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P_g(k)=\sum_{k'\ge k}^{k_c}P(k')\dbinom{k'}{k}(1-g)^kg^{k'-k}
P_g(k)=\sum_{k'\ge k}^{k_c}P(k')\dbinom{k'}{k}(1-g)^kg^{k'-k}
\end{equation}
\end{equation}
Equation (64) can be readily solved in the case of scale-free networks. For a degree exponent $\gamma=3$, the immunization threshold reads 值$g_c(\lambda)\simeq exp[-2/(m\lambda)]$, where $m$ is the mini�mum degree in the network. This result highlights the convenience of targeted immunization, with an immunization threshold that is exponentially small over a large range of the spreading rate $\lambda$. A similar effect can be obtained with a proportional immunization strategy <ref name="Pastor2002b"></ref> [see also Dezsö and Barabási (2002)<ref>Dezső Z, Barabási A L. Halting viruses in scale-free networks[J]. Physical Review E, 2002, 65(5): 055103.</ref> for a similar approach involving the cure of infected individuals with a rate proportional to their degree], in which nodes of degree $k$ are immunized with probability $g_k$, which is some increasing function of $k$. In this case, the infection is eradicated when $g_k\ge1-1/(\lambda k)$, leading to an immunization threshold <ref name="Pastor2002b"></ref>
Equation (64) can be readily solved in the case of scale-free networks. For a degree exponent $\gamma=3$, the immunization threshold reads 值$g_c(\lambda)\simeq exp[-2/(m\lambda)]$, where $m$ is the minimum degree in the network. This result highlights the convenience of targeted immunization, with an immunization threshold that is exponentially small over a large range of the spreading rate $\lambda$. A similar effect can be obtained with a proportional immunization strategy <ref name="Pastor2002b"></ref> [see also Dezsö and Barabási (2002)<ref>Dezső Z, Barabási A L. Halting viruses in scale-free networks[J]. Physical Review E, 2002, 65(5): 055103.</ref> for a similar approach involving the cure of infected individuals with a rate proportional to their degree], in which nodes of degree $k$ are immunized with probability $g_k$, which is some increasing function of $k$. In this case, the infection is eradicated when $g_k\ge1-1/(\lambda k)$, leading to an immunization threshold <ref name="Pastor2002b"></ref>
在无标度网络的情况下，上述免疫阈值的隐式方程很容易求解。当网络度分布指数$\gamma=3$时，免疫阈值$g_c(\lambda)\simeq exp[-2/(m\lambda)]$，其中$m$是网络中最小的度值。该结果突显出了目标免疫的便利性，其免疫阈值在传播速率$\lambda$的较大范围内呈指数减小。此外，采用比例免疫的策略也可以获得类似的效果（Pastor-Satorras和Vespignani，2002b）[另见Dezsö和Barabási（2002年）的相似方法，即以感染者的度值为偏好来治愈感染者的方案]， 其中度值为$k$的节点以概率$g_k$被免疫，$g_k$是$k$的某种递增函数。在这种情况下，当$g_k\ge1-1/(\lambda k)$时，感染被完全消除，从而达到了免疫阈值（Pastor-Satorras and Vespignani，2002b）
在无标度网络的情况下，上述免疫阈值的隐式方程很容易求解。当网络度分布指数$\gamma=3$时，免疫阈值$g_c(\lambda)\simeq exp[-2/(m\lambda)]$，其中$m$是网络中最小的度值。该结果突显出了目标免疫的便利性，其免疫阈值在传播速率$\lambda$的较大范围内呈指数减小。此外，采用比例免疫的策略也可以获得类似的效果（Pastor-Satorras和Vespignani，2002b）[另见Dezsö和Barabási（2002年）的相似方法，即以感染者的度值为偏好来治愈感染者的方案]， 其中度值为$k$的节点以概率$g_k$被免疫，$g_k$是$k$的某种递增函数。在这种情况下，当$g_k\ge1-1/(\lambda k)$时，感染被完全消除，从而达到了免疫阈值（Pastor-Satorras and Vespignani，2002b）