# Transportation Geography and Network Science/Road networks

## Background

The road network is the system of interconnected roads designed to accommodate wheeled road-going vehicles and pedestrian traffic [1].

The role of the road network is to allow the movement of goods, services and people [2]. The road network generally forms the most basic level of transport infrastructure within urban areas and will link with all other areas, both within and beyond the boundaries of the urban area [1]. A network is made up of road corridors that perform different functions, known as the road hierarchy [2].

A road network can be divided into parts such as [1]:

• Intersections
• Motorways
• Bicycle Lanes
• Footpaths and Pedestrian Areas
• Pedestrian Crossings
• Bridges and Tunnels

Mobility and Accessibility Compression of the Hierarchy of Roads [3]

A functional hierarchy is the most common type which ranks roads according to how the roads are expected to function with respect to local through-traffic. In doing so, it recognises that the roads form part of an interconnected network and addresses the competing road uses of mobility and access [4]. While sources differ on the exact nomenclature, the basic hierarchy comprises freeways, arterials, collectors, and local roads [3]. A system of interconnected roads with the same hierarchy forms a network with relative hierarchy.

Hierarchy United States [5] United Kingdom [6] China Mainland [7] NSW, Australia [8]
Freeways Interstate Highway System Motorway System National Expressway System and Provincal Expressway System Motorways System
Arterials Interstate Business Routes System, US Numbered Highway System and State Highway System A Road System and B Road System National Highway System and Provincial Road System A Routes System and B Routes System

Long distance travel in a mature road network typically has the following pattern:

Origin - Local Roads - Collectors - Arterials - Freeways - Arterials - Collectors - Local Roads - Destination


An example of this is travelling from Sydney Showground in Sydney Olympic Park to the University of Sydney Quadrangle Clocktower:

Sydney Showground Grand Parade Australia Ave Homebush Bay Dr (A3) Western Motorway (M4) Parramatta Road (A22) Ross St and St Johns Road Derwent Street and University Avenue University of Sydney Quadrangle Clocktower
File:SS-USYDCT.png
Travel from Sydney Showground to the University of Sydney Quadrangle Clocktower

Grid Plan Example - Commissioners' Plan of 1811 for Manhattan

Organic Street Pattern Example - Map of London in 1593

Radial Street Pattern Example - The Place Charles de Gaulle, Paris

Planned Irregular (Cul-de-sac) Pattern - A plan of Village Homes in Davis, California

### Grid Street Pattern

The grid pattern, grid street pattern, or gridiron pattern is a type of city plan in which streets run at right angles to each other, forming a grid. The infrastructure cost for regular grid patterns is generally higher than for patterns with discontinuous streets. This geometry helps with orientation and wayfinding and its frequent intersections with the choice and directness of route to desired destinations. All streets in a grid are accessible to traffic (non-hierarchical).

Typical uniform grids are unresponsive to topography. In a modern context, steep grades limit accessibility by car, and more so by bicycle, on foot, or wheelchair, particularly in cold climates. The same inflexibility of the grid leads to disregarding environmentally sensitive areas such as small streams and creeks or mature woodlots in preference for the application of the immutable geometry.

The frequency of intersections in grid patterns, however, becomes also a disadvantage for pedestrians and bicycles. It disrupts the relaxed canter of walking and forces pedestrians repeatedly onto the road, a hostile, anxiety-generating territory. People with physical limitations or frailties, children and seniors, for example, can find a regular walk challenging. For bicycles, this disadvantage is accentuated as their normal speed is at least double that of pedestrians[9].

### Organic Street Pattern

Organic streets can come from disorganized, fast-paced development, as is common in today’s slums; or it can grow slowly over time, as one plot of land is divided, and streets are built over the new boundary, like many of the old towns of England and other European powers. The Organic street pattern is typically unique. This geometry makes traffic congestion occurs a lot more often, and road users easily to get lost[10].

In a radial street pattern, the road is spread out from a central point, similar to a spider's web. The radial street pattern could be largely found in Paris. The radial street pattern makes the traffic flow easier, but slower as there are no shortcuts. All roads lead to a central point, which generally causes traffic jams near the centre[10].

