Transportation Geography and Network Science/Characterizing Graphs

beta index

The beta index ( $\beta$ ) measures the connectivity relating the number of edges to the number of nodes. It is given as:

$\beta ={\frac {e}{v}}$

where e = number of edges (links), v = number of vertices (nodes)

The greater the value of $\beta$ , the greater the connectivity. As transport networks develop and become more efficient, the value of $\beta$ should rise.

cyclomatic number

The cyclomatic number ( $u$ ) is the maximum number of independent cycles in a graph.

$u=e-v+p$

where p = number of graphs or subgraphs.

alpha index

The alpha index ( $\alpha$ ) is the ratio of the actual number of circuits in a network to the maximum possible number of circuits in that network. It is given as:

$\alpha ={\frac {u}{2v-5}}$

Values range from 0%—no circuits—to 100%—a completely interconnected network.

gamma index

The gamma index ( $\gamma$ ) measures the connectivity in a network. It is a measure of the ratio of the number of edges in a network to the maximum number possible in a planar network ( $3(v-2)$ )

$\gamma ={\frac {e}{3(v-2)}}$

The index ranges from 0 (no connections between nodes) to 1.0 (the maximum number of connections, with direct links between all the nodes).

Completeness

The number of links in a real world network is typically less than the maximum number of links and the completeness index used here captures this difference. This measure is estimated at the metropolitan level.

$\rho _{complete}={\frac {e}{e_{max}}}={\frac {e}{{v^{2}}-{v}}}$

$e$ refers to the number of links or street segments in the network and $v$ refers to the number of intersections or nodes in the network. Compare with the $\gamma$ index above.

König number

The König number (or associated number) is the number of edges from any node in a network to the furthest node from it. This is a topological measure of distance, in edges rather than in kilometres. A low associated number indicates a high degree of connectivity; the lower the König number, the greater the Centrality of that node.