Impact of Rapid Urbanization on Vulnerability of Land System from Complex Networks View: A Methodological Approach

Table 1

Mathematical definitions of complex network measures and their representation in land systems.

Measure	Symbol	Definition	Meaning in land systems
Node		A basic element that comprises a network, such as a human in social networks, an airport in air-flights networks.	Each land use type in the land system is identified as a node in the network.
Edge (link)		In a directed network, edge denotes the connection status from node and ; = 1 if there is a link pointing from to ; = 0 otherwise.	The conversion from land use type to during a period; = 1 if there exists the conversion from land use type to ; = 0 otherwise.
Weight		In a directed network, weight of the link from node to node ; weight is often given by length, thickness, capacity, load, or strength associated with an edge.	Weight is given by land area converted from land use type to during a period.
Adjacency matrix		A matrix made up by all links between each pair of nodes. For unweighted networks, the adjacency matrix is made up by . For weighted networks, it is formed by . The diagonal of the adjacency matrix contains zeros.	A matrix made up by all land use conversions that take place between each pair of land use types during a period.
Degree		Number of links connected to a node .	Number of land use conversions linked to land use type .
In-degree		Number of ingoing links to node .	Number of land use conversions from other land use types to land use type .
Out-degree		Number of outgoing links from node .	Number of land use conversions from land use type to other land use types.
Average node degree		Average number of links of all nodes in the network.	Average number of all land use conversions in the land system network during a period, an indicator of network “active” degree.
Node strength		In a weighted graph, the node strength denotes total weights linked to node , which integrates the information on the number (degree) and the weights of links incident in node .	The total land area involved in land use conversions related to land use type . This indicator can identify the most “active” land use type that has the largest total area losses and gains
In-strength		Total weights of ingoing links.	The total land area transfer from other land use types to land use type . This indicator can identify the land use type that has the largest area gains.
Out-strength		Total weights of outgoing links.	The total land area transfer from land use type to other land use types. This indicator can identify the land use type that has the largest area losses.
The division of the out-strength and in-strength		This indicator is used to judge whether node has more in-strength or out-strength.	If > 1, land use type is output-dominated; If = 1, the area of land use type is in a dynamic balance status; If < 1, land use type is input- dominated.
Average node strength		Average node strength of all nodes in the network.	Average land area involved in all land use conversions in the land system network during a period.
Shortest path length		Number of edges along the shortest path connecting nodes and .	Minimum sum of conversions that are needed to connect land use type and type ; reflect the internal structural stability of the land system. A smaller value indicates the transitions between land use type and type are easier.
Average shortest path length		Average number of edges it takes to get from one node of the network to another.	Average number of conversions it takes to link one land use type to another in the land system.
Node betweenness		The ratio of number of edges passing through node to connect two nonadjacent nodes (e.g., and ) to the total number of geodesics connecting and in the network. An indicator to measure the importance of a node in the network.	The ratio of links passing through land use type to connect two land use types (e.g., and ) that have no direct conversions, to the total number of links connecting and in the network. An indicator to measure the importance of a land use type in the land system.
Clustering coefficient		In a weighted network, clustering coefficient is the ratio of the number of edges among the neighbors of node to the maximum number of possible edges. It measures the centralization degree of a system, that is, the degree to which nodes in a network tend to cluster together.	It measures whether the network structure of the land system is more clustered.
Load		Load is the initial quantity of some type of information or resource carried by a node in complex systems, such as traffic load in transport network; power load in electric network; and information load in virtual network. It is one type of weight.	Load associated with each land use type is represented by land area.
Maximum load capacity		Each node is also associated with a maximum load that the node can handle.	To maintain the diversity of landscape as well as the important functions provided by different land uses, one land use type cannot undergo unlimited area increase. In reality, the upper limit for land use types is often regulated by land use planning, city planning, and natural resources protection plans.
Topological efficiency		The topological efficiency is defined as the inverse of geodesic distance; it measures the efficiency of information transmission among nodes in a network. The network has larger efficiency if it has smaller geodesics between pairs of nodes.	Land systems with high topological efficiency will be more resilient to larger area shrinkage in key land use types since there are more conversions among land use types in the network. Thus, the attacked land use types with area shrinkages will be more likely to be compensated by the conversions from other land use types, which lowers the negative effects of intentional attacks.