# Transportation Geography and Network Science/Characterizing Graphs

## beta index

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

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

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

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

## cyclomatic number

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

${\displaystyle u=e-v+p}$

where p = number of graphs or subgraphs.

## alpha index

The alpha index (${\displaystyle \alpha }$ ) 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:

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

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

## gamma index

The gamma index (${\displaystyle \gamma }$ ) 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 (${\displaystyle 3(v-2)}$ )

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

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.

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

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

## eta index

The eta index (${\displaystyle \eta }$ ) measure the length of the graph over the number of edges.

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

## theta index

The theta index (${\displaystyle \theta }$ ) measure the traffic (Q(G)) per vertex.

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

## iota index

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

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

${\displaystyle W(G)=1,\forall o=1}$

${\displaystyle W(G)=\sum _{e}2*o,\forall o>1}$

Source: [1]