Smilkov Daniel, Kocarev Ljupco
Macedonian Academy for Sciences and Arts, Skopje, Macedonia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Jan;85(1 Pt 2):016114. doi: 10.1103/PhysRevE.85.016114. Epub 2012 Jan 24.
The influence of the network's structure on the dynamics of spreading processes has been extensively studied in the last decade. Important results that partially answer this question show a weak connection between the macroscopic behavior of these processes and specific structural properties in the network, such as the largest eigenvalue of a topology related matrix. However, little is known about the direct influence of the network topology on the microscopic level, such as the influence of the (neighboring) network on the probability of a particular node's infection. To answer this question, we derive both an upper and a lower bound for the probability that a particular node is infective in a susceptible-infective-susceptible model for two cases of spreading processes: reactive and contact processes. The bounds are derived by considering the n-hop neighborhood of the node; the bounds are tighter as one uses a larger n-hop neighborhood to calculate them. Consequently, using local information for different neighborhood sizes, we assess the extent to which the topology influences the spreading process, thus providing also a strong macroscopic connection between the former and the latter. Our findings are complemented by numerical results for a real-world email network. A very good estimate for the infection density ρ is obtained using only two-hop neighborhoods, which account for 0.4% of the entire network topology on average.
在过去十年中，网络结构对传播过程动态的影响已得到广泛研究。部分回答该问题的重要结果表明，这些过程的宏观行为与网络中的特定结构属性之间存在微弱联系，例如与拓扑相关矩阵的最大特征值。然而，关于网络拓扑在微观层面的直接影响，例如（相邻）网络对特定节点感染概率的影响，人们了解甚少。为回答这个问题，对于两种传播过程情形：反应过程和接触过程，我们在易感-感染-易感模型中推导了特定节点被感染概率的上下界。这些界是通过考虑节点的n跳邻域得出的；使用更大的n跳邻域来计算时，界会更紧密。因此，利用不同邻域大小的局部信息，我们评估了拓扑对传播过程的影响程度，从而也在前者和后者之间建立了紧密的宏观联系。我们的发现通过一个真实世界电子邮件网络的数值结果得到补充。仅使用两跳邻域就能获得对感染密度ρ的非常好的估计，两跳邻域平均占整个网络拓扑的0.4%。