### Planned Irregular (Cul-de-sac) Pattern

Planned irregular pattern includes a significant amount of Cul-de-sac (dead-end road) in the network. The network has less intersection, and the road in the network curve a lot. This pattern could be largely found in modern cities and newer suburbs. The planned irregular pattern helps to improve the flow of traffic. Road in the network is quieter as there are fewer intersections and less through-traffic. The nature of topography could be nicely accommodated in this pattern. It is easy for the road users to get lost in an irregular pattern, and this pattern is not easy to subdivide to expand[10].

### Topological Measurement

${\displaystyle e}$ : Number of edges (road segments)
${\displaystyle v}$ : Number of vertices (nodes), including road intersections, travel origins and destinations
${\displaystyle g}$ : Number of maximally connected components (a planar network may be unconnected but may consist of connected pieces, which are called 'maximally connected components' or 'con-nected components'.)

The cyclomatic number indicates the number of circuits in a network.

${\displaystyle \alpha }$  index is the ratio between the actual number of circuits in the network and the maximum number of circuits.
${\displaystyle \beta }$  index is the ratio between the number of links and the number of nodes.
${\displaystyle \gamma }$  index is the ratio between the actual number of links and the maximum number of possible links in the network

Values for ${\displaystyle \alpha }$  index and ${\displaystyle \gamma }$  index varies between 0 to 1. A higher value of cyclomatic, ${\displaystyle \alpha }$ , ${\displaystyle \beta }$  and ${\displaystyle \gamma }$  represents a more connected network [11].

### Heterogeneity Measurement

Heterogeneity is a common feature of many complex networks. A statistically collective measure of entropy (${\displaystyle H}$ ) can be used to evaluate the link‐based heterogeneity of road networks:

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

Where:

${\displaystyle X}$  represents the road network system
${\displaystyle m}$  is a road link subset base on different road properties such as functional type, traffic volume or LOS.
${\displaystyle p_{i}}$  as the frequency of road links in the ${\displaystyle i}$ th subset over the total number of links

The entropy value of a homogenous group is zero. A positive entropy measure indicates that there exists heterogeneity in a network with more than one group of links. A larger entropy measure indicates a greater heterogeneity of the network [11].

### Connection Patterns

Firstly, circuit blocks need to be identified from a road network base on the processor in the figure below.

Identification of Circuit Blocks in a Connected Graph

A circuit block is defined as a block that contains at least one circuit and contains neither bridges nor articulation points. If a circuit block contains only one circuit, it is defined as a ring; if it contains more than one circuit, it is defined as a web.

'Ringness' of a network could be measured by:

${\displaystyle \Phi _{ring}={\frac {Total\;Length\;of\;Arterials\;on\;Rings}{Total\;Length\;of\;Arterials}}}$

'Webness' of a network could be measured by:

${\displaystyle \Phi _{web}={\frac {Total\;Length\;of\;Arterials\;on\;Webs}{Total\;Length\;of\;Arterials}}}$

The concepts of 'circuitness' and 'treeness' can be defined as:

${\displaystyle \Phi _{circuit}=\Phi _{ring}+\Phi _{web}}$
${\displaystyle \Phi _{tree}=1-\Phi _{circuit}}$

These ratios range from 0 to 1, indicating to what extent arterials are connected as circuits or trees. A high ratio of treeness indicates a branching structure while a high ratio of circuitness indicates a circuit network.

Beltway in a road network is defined as one dominant circuit block. If the dominant block is a ring, the ring is identified as the beltway; if the dominant block is a web, the beltway is defined as the envelope of the circuit block. The 'beltness' can be measured by:

${\displaystyle \Phi _{belt}={\frac {Length\;of\;the\;beltway}{Total\;Length\;of\;Arterials}}}$

An urban arterial network may have multiple concentric beltways around its CBD(s). An inner beltway can be identified by breaking its outer beltway and repeating the above procedures. These steps can be repeated until no dominant circuit block is found. The beltness of each beltway can be calculated [11].

