# Transportation Geography and Network Science/Hierarchy

Random vs. Scale Free Network

Heterogeneity indicates the differentiation of components (nodes or links) of a network with regard to their importance to the network. Network scientists need to recognize the link-centric nature of road networks.

A random network may have a bell-curve distribution of node degree, while a scale-free network will have a power-law distribution.

## Measures of Heterogeneity

Road networks are heterogeneous, considering the differentiated functional designs and operational performance of hierarchical roads. The entropy measure of heterogeneity:

$H(X)=-\sum\limits_{i=1}^{m}{p_{i}\log _{2}(p_{i})}$

Where $m$ is the number of subsets in the system $X$, and $p_i$ is the proportion of agents in the $i^{th}$ subset.

Individual links can be grouped into subsets based on different road properties such as functional type, traffic volume, and level of service.

The hierarchy of roads categorizes roads according to their functions and capacities. While sources differ on the exact nomenclature, the basic hierarchy comprises freeways, arterials, collectors, and local roads.

### Freeways

At the top of the hierarchy are limited access roads freeways or motorways, including most toll roads. These roads provide largely uninterrupted travel, often using partial or full access control, and are designed for high speeds. Some freeways have collector/distributor lanes (also known as local lanes) which further reduce the number of access ramps that directly interface with the freeway; rather, the freeway periodically interfaces with these parallel roadways, which themselves have multiple on and off-ramps. These allow the freeway to operate with less friction at an even higher speed and with higher flow. Often freeways are included in the next category, arterials.

### Arterials

Arterials are major through roads that are expected to carry large volumes of traffic. Arterials are often divided into major and minor arterials, and rural and urban arterials.

In some places there are large divided roads with few or no driveways that cannot be called freeways because they have occasional at-grade intersections with traffic lights that stop traffic (expressways in California, dual carriageways in UK ) or they are just too short (superarterials in Nevada). Such roads are usually classified as arterials.

Frontage roads are often used to reduce the conflict between the high-speed nature of an arterial and property access concerns.

### Collectors

Collectors (not to be confused with collector/distributor roads, which reduce weaving on freeways), collect traffic from local roads, and distribute it to arterials. Traffic using a collector is usually going to or coming from somewhere nearby.

At the bottom of the hierarchy are local streets and roads. These roads have the lowest speed limit, and carry low volumes of traffic. In some areas, these roads may be unpaved.

## Discontinuity

Discontinuity measures the number of changes in street hierarchy experienced along a path. The relative discontinuity measure is discontinuity divided by the trip length [1].

$y_a = |{k_1} - {k_2}|$

The discontinuity of the trip along the shortest path, P, is estimated as: $Y(P) = \sum_{a\epsilon{P}} y_a$

The relative discontinuity is then estimated as: $Y'(P) = Y(P)/l(P)$

where

• $Y(P)$ = discontinuity of the trip along the shortest path
• $l(P)$ = Total length (km) of trip along the shortest path

## References

1. Xie, Feng and David Levinson (2007) Measuring the Structure of Road Networks. Geographical Analysis 39(3) pp.336-356