The theory of pattern formation on directed networks
The theory of pattern formation on directed networks
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DOI：
10.1038/ncomms5517
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发表时间：
2014-07-01
影响因子：
16.6
通讯作者：
Fanelli, Duccio
中科院分区：
文献类型：
--
作者：
Asllani, Malbor;Challenger, Joseph D.;Fanelli, Duccio
近年来，网络上的动态过程引起了人们的广泛关注。定义在对称网络上的反应扩散系统中的模式形成理论由于其在广泛的学科中的应用而经常被研究。在这里，我们将理论扩展到有向网络的情况，有向网络存在于许多不同的领域，如神经科学、计算机网络和交通系统。由于网络拉普拉斯的结构，色散关系既有实部又有虚部，这与对称的无向网络的情况不同。由于网络的拓扑结构，齐次不动点可能变得不稳定，从而导致一类新的不稳定，这在无向图上是不能诱导的。线性稳定性分析的结果允许对不稳定区域进行分析跟踪。数值模拟显示了行波或准静态模式，这取决于底层图形的特征。
Dynamical processes on networks have generated widespread interest in recent years. The theory of pattern formation in reaction-diffusion systems defined on symmetric networks has often been investigated, due to its applications in a wide range of disciplines. Here we extend the theory to the case of directed networks, which are found in a number of different fields, such as neuroscience, computer networks and traffic systems. Owing to the structure of the network Laplacian, the dispersion relation has both real and imaginary parts, at variance with the case for a symmetric, undirected network. The homogeneous fixed point can become unstable due to the topology of the network, resulting in a new class of instabilities, which cannot be induced on undirected graphs. Results from a linear stability analysis allow the instability region to be analytically traced. Numerical simulations show travelling waves, or quasi-stationary patterns, depending on the characteristics of the underlying graph.