### Continuity

The dicontinuity of a road network can be measured by:

${\displaystyle Y={\frac {\sum _{all(R,S)}Y(P_{RS})*q_{RS}}{\sum _{all(R,S)}l(P_{RS})*q_{RS}}}}$

Where ${\displaystyle P_{RS}}$  the shortest path between a given O–D node pair ${\displaystyle (R,S)}$ , ${\displaystyle q_{RS}}$  is the number of trips between the origin and the destination, and ${\displaystyle l_{RS}}$  is the length of the shortest path. ${\displaystyle Y_{RS}}$  is the discontinuity of trip from R to S, and can be calculated by:

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

Where ${\displaystyle a}$  is the upstream link.

The discontinuity of a movement can be calculated by:

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

Where ${\displaystyle k_{1},k_{2}}$  is the hierarchies of the upstream and downstream links [11].

## Road Networks Travel Reliability Measurement

### Travel Time Window

The travel time window is a simple statistic which considers the travel time range a standard deviation, ${\displaystyle \sigma }$ , either side of the mean travel time, ${\displaystyle \mu }$

${\displaystyle Travel\;Time\;Window=\mu _{travel\;time}\pm \sigma _{travel\;time}}$

This metric can be applied to different spatial and temporal segmentations as well as applied across both public and private transport modes. The weakness of this measure is its independence, the change in travel time window for a particular link or route may be useful but it is difficult to compare routes and links. Furthermore, the measure does not indicate whether the travel time window is acceptable or unacceptable for a user and thus may not provide a beneficial measure for practitioners. Finally, the standard deviation is not sensitive to the tail of the distribution, so this measure conveys information about the typical range rather than the likelihood of encountering an especially bad travel time [12].

### Coefficient of Variation

The coefficient of variation, is a standardised measure of dispersion and is the ratio of the standard deviation of a sample (${\displaystyle \sigma }$ ) to the mean (${\displaystyle \mu }$ ). In the context of travel time variability, the Coefficient of Variation Equation is presented below:

${\displaystyle Coefficient\;of\;Variation={\frac {\sigma _{travel\;time}}{\mu _{travel\;time}}}}$

The standardised approach using the coefficient of variation is an improved measure compared to using standard deviation as it is not dependent on the trip length. Furthermore it can also be quantified for road segments, entire road corridors and for a network as a whole. The coefficient of variation can be thought of as a fractional uncertainty. It is easy-to-calculate and accounts for trip length, but it builds on the (probably incorrect) assumptions that a travel time is equally likely to be above and below the expectation value and that a traveller’s tolerance for uncertainty scales linearly with the duration of the journey. Additionally, the weakness mentioned for travel time window regarding insensitivity to the tail of the distribution holds for coefficient of variation as well [12].

### Variability Index

The variability index compares peak and off-peak driving conditions and can be calculated as shown below:

${\displaystyle Variability\;Index={\frac {{CI}_{95\%Upper,Peak}-{CI}_{95\%Lower,Peak}}{{CI}_{95\%Upper,Off\;Peak}-{CI}_{95\%Lower,Off\;Peak}}}}$

Where:

${\displaystyle {CI}_{95\%Upper,Peak}}$  = The upper bound of the 95% confidence interval of travel time during the peak period
${\displaystyle {CI}_{95\%Lower,Peak}}$  = The lower bound of the 95% confidence interval of travel time during the peak period
${\displaystyle {CI}_{95\%Upper,Off\;Peak}}$  = The upper bound of the 95% confidence interval of travel time during the off peak period
${\displaystyle {CI}_{95\%Lower,Off\;Peak}}$  = The lower bound of the 95% confidence interval of travel time during off peak period

