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The structure and properties of public transportation networks have great implications in urban planning, public policies, and infectious disease control. This study contributes a weighted complex network analysis of travel routes on the national highway network of Pakistan. The network is responsible for handling 75 percent of the road traffic yet is largely inadequate, poor, and unreliable. The highway network displays small world properties and is assortative in nature. Based on the betweenness centrality of the nodes, the most important cities are identified as this could help in identifying the potential congestion points in the network. Keeping in view the strategic location of Pakistan, such a study is of practical importance and could provide opportunities for policy makers to improve the performance of the highway network.

With the expansion of world economy and international trade, the role of transportation infrastructure is of crucial importance to the development of a country and is an important indicator of its economic growth. Land, water, and air transportation forms the backbone of an economy by supporting the movement of information, people, and goods. Previously, econometric models were applied to estimate the effect of infrastructure on the international trade; however, such a study did not provide sufficient insight into infrastructure itself. It is during the recent few years that different transportation systems are studied in much more detail using the complex network analysis. Key studies include but are not limited to the US, China, India, and worldwide air networks [

Pakistan lies at the crossroads between South Asia, Central Asia, and Western Asia. The location provides Pakistan with a valuable opportunity to enhance its economy by providing logistic routes and services to the landlocked central Asian countries and to act as a bridge. On the contrary, the logistic network of Pakistan is largely inadequate with total length of the national highways standing at roughly 8,780 kilometers, accounting for only 3 percent of the entire road network but handling 75 percent of the road traffic in the country [

The rest of the paper is as follows. Section

Data for the movement of passengers were provided by the National Transport Research Center (NTRC) and the Provincial Public Transport Authorities of Pakistan. The data was transformed and travel for each day was represented as a weighted graph

Table

Statistical properties of Pakistan national highway network.

Property | Value |
---|---|

Nodes, |
266 |

Edges, |
4802 |

Average path length, |
2.49 |

Average clustering coefficient, |
0.81 |

Average weighted clustering coefficient, |
0.82 |

Diameter, |
5 |

Average degree | 36.1 |

Degree range | (1, 109) |

Average weight, |
515.7 |

Weight range | (110, 1175) |

Average strength, |
18,617.9 |

Strength range | (150, 69,996) |

Assortativity, |
0.48 |

The network has 266 nodes with 4802 edges. The average shortest path length (the minimum number of edges passed through to get from one node to another) between one node and all other nodes of the network is calculated using the following equation:

Weight distribution.

The degree of a node, a measure of its connectivity, is defined as the fraction of nodes with degree

The average degree of the whole graph can be obtained using the following equation:

Subsequently, a node’s strength is simply the sum of the weights on the edges incident upon it and is given by

The average strength of the whole graph can then be obtained using the following equation:

The network possesses a high average degree of 36.1, indicating high connectivity among the nodes. The degree distribution of the network is presented in Figure

Degree distribution.

Strength as a function of degree.

The clustering coefficient of a node

Using the above equation, the average clustering coefficient (

Unlike the clustering coefficient, the weighted clustering coefficient

The average weighted clustering coefficient can thus be represented by the following mathematical expression:

If the weighted clustering coefficient is equal to the clustering coefficient of the network, (

Clustering spectrum.

Another important topological characteristic of a network that is examined is the degree-degree correlation between connected nodes. A given network is said to be assortative if the high-degree nodes have a tendency to connect to other high-degree nodes. Similarly, disassortative networks are where low-degree nodes tend to connect to high-degree nodes. Newman introduced a summary statistic for assortativity (

This statistic lies in between the range of

If

Average degree of nearest neighbors of nodes with degree

The betweenness centrality measure is used to identify the nodes with high congestion [

Top ten cities identified based on betweenness centrality.

Betweenness centrality | City |
---|---|

0.089434 | Quetta |

0.070581 | Peshawar |

0.067116 | Karachi |

0.052508 | Naushahro Feroze |

0.049318 | Hoshab |

0.046812 | Mastung |

0.044426 | Hyderabad |

0.038023 | Loralai |

0.037758 | Sibi |

0.034866 | Jamshoro |

Betweenness centrality.

Transportation networks, whether being land, air, or sea, communicate the development level of a country and can rightly be described as forming the backbone of economic development. Along with other tools, complex network methodologies have been extensively used to analyze transportation networks. As an addition to the theory and application of complex networks, the weighted national highway network of Pakistan is analyzed using complex network theory. The PNHN is a highly clustered network where the degree distribution is neither normal nor power law. The small world properties and assortative mixing of the highway network are evident from the calculated properties. It is interesting to note that the topological properties of the PNHN are largely similar to those of the Indian highway network. Furthermore, using betweenness centrality, the cities with potential traffic congestion are also identified.

The beauty of complex network theory is that it is a powerful tool with limitless application possibilities. Although the analysis was performed taking daily average number of passengers as edge weights, it would also be interesting to conduct a much larger study using data for weeks or months. Similarly, subject to availability of data, weighted network analysis using movement of traffic or better yet the flow of goods in terms of TEU (twenty foot equivalent unit) from one city/district to other could provide useful insight into the logistics aspect of the network. Such a study would clearly highlight the network and its topological features in much more detail and help the policy makers to further enhance the infrastructure to achieve efficient flow.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research is supported by the 2011 Founded Project of the National Natural Science Foundation of China (71171084), the 2011 Research Fund for the Doctoral Program of Higher Education of China (20110172110010), and the Fundamental Research Funds for the Central Universities (2012, x2gsD2117850).