成过程存在明显差异：在密集的随机图中\（p _ + \）可能接近一半，而在稀疏的随机图中\（p _ + \）必须消失。还讨论了G中动态垄断的规模。