# Transportation Geography and Network Science/Connectivity

Networks have a variety of connection patterns Strategic links (arterials) connect contiguously in most road networks. The geometric patterns of arterials at a collective level play a profound role in shaping traffic pattern and urban landscape. Existing graph theoretic measures of network topology and connectivity do not identify and quantify connection patterns of networks, such as shown in the image on the right.

## Network Structure Measures

${\displaystyle \varphi _{ring}={\frac {Total\ length\ of\ arterials\ on\ rings}{Total\ length\ of\ arterials}}}$

${\displaystyle \varphi _{web}={\frac {Total\ length\ of\ arterials\ on\ webs}{Total\ length\ of\ arterials}}}$

${\displaystyle \varphi _{circuit}=\varphi _{ring}+\varphi _{web}}$

${\displaystyle \varphi _{tree}=1-\varphi _{circuit}}$

${\displaystyle \varphi _{belt}={\frac {Length\ of\ the\ beltway}{Total\ length\ of\ arterials}}}$

## Discontinuity

Discontinuity examines travelers’ perceptions of inconvenience associated with transferring between different levels of roads.

${\displaystyle y_{a}=\left|k_{1}-k_{2}\right|}$

${\displaystyle a}$ = upstream link ${\displaystyle k_{1}}$, ${\displaystyle k_{2}}$ = hierarchies of the upstream link and downstream link

The discontinuity of a trip from R to S:

${\displaystyle Y(P_{RS})=\sum \limits _{a\in P_{RS}}{y_{a}}}$

The discontinuity of a network:

${\displaystyle Y={\frac {\sum \limits _{\forall \left(R,S\right)}{Y\left(P_{RS}\right)}q_{RS}}{\sum \limits _{\forall \left(R,S\right)}{l\left(P_{RS}\right)}q_{RS}}}}$

${\displaystyle P_{RS}}$ = The shortest path between any given O-D pair; ${\displaystyle q_{RS}}$ = O-D trips; ${\displaystyle l(P_{RS})}$ = The length of the shortest path. ${\displaystyle Y(P_{RS})}$= The discontinuity along the shortest path.

The measure of discontinuity evaluates the quality of a road network based on the travelers’ perception, which can be extended to account for many “discontinuous” factors, such as the delay at traffic signals or ramp meters, the toll paid entering a toll road, etc.

These measures of aggregate network properties can be used to provide common yardsticks to compare contemporary network structures, and to trace the structural change of networks over time. They could also become useful guidance for urban planners in the design of collective patterns of urban roads.