无标度图和Erdens-Rényi随机图中的可逆级联
On reversible cascades in scale-free and Erdős-Rényi random graphs
摘要：
考虑以下在具有直径$Δ$的简单无向图$G(V,E)$上的级联过程。在第0轮中，顶点的集合$S\subseteq V$（称为种子）处于活动状态。在循环$i+1,$ $i\in\mathbb{N},$中，如果在循环$i$中至少一部分相邻顶点的$ρ\in(\,0,1\,]$是活动的，则非孤立顶点被激活;否则它被去激活。对于$k\in\mathbb{N},$，令$\text{min-seed}^{(k)}(G,ρ)$为在舍入$k$中或之前激活所有顶点所需的最小种子数。本文导出了$\text{min-seed}^{(k)}(G,ρ)$的上界。特别地，如果$G$是连通的，并且存在常数$C>0$和$γ>2$，使得对于所有$k\in\mathbb{Z}^+,$，$G$中的度- $k$顶点的分数至多为$C/k^γ$，则$\text{min-seed}^{(Δ)}(G,ρ)=O(\lceilρ^{γ-1}\,|\,V\,|\rceil)$。此外，对于$n\in\mathbb{Z}^+,$ $p=Ω((\ln{(e/ρ)})/(ρn))$和概率$1-\exp{(-n^{Ω(1)})}$在Erdens-Rényi随机图$G(n,p),$ $\text{min-seed}^{(1)}(G(n,p),ρ)=O(ρn)$。
英文摘要：
Consider the following cascading process on a simple undirected graph $G(V,E)$ with diameter $Δ$. In round zero, a set $S\subseteq V$ of vertices, called the seeds, are active. In round $i+1,$ $i\in\mathbb{N},$ a non-isolated vertex is activated if at least a $ρ\in(\,0,1\,]$ fraction of its neighbors are active in round $i$; it is deactivated otherwise. For $k\in\mathbb{N},$ let $\text{min-seed}^{(k)}(G,ρ)$ be the minimum number of seeds needed to activate all vertices in or before round $k$. This paper derives upper bounds on $\text{min-seed}^{(k)}(G,ρ)$. In particular, if $G$ is connected and there exist constants $C>0$ and $γ>2$ such that the fraction of degree-$k$ vertices in $G$ is at most $C/k^γ$ for all $k\in\mathbb{Z}^+,$ then $\text{min-seed}^{(Δ)}(G,ρ)=O(\lceilρ^{γ-1}\,|\,V\,|\rceil)$. Furthermore, for $n\in\mathbb{Z}^+,$ $p=Ω((\ln{(e/ρ)})/(ρn))$ and with probability $1-\exp{(-n^{Ω(1)})}$ over the Erdős-Rényi random graphs $G(n,p),$ $\text{min-seed}^{(1)}(G(n,p),ρ)=O(ρn)$.
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