Last modified on 13 May 2011, at 20:18

Transportation Geography and Network Science/Characterizing Graphs

beta indexEdit

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 numberEdit

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 indexEdit

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 indexEdit

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).

CompletenessEdit

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 numberEdit

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.

eta indexEdit

The eta index (\eta) measure the length of the graph over the number of edges.

\eta=\frac{L(G)}{e}

theta indexEdit

The theta index (\theta) measure the traffic (Q(G)) per vertex.

\theta=\frac{Q(G)}{v}

iota indexEdit

The iota index (\iota) measures the ratio between the length of its network and its weighted vertices.

 \iota=\frac{L(G)}{W(G)}

 W(G)=1,\forall o=1

 W(G)=\sum_{e}2*o,\forall o>1

Source: [1]