Lambiotte R, Sinatra R, Delvenne J-C, Evans T S, Barahona M, Latora V
Department of Mathematics, Imperial College London, London, United Kingdom.
Phys Rev E Stat Nonlin Soft Matter Phys. 2011 Jul;84(1 Pt 2):017102. doi: 10.1103/PhysRevE.84.017102. Epub 2011 Jul 25.
The behavior of complex systems is determined not only by the topological organization of their interconnections but also by the dynamical processes taking place among their constituents. A faithful modeling of the dynamics is essential because different dynamical processes may be affected very differently by network topology. A full characterization of such systems thus requires a formalization that encompasses both aspects simultaneously, rather than relying only on the topological adjacency matrix. To achieve this, we introduce the concept of flow graphs, namely weighted networks where dynamical flows are embedded into the link weights. Flow graphs provide an integrated representation of the structure and dynamics of the system, which can then be analyzed with standard tools from network theory. Conversely, a structural network feature of our choice can also be used as the basis for the construction of a flow graph that will then encompass a dynamics biased by such a feature. We illustrate the ideas by focusing on the mathematical properties of generic linear processes on complex networks that can be represented as biased random walks and their dual consensus dynamics, and show how our framework improves our understanding of these processes.
复杂系统的行为不仅取决于其互连的拓扑结构，还取决于其组成部分之间发生的动态过程。对动力学进行忠实建模至关重要，因为不同的动态过程可能会受到网络拓扑的影响非常不同。因此，对这类系统的全面表征需要一种同时涵盖这两个方面的形式化方法，而不是仅依赖于拓扑邻接矩阵。为了实现这一点，我们引入了流图的概念，即动态流嵌入到链路权重中的加权网络。流图提供了系统结构和动力学的综合表示，然后可以使用网络理论的标准工具进行分析。相反，我们选择的结构网络特征也可以用作构建流图的基础，该流图随后将包含受此类特征影响的动力学。我们通过关注复杂网络上通用线性过程的数学性质来说明这些想法，这些过程可以表示为有偏随机游走及其对偶共识动力学，并展示我们的框架如何提高我们对这些过程的理解。
