稀疏随机图扩散的中心极限定理
A Central Limit Theorem for Diffusion in Sparse Random Graphs
摘要：
研究了由配置模型构造的固定度稀疏随机图的自举渗流和自举扩散。每个节点都有两种状态：活动或非活动。我们假设为每个节点分配一个非负（整数）阈值。扩散过程由阈值为零的节点子集发起，该节点子集由初始激活的节点组成，而每个其他节点都是不活动的。随后，在每一轮中，如果具有阈值$θ$的不活动节点至少激活了其邻居中的$θ$，则它也变为活动的并且永远保持活动。重复此操作，直到不再激活节点。本文的主要结果为激活节点的最终规模提供了一个中心极限定理。也就是说，在适当的程度和阈值分布的假设下，我们证明了激活节点的最终大小具有渐近高斯波动。
英文摘要：
We consider bootstrap percolation and diffusion in sparse random graphs with fixed degrees, constructed by configuration model. Every node has two states: it is either active or inactive. We assume that to each node is assigned a nonnegative (integer) threshold. The diffusion process is initiated by a subset of nodes with threshold zero which consists of initially activated nodes, whereas every other node is inactive. Subsequently, in each round, if an inactive node with threshold $θ$ has at least $θ$ of its neighbours activated, then it also becomes active and remains so forever. This is repeated until no more nodes become activated. The main result of this paper provides a central limit theorem for the final size of activated nodes. Namely, under suitable assumptions on the degree and threshold distributions, we show that the final size of activated nodes has asymptotically Gaussian fluctuations.
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