# A Method to Optimize Routing Paths for City-Pair Airlines on Three-Layer Air Transport Networks
## 1 Introduction
### Related Work  
As mentioned in [7], there are many previous works focused on the modelling of air transportation systems. A pioneering study [8] presents the China air network with graph theory and analyses its part features of small worlds and of scale-free networks. Based on this initial study, many researchers [9; 10] have applied weighted networks [11] or complex networks [12] to analyse existing air transportation systems, where each node represents one airport (or a city), and an edge exists if there is a flight between two airports. In [13], the authors exhaustively analysed the worldwide airport network and found out that it should be a small-world network with a power-law degree distribution [14]. [PERSON] et al. [15] depicted three different layers of the world airline network based on the k-core decomposition algorithm. [PERSON] [16] described the Airport Network of India as a complex weighted network, which is a small-world network with a truncated power-law degree distribution and has a signature of hierarchy. In [17], the authors investigated the structure and evolution of the Brazilian Airport in terms of four aspects: routes, connections, passengers, and cargo. [PERSON] [18] presented an exploratory analysis of the evolution of the US airport network from 1990 to 2010. In [19], the multi-layer structure of the European airport network, composed of connections and flights between European cities, was decomposed using the k-core decomposition of the connections network. In [8], the authors showed that the topological structure of the Chinese airport network shares two feature of small worlds, a short average path length and a high degree of clustering. [PERSON] et. al [20] analysed the evolution of Chinese airport network via complex network theory and found out that the topology remains steady through the past several years, but the cargo traffic and passengers have been growing in an exponential rate. [PERSON] et.al [21] proposed a k-core decomposition method to decompose the Chinese airline network into three layers, core layer, bridge layer, and periphery layer, which revealed the unique demographic and geographical properties of China. In [22], the authors carried out an empirical structural analysis of the Chinese airline network with complex network theory. [PERSON] et al. [23] made comparisons between airport networks of Australia, Brazil, Canada, China, India, Russia, US, and Europe in the network properties and found that all of eight networks clearly share small-world properties. However, the airport network shows only one perspective of the air transportation systems. From another perspective, it is necessary to consider where aircraft actually fly through the airspace. As the backbone of the air transport network, the air route network has been extensively studied as well. [PERSON] et al. [24] studied the Chinese air route network under the framework of complex networks and made some comparisons between the basic network features of the air route network and the airline network. The authors illustrated that the topological structure of the air route network is more homogeneous than that of the airport network, while the distribution of the flight flows in the air route network is rather heterogeneous. In their following work [25], the authors proposed a novel Memetic Algorithm for air route networks design. [PERSON] et al. [26] proposed a memetic algorithm to explore the robustness of the Chinese air route network, which identified the vital edges that form the backbone of Chinese air transportation system. [PERSON] et al. [27] analysed the air navigation route system from fifteen different countries in terms of five metrics using complex network theory and found that air navigation route networks for all fifteen countries are rather heterogeneous. [PERSON] et al. [28] analysed the European air transportation system using the two network layers, including the air route network and the airport network.