图与复杂网络中的通信融合
Communication Melting in Graphs and Complex Networks
摘要：
复杂网络是许多复杂系统中相互作用的代表图。通常，这些交互是系统单元之间的通信/扩散通道的抽象。真实的复杂网络，例如交通网络，揭示了由系统的动态应力控制的不同操作阶段。在这里，我们将展示如何，可通信性，拓扑描述符，揭示了这些扩散路径的网络功能的效率，可以用来揭示上述的过渡。通过考虑在给定的温度（应力）的图形/网络中的节点和边缘的振动模型，我们表明，通信功能起着谐振子网络的热格林函数的作用。之后，我们分析证明了存在一个普遍的相变的通信结构的简单图。这种转变类似于固体中发生的熔化过程。例如，类似于晶体的规则样图在较低温度下熔化，并且比类似于非晶态固体的随机空间图在连接到断开结构之间显示出更尖锐的过渡。最后，我们研究计算这个图熔化过程中的一些现实世界的网络和观察到的速度熔化的图形的变化，无论是作为一个指数或作为幂律与逆温度。在局部水平上，我们发现节点熔化的主要驱动因素是对应节点的特征向量中心性，特别是当逆温度的临界值接近零时。这些普遍的结果揭示了网络上的许多动态扩散过程，这些过程表现为交通堵塞，通信丢失或故障级联。
英文摘要：
Complex networks are the representative graphs of interactions in many complex systems. Usually, these interactions are abstractions of the communication/diffusion channels between the units of the system. Real complex networks, e.g. traffic networks, reveal different operation phases governed by the dynamical stress of the system. Here we show how, communicability, a topological descriptor that reveals the efficiency of the network functionality in terms of these diffusive paths, could be used to reveal the transitions mentioned. By considering a vibrational model of nodes and edges in a graph/network at a given temperature (stress), we show that the communicability function plays the role of the thermal Green's function of a network of harmonic oscillators. After, we prove analytically the existence of a universal phase transition in the communicability structure of every simple graph. This transition resembles the melting process occurring in solids. For instance, regular-like graphs resembling crystals, melts at lower temperatures and display a sharper transition between connected to disconnected structures than the random spatial graphs, which resemble amorphous solids. Finally, we study computationally this graph melting process in some real-world networks and observe that the rate of melting of graphs changes either as an exponential or as a power-law with the inverse temperature. At the local level we discover that the main driver for node melting is the eigenvector centrality of the corresponding node, particularly when the critical value of the inverse temperature approaches zero. These universal results sheds light on many dynamical diffusive-like processes on networks that present transitions as traffic jams, communication lost or failure cascades.
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