# 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.

## eta index

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

$\eta ={\frac {L(G)}{e}}$ ## theta index

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

$\theta ={\frac {Q(G)}{v}}$ ## iota index

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: