Transportation Geography and Network Science/Flows on Networks

Flows

[Figure 1. Interstate 80

Flow in network science is a broadly used concept. Flows apply to all network types and takes on different meanings. A broad definition for flow is the quantity of movements past a point during a time period movements. The entity of movement can be a liquid, a solid, a gas or even a concept. Types of networks and examples of related flows include:

• Computer networks: Flow of binary units (bits), 1 mbps = 1,000,000 bits per second

• Hydraulic networks: Flow of a liquid: gallons per minute or barrels per day (oil)

• Shipping network: Flow of a commodity: gross annual tonnage

• Electrical networks: Flow of electrons, 1 Ampere = 6.241 X 1018 electrons (1 Coulomb) per sec second

In the topic of interest, transportation networks, vehicle movements are the focus and measured in vehicles per hour (vph). Vehicular flow is dependent on vehicles density and speed. The equation to measure flow is:

q=k*v

where:

q=vehicle flow (vehicles per hour

k=density (vehicles per mile)

v=average speed (mph)

A related measure used by traffic engineers is peak hour flow (PHF). The PHF equation is:

PHF = (peak 15-minute vehicle count) * 4

To calculate, simply multiply the highest 15-minute volume by four. This is a useful measurement for planners and engineers when designing for busiest-case scenario. A situation where PHF is useful is when normal flows may be relatively low, but local activities cause traffic patterns fluctuate causing periodic “spikes” that may not be reflected during an hourly count.

The relationship between vehicle flow, density and average speed can be expressed graphically in what is known as Greenshield's model. As demonstrated in the top left figure showing the flow-density relationship, there is a critical density where any addition vehicles in a segment will decrease total vehicles flow. This is due to the incremental decrease in speed for each vehicle added as shown in the lower left graph.

Nature of flows

Flow behave differently depending on the network type. In electrical and hydraulic networks for example, flows are instantaneous throughout a link. Any entering electrons or liquid at one end of a link will simultaneously force out the same amount at the other end. In road networks, flows have a wavelike pattern. There is a delay from the a change in speed at any given time. A change in speed also occurs at different locations on a link. This is referred to as a shockwave which is expressed in the equation:

$v_{w}={\frac {q_{2}-q_{1}}{k_{2}-k_{1}}}\,\!$

where:

$v_{w}$= the propagation velocity of the shockwave (mph)

$q_{2}$ = flow prior to change in conditions (vph)

$q_{1}$ = flow after change in conditions (vph)

$k_{2}$ = traffic density prior to change in conditions (vpm)

$k_{1}$ = traffic density after change in condition (vpm)

Figure 3 reveals graphically the delay in a change in velocity with respect with to time. Here, the top line representing the first vehicle leads the remaining seven. Note the increase in delay for each subsequant vehicle after the changes in speed of vehicle 1. Figure 4 illustrates shockwave behavior with the free-flowing green vehicles coming upon denser conditions. Note the positive or negative sign for shockwave velocity. A positive number indicates a shockwave moving in the direction of traffic as a negative number indicates a counterflow direction.

Figure 4. Illustration of a shockwaves between two traffic flows

Figure 3 Time-Space Diagram illustrating vehicle speed, headway and spacing relative to each other.

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Walks

To further understand the behavior of flows, an overview of walks in a network setting is explored. A walk is a sequence of arcs (and nodes) It also could be thought of as a trajectory consisting of multiple steps There are two types of walks to consider, random and directed walks.

Random Walks

Figure 5. An animated example of a Brownian motion-like random walk on a torus.

Random walks consist of steps in which each step is independent of the prior step. An arc or node may be utilized multiple times. There is no defined destination and the direction of each step is random. This walk is sometimes referred to as the “drunkard’s walk.” The program (to be installed) is a two-dimensional illustration of a pixel which has eight possible directions for each successive step. Examples of random walks include an animal or insect foraging for food, stock market trends and a gambler’s funds. Random walks can be used to describe flows in a road network. Though a driver does not retrace a path (arc) or return to a just-visited intersection (node), random walks still play a role in road networks. For starters, when a driver is new to an area and seeks to depart a location (e.g. a downtown setting after taking in a large event), a driver simply wishes to leave the area and may randomly try different roads. Also playing a role is the early establishment of the road itself. Many of today’s highways are predated by paths used by horse and buggy. Even before horses and European settlers, native peoples established foot paths which some may be the result of trial and error foraging, which is a random walk.

Directed Walks

Figure 6. A Directed Walk

Directed walks consist of steps which do not retract to once-visited arcs or nodes. These walks are oriented towards a final destination. The figure at right illustrates a desired destination where the walk eventually terminates at the desired node. Examples of directed walks include electricity, river flows, migrating animals and internet communications.

Directed walks are more applicable to flows in transportation networks than random walks. Trips by motor vehicle typically have a destination and no road segment (arc) or intersection (node) is used more than once in a walk. The establishing of a highway is also typically a directed walk. Although the original path may be the result of random walks centuries ago, the new highway will connect the same nodes but often will select a slightly different route, depending on a number of factors such as topography, historical sites, environmental concerns, and the locations of existing buildings. Also, some of the original paths predating today’s highways may have been decided by directed walks. Though the first paths may be footpaths, many of these paths connected to a destination such as a neighboring community or body of water.