The index presents the ratio of the range of travel times at the 95% confidence interval between peak and off peak conditions. Values approaching 1 indicate a reliable road link or system as both off-peak and peak 95th percentile travel times are similar. Whereas, a value in excess of 1 indicates a less reliable road link or system as there is a greater difference in the travel time range. The Variability Index is useful in comparing the severity of peak period traffic conditions. The Variability Index is also more sensitive to the tails of the distribution, which are important in determining the traveller’s user experience. A disadvantage of this statistic is an ambiguity between reliability congested and reliably uncongested facilities. Consider a road with a variability index approaching 1 because the road is near capacity at all times of day and overall travel times are far greater than free flow travel times. In this case, presence of traffic incident could potentially result in incredibly unreliable conditions. The Variability Index is very sensitive to the definition of the peak period, and it combines day-to-day unpredictability and time-of-day variability into one measure. Thus, without detailed data which captures both recurrent and non-recurrent congestion at a fine temporal scale, the index may be biased [12].

### Buffer Time Measures

Buffer measures are derivatives of how much extra time is required for on-time arrival at a destination. Indicators in this category include buffer time, buffer time index and planning time index. The greatest advantage of buffer time indicators is they are more intuitive and better reflect traveller behaviour compared to statistical range measures, which require a familiarity with statistics concepts like standard deviation. Users are comfortable with allocating a time period related to travel as long as on-time arrival can be guaranteed at a certain level of probability. Buffer time could be defined as[12]:

${\displaystyle Buffer\;Time=95th\;percentile\;travel\;time-Average\;travel\;time}$

The buffer time is the amount of additional time which must be allocated to be 95% certain of achieving on-time arrival. This measure considers the travel time budget a user would allocate for a particular trip. As this is related to a particular trip, comparability is affected by trip length and is generally route specific. Buffer Time Index (BI) is a weighted buffer time and is described in the equation below[12]:

${\displaystyle BI={\frac {95th\;percentile\;travel\;time-Average\;travel\;time}{Average\;travel\;time}}}$

Lower values of BI suggest more reliable road links or routes. Standardisation based on the mean travel time improves the ability to compare between different parts of the road network. For an example consider a route, Route A, with a mean travel time of 40 minutes and 95th percentile travel time of 50 minutes. Also consider Route B, with a mean travel time of 10 minutes and 95th percentile travel time of 20 minutes. It is clear that Route B is less reliable than Route A, however both routes have a buffer time value of 10 minutes. The buffer time index for Route A (BI = 0.25) is lower than Route B (BI = 0.5) highlighting the benefit of standardisation[12].

Planning Time Index (PTI) compares the 95th percentile travel time to free flow travel time:

${\displaystyle PTI={\frac {95th\;percentile\;travel\;time}{Free\;flow\;travel\;time}}}$

Similar to the BI, the PTI is also a standardised measure of buffer time which enhances the metrics suitability when conducting comparative assessments. Comparing to the coefficient of variation, PTI presents a normalised assessment of reliability across links or routes with the added feature of sensitivity to the tail of the distribution. With roughly 20 working days per month, measures that build on the 95th percentile travel time are often interpreted relating to the travel time experienced on the worst commute of the month. While the buffer measures are directly measured from travel time distributions, they offer more natural interpretations than those measures based on statistical width[12].

### Florida Reliability Method

The Florida Reliability Method was developed through Florida’s Mobility Performance Measures Program completed by the Florida Department of Transportation (FDOT). The method measures reliability as the percent of trips on a corridor that are less than or equal to the expected travel time including a buffer, as shown in the equation below [12]:

${\displaystyle Florida\;Reliability\;Statistic=1-CDF(E(TT))}$

Where:

${\displaystyle CDF(E(TT))}$  = The proportion of trips that have a travel time ${\displaystyle (TT)}$  which is less than the expected travel time ${\displaystyle (E(TT))}$ .
${\displaystyle E(TT)}$  = The expected travel time is sum of the mean travel time and a buffer time that is deemed acceptable by either the transport authority or the road user.

Complexity of the approach arises from the ambiguity of setting a benchmark expected travel time. Similar to the buffer metrics, this method accounts for distance and geometric variances of links and routes as it is based on occurrences of failure/success in achieving a targeted travel time and not the quantification of travel time values [12].

