Collective relaxation dynamics of small-world networks
Collective relaxation dynamics of small-world networks
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DOI：
10.1103/physreve.91.052815
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发表时间：
2015-05-27
影响因子：
2.4
通讯作者：
Timme, Marc
中科院分区：
文献类型：
--
作者：
Grabow, Carsten;Grosskinsky, Stefan;Timme, Marc
复杂网络呈现出广泛的集体动力学现象，包括同步、扩散、松弛和协调过程。它们的渐近动力学一般由局部雅可比、图拉普拉斯或类似的线性算子来刻画。具有规则、小世界和随机连通性的网络的结构已经被很好地理解，但它们的集体动力学性质在很大程度上仍然是未知的。在这里，我们提出了一个两阶段平均场理论来推导网络谱的解析表达式。一个单一的公式涵盖了Watts-Stogue atz网络中从规则到小世界到强随机化的拓扑结构，解释了网络大小N、平均度k和拓扑随机性Q的同时依赖性。我们给出了第二大和最小本征值的简化解析预测，数值检验证实了我们对零拓扑随机性Q、小拓扑随机性Q和中等拓扑随机性Q的理论预测，包括整个小世界区域。对于一阶的大Q，我们应用了标准的随机矩阵理论，从而涵盖了从规则到随机的网络拓扑的全部范围。这些结果有助于我们从解析和力学上理解网络动力系统的集体松弛现象。
Complex networks exhibit a wide range of collective dynamic phenomena, including synchronization, diffusion, relaxation, and coordination processes. Their asymptotic dynamics is generically characterized by the local Jacobian, graph Laplacian, or a similar linear operator. The structure of networks with regular, small-world, and random connectivities are reasonably well understood, but their collective dynamical properties remain largely unknown. Here we present a two-stage mean-field theory to derive analytic expressions for network spectra. A single formula covers the spectrum from regular via small-world to strongly randomized topologies in Watts-Strogatz networks, explaining the simultaneous dependencies on network size N, average degree k, and topological randomness q. We present simplified analytic predictions for the second-largest and smallest eigenvalue, and numerical checks confirm our theoretical predictions for zero, small, and moderate topological randomness q, including the entire small-world regime. For large q of the order of one, we apply standard random matrix theory, thereby overarching the full range from regular to randomized network topologies. These results may contribute to our analytic and mechanistic understanding of collective relaxation phenomena of network dynamical systems.