Gibrat Fluctuation And Optimal Navigation Of The Time-varying Complex Systems

Our ability to predict and control complex systems depends on the proper recognition of their internal organizing mechanisms. Since the discovery of the small-world and scale-free properties which dominates the real world, the complexity science has been rapidly developed and gradually approaches the core principles of the complex systems. For a long time, the dynamic analysis of complex systems is based on some simplified static or quasi-static models, while the fluctuation nature of their topologies is always neglected. In fact, when the time scale of the transport dynamics is comparable to that of the topological fluctuation, both the fundamental metrics such as network connectivity, shortest path, clustering and some transport dynamics such as searching and navigation will go beyond our traditional knowledge. Hence a deep understanding of the fluctuation property and ongoing dynamic behavior of the time-varying complex systems is of great significance in practical aspects such as propagation control and network optimization.In this paper, we focus on the topological fluctuations of the complex systems and the nontrivial influence of the time-ordered topologies on the navigation process. Our main contributions are as follows:1) Empirically analyze the fluctuation properties on different time-scales and theoretically discuss the origin of Gibrat’s law of the Internet.We analyze the microscopic fluctuation properties of the Internet autonomous systems. The non-Gaussian behavior of node degree growth rate distribution and the crossover transition from the PA phase to Gibrat phase with increasing time scales indicate a non-uniform evolution of the Internet. The linear correlation between the fluctuation of the degree increment of a node and that of its neighbors indicates a locally rapid increase in the Internet. The degree fluctuation of the central node induces the degree synchronous updating of its surrounding neighbors. Moreover, the behavior of topological fluctuations depends on the distribution symmetry of the correlation between the fluctuations of node degree increment. When the correlation coefficients are symmetrically distributed, the Internet evolves as the PA model, while the symmetry breaking of the correlation leads to a macroscopic positive correlation and contributing to the emergence of Gibrat’s law. Different to the traditional mean-field description, this framework not only provides a strong theoretical support for understanding the microscopic dynamics of the networked systems, but also points out a new direction for the accurate modeling of the Internet.2) Propose two typical models of time-varying small-world networks and discuss their optimal navigation structures under temporal dimension.The energy-dissipation system and transportation system are two typical kinds of complex systems with both spatial and time-varying properties. Considering the spatial constraint on interaction dynamics in the real networks, we propose two kinds of time-varying spatial small-world networks from the aspect of energy dissipation and activity driven, respectively. By Monte Carlo simulation and Master Equation, we discuss how the time-ordered topologies influence the optimal navigation structure for the first time. Being based on global or local information, the optimal navigation properties greatly deviate from that of the static navigation models. Specially, there is a unique restriction between the global optimal condition and the coupling strength between the temporal and spatial effects in time-varying transportation networks, which unveils a new optimization principle in route design and schedule arrangement.3) Empirically analyze the temporal-spatial distribution properties of the airline systems and propose a flight optimization framework.We construct two time-varying airline networks of different sizes based on the flight departure time of the British Airways and Austria Airlines, respectively. The empirical analysis unveils, for the first time, the temporal-spatial correspondence between the flight frequency and route geometric distances. Applying this feature into the cost-minimized optimization model, we propose an airline optimization framework based on the optimal navigation structure of time-varying spatial networks. This framework can directly predict the overall optimal distribution of flight distances and corresponding flight frequencies only with the information of the passenger flow assignment, thus taking full advantages of the structural convenience. It also provides a new perspective for the overall scheduling of other transportation systems.