### Skewness Metrics

The skewness statistic, ${\displaystyle \lambda ^{skew}}$  , indicates the asymmetry of the travel time dataset. It is calculated as the ratio of difference of the 90th percentile and 50th percentile travel time and the difference between the 50th percentile and 10th percentile travel time [12].

${\displaystyle \lambda ^{skew}={\frac {T_{90}-T_{50}}{T_{50}-T_{10}}}}$

The skewness statistic quantifies the comparison between the width of the right (slow) and left (fast) sides of the travel time distribution. It is a valuable counterpart to the statistical width measures because it explicitly addresses the asymmetry of the distribution and the importance of the tail of slow travel times to travellers [12].

The width of travel time, ${\displaystyle \lambda ^{var}}$ , is defined as the ratio of the difference of the 90th and 50th percentile travel time to the 50th percentile travel time. This ratio provides a standardised range of travel time that is defined by percentiles rather than mean and standard deviation [12].

${\displaystyle \lambda ^{var}={\frac {T_{90}-T_{10}}{T_{50}}}}$

Unlike the skewness statistic, this metric does not address asymmetry, but it is sensitive to the tails and it accurately describes a confidence interval (80% of observations fall within the width) normalised by the expected value (the median or 50th percentile travel time) [12].

'A Guidebook for Performance-Based Transportation Planning' from the US National Cooperative Highway Research Program provided a structured inventory of performance measures used in eight categories representing typical agency goals[13]:

1. Accessibility:
• Average travel time from facility to destination (by mode)
• Average travel time from facility to major highway network
• Average trip length
• Overall mode split
• Mode split by facility or route
• Number of structures with vertical (or horizontal) clearance less than X ft.
• Bridge weight limits
2. Mobility
• Origin-destination travel times
• Total travel time
• Average travel time from facility to destination
• VMT by congestion level
• Lost time due to congestion
• Delay per VMT
• Level of service
• Intersection level or service
• Volume/capacity ratio
3. Economic Development
• Direct jobs supported or created
• Economic costs of accidents
• Economic costs of lost time
• Indirect jobs supported or created
4. Quality of Life
• Lost time due to congestion
• Accidents (or injuries or fatalities) per VMT
• Customer perception of safety in system
• Tons of pollution (or vehicle emissions) generated
5. Environmental and Resource Conservation
• Overall mode split
• Tons of pollution (or vehicle emissions) generated
• Fuel usage
• Number of accidents involving hazardous waste
6. Safety
• Number of accidents per VMT
• Number of accidents per year
• Number of accidents per trip
• Number of accidents per capita
• Number of accidents per ton-mile traveled
• Response time to incidents
• Customer perception of safety while in system
• Accidents (or injuries or fatalities) per VMT
• Percentage of highway mainline pavement (or bridges) rated good or better
• Average response time for emergency services
• Number of accidents involving hazardous waste
7. Operational Efficiency
• Origin-destination travel times
• Total travel times
• Average travel time from facility to destination
• Average travel time from facility to major highway network
• Volume/capacity ratio
• Overall mode split
• Cost per ton-mile
• Average vehicle occupancy
8. System Condition and Performance
• Percent of roadway/bridge system below standard condition
• Age distribution
• System Preservation
• Percentage of highway mainline pavement (or bridges) rated good or better

## References

1. a b c "Road Network". URBAN Securipedia. November 2013. Retrieved 2020-09-10.
2. a b "Western Brisbane Transport Network Investigation". Queensland Government - Queensland Transport. January 2008. Retrieved 2020-09-14.
3. a b c "Hierarchy of Roads". Wikipedia. January 2020. Retrieved 2020-09-14.
4. "Assessing the Feasibility of a National Road Classification". ICSM Bi-Annual Meeting. October 2006. Retrieved 2020-09-14.
5. "Numbered Highways in the United States". Wikipedia. September 2020. Retrieved 2020-09-15.
6. "Roads in the United Kingdom". Wikipedia. August 2020. Retrieved 2020-09-15.