CONNECTIVITY “The connectivity of a network may be defined as the degree of completeness of the links between nodes” (Robinson and Bamford, 1978).

 Transport is clearly a factor of fundamental importance in all- economic activities and the cost of transport is one of the most significant variables affecting the market price of any commodity.  Economic development of a region may be said to require a certain level of transport provision in order to maximize its development potential.  In macro economic benefits of infrastructural investment, Keynes pointed out that the effective demand is the tool to stimulate the employment. WHAT IS CONNECTIVITY? o Connectivity can be defined as the existence of a connection or link between two points of network, which relates to the existence of direct or indirect links. o The concept of inter-connectivity, however, varies from connectivity. o Connectivity is a property of two nodes, while, inter-connectivity is a property of at least three nodes of a link.

 The structure of transportation networks, developed several descriptive indices for measuring the connectivity of networks, i.e., beta, gamma, alpha indices and cyclomatic number. Network analysis is an important aspect of transport geography because it involves the description of the disposition of nodes and their relationships and line or linkage of distribution. When a network is abstracted as a set of edges that are related to set of vertices (nodes), a fundamental questions is the degree to which all pairs of vertices are interconnected.  When examining a transportation network, a geographer is also interested in node-linkages association in terms of accessibility. The structure of a network, changes in response to the addition of new linkages or the improvement of existing linkages. The measurement of connectivity is based on graph theory. Many of them were initially developed by Kansky and can be used for:  Expressing the relationship between values and the network structures they represent.Comparing different transportation networks at a specific point in time.Comparing the evolution of a transport network at different points in time.

Outside the description of the network size by the number of nodes and edges, and its total length and traffic, several measures are used to define the structural attributes of a graph; the diameter, the number of cycles and the order of a node. Diameter (d) The length of the shortest path between the most distanced nodes of a graph is the diameter. d measures the extent of a graph and the topological length between two nodes. The diameter enables to measure the development of a network in time. The higher diameter, the less linked a network tends to be. In the case of a complex graph, the diameter can be found with a topological distance matrix (Shamble distance), which computes for each node pair its minimal topological distance. Graphs which extent remains constant, but with a higher connectivity, have lower diameter values. Planar networks often have a large diameter due to the presence of many intermediate stops between two distant nodes.

Number of Cycles (u). The maximum number of independent cycles in a graph. This number (u) is estimated through the number of nodes (v), links (e) and of sub-graphs (p). Trees and simple networks have a value of 0 since they have no cycles. The more complex a network is, the higher the value of u, so it can be used as an indicator of the level of development and complexity of a transport system.

A measure of the efficiency of a transport network in terms of how well it overcomes distance or the friction of distance. The closer the detour index gets to 1, the more the network is spatially efficient. Networks having a detour index of 1 are rarely, if ever, seen and most networks would fit on an asymptotic curve getting close to 1, but never reaching it. For instance, the straight distance, D(S), between two nodes may be 40 km but the transport distance, D(T); real distance, is 50 km. The detour index is thus 0.8 (40 / 50). The complexity of the topography is often a good indicator of the level of detour. In order to derive a measure of relative efficiency, the Detour Index Relative Efficiency(Erel) is thus the ratio between the Detour Index calculated from the original network and the Detour Index calculated either from the MST (minimum spanning tree) or the GT (greedy triangulation). DETOUR INDEX.

Network Density Measures the territorial occupation of a transport network in terms of km of links (L) per square kilometers of surface (S). The higher it is, the more a network is developed.

Pi Index The relationship between the total length of the graph L(G) and the distance along its diameter D(d). It is labeled as Pi because of its similarity with the real Pi value (3.14), which is expressing the ratio between the circumference and the diameter of a circle. A high index shows a developed network. It is a measure of distance per units of diameter and an indicator of the shape of a network. Pi Index L(G)D(d)Pi 130 km75 km km75 km3.266

ETA INDEX Average length per link. Adding new nodes will cause a decrease of Eta as the average length per link declines.

THETA INDEX Measures the function of a node, which is the average amount of traffic per intersection. The higher theta is, the greater the load of the network. The measure can also be applied to the number of links (edges).

BETA INDEX Measures the level of connectivity in a graph and is expressed by the relationship between the number of links (e) over the number of nodes (v). Trees and simple networks have Beta value of less than one. A connected network with one cycle has a value of 1. More complex networks have a value greater than 1. Complex networks have a high value of Beta. The rich-club coefficient is the Beta index applied to relations among larger order (degree) nodes; it verifies whether the connectivity is higher among larger degree nodes than for the whole network.

ALPHA INDEX A measure of connectivity which evaluates the number of cycles in a graph in comparison with the maximum number of cycles. The higher the alpha index, the more a network is connected. Trees and simple networks will have a value of 0. A value of 1 indicates a completely connected network. Measures the level of connectivity independently of the number of nodes. It is very rare that a network will have an alpha value of 1, because this would imply very serious redundancies. This index is also called Meshedness Coefficient in the literature on planar networks. (for planar graphs) (for non-planar graphs)

GAMMA INDEX (G) A measure of connectivity that considers the relationship between the number of observed links and the number of possible links. The value of gamma is between 0 and 1 where a value of 1 indicates a completely connected network and would be extremely unlikely in reality. Gamma is an efficient value to measure the progression of a network in time. (for planar graphs) (for non-planar graphs)

AVERAGE SHORTEST PATH LENGTH (S) A measure of efficiency that is the average number of stops needed to reach two distant nodes in the graph. The lower the result, the more efficient the network in providing ease of circulation. In comparison, the diameter is the maximum length of all possible shortest paths. SHIMBEL INDEX (or Shimbel distance, nodal accessibility, nodality) A measure of accessibility representing the sum of the length of all shortest paths connecting all other nodes in the graph. The inverse measure is also called closeness centrality or distance centrality.

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