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: