Transportation Geography and Network Science/Accessibility

Accessibility refers to the ease of reaching destinations. People who are in places that are highly accessible can reach many other activities or destinations quickly, people in inaccessible places can reach fewer places in the same amount of time.

A measure that is often used is to measure accessibility in a transportation analysis zone i is:

$$ Accessibility_i = \sum_j {Opportunities_j } \times f\left( {C_{ij} } \right) $$ where:
 * $$i$$ = index of origin zones
 * $$j$$ = index of destination zones
 * $$f\left( {C_{ij} } \right)$$ = function of generalized travel cost (so that nearer or less expensive places are weighted more than farther or more expensive places).

This is sometimes in contrast with Access which is a binary indicator of whether a destination can be reached on a network.

Network Size
We use the Figure to first introduce accessibility as a network concept. As shown on the top, there are two cities (or nodes), city A and city B. There are therefore two travel markets: A-B and B-A. The middle case adds one city, and one link, but greatly increases the number of travel markets: A-B and B-A remain, but A-C and C-A, and B-C and C-B are added (we increased by four markets to a total of six). One link tripled the number of Origin-Destination (O-D) pairs served. The bottom case adds one more link (for a total of 3), but the number of markets again increases significantly: we still have A-B, B-A, A-C, C-A, B-C, and C-B; but now we also have A-D, D-A, B-D, D-B, and C-D and D-C. The number of markets doubled (we increased by 6 markets to a total of 12).



This phenomenon, dubbed the “Law of the Network” (and in a computer networking context, Metcalfe’s Law) can be expressed as

$$S=N(N-1)$$

Where:

S = the size of the network (number of markets)

N = the number of nodes

(To illustrate: with 2 nodes: S = 2*1 = 2, with 3 nodes: S = 3*2 = 6, with 4 nodes: S = 4*3 =12, etc.)

The value of S grows non-linearly as nodes are added to the network, until all nodes are connected. Clearly there is increasing value to the network as it gets larger. Since people are willing to pay more for goods of higher value, we would expect that people would pay more to belong to a larger network (live in a larger city).

Accessibility (A) differs from Network Size (S), in that accessibility multiplies each interaction by a function of the travel cost, such that far away interactions have less weight than nearby interactions. Accessibility also replaces the simple measure, number of nodes, with a slightly more sophisticated measure, e.g. number of jobs, to measure employment accessibility (or number of workers to measure labor force accessibility). This allows us to see how well the system connects workers with jobs.

Specification
The isochronic or cumulative opportunity measure is one of the basic and early measures discussed in the literature (Vickerman, 1974; Wachs & Kumagai, 1973). This approach counts the number of potential opportunities that can be reached within a predetermined travel time (or distance).

$$A_i = Sum_j B_j * a_j$$

Where

$$A_i $$ 		Accessibility measured at point i to potential activity in zone j

$$ a_j$$		Opportunities in zone j

$$ B_j$$		A binary value equals to 1 if zone is within the predetermined threshold and 0 otherwise For instance this measure can be used to identify the number of recreational opportunities around a residential location i that are within 400 meters (approximately one quarter mile) of network distance (zone j). This measure does not account for the size of the facility (attractiveness) or the impedance of reaching it (cost). It is widely used in hedonic modeling to control for access to neighborhood amenities. It is simple to understand and calculate, but makes an artificial distinction that opportunities 399 meters away are valuable, while those 401 meters away have no value.

Illustration


The isochronic or cumulative opportunity measure counts the number of potential opportunities that can be reached within a predetermined travel time (or distance). The Figure shows the cumulative opportunity measure of accessibility to jobs for the Twin Cities metropolitan region measured at 10 minutes of travel time during the morning peak hour from origins.

Planners and non-professionals can easily interpret this measure. A main point of weakness of the measure is that it does not account for people’s actual choices of residence and employment location. Also it equally weights people within the same bin of travel time without considering the attractiveness of the areas where they reside or where they are employed. A similar measure can be produced for various time ranges. It is noticed from the figures the increase in the number of opportunities with the increase in travel time. Around 70% of the Transportation Analysis Zones had more than 1,281,710 jobs within a 50 minute travel time, which indicates the current level of accessibility in the region.

Similarly cumulative opportunity measures can be produced for other destinations (besides jobs).

Specification
The gravity-based measure discussed in (Hansen, 1959) is still the most widely used general method for measuring accessibility, although it is more complex in calculations and has some points of weaknesses.

$$A_{im}=\sum\limits_{j}^ – {O_{j}f(C_{ijm})}$$

$$A_{im}=\sum\limits_{j}^ – {O_{j}C_{ijm}^{-2}}$$

$$A_{im}=\sum\limits_{j}^ – {O_{j}\exp (\theta C_{ijm})}$$

Where

$$A_{im}$$ 		Accessibility at point i to potential activity at point j using mode m

$$O_{j}$$		The opportunities at point j

$$f(C_{ijm})$$	The impedance or cost function to travel between i and j using mode m

$$\exp (\theta C_{ijm})$$	Negative exponential function to travel between i and j using mode m

The differences between various studies of accessibility that utilize this method are mainly in functional forms that measure the cost to move between origin and destination and how opportunities are calculated. The opportunities can be the frequency of bus service when measuring accessibility to transit service, while it can be the number of employees when measuring the accessibility to work, or park size when measuring accessibility open space. The accessibility measure is expected to increase with the increase in the opportunity measure. The summation is used so as to include all potential sites j that might encompass desired activities. In other words, if we are measuring accessibility to the Mall of America in the Twin Cities, the total number of individual sites j (denoted with a capital J) will be equal to one since only one Mall of America exists in the Twin Cities. Meanwhile measuring accessibility to shopping malls in the same region will require calculating the previous function to all shopping malls in the region, while using a factor such as number of stores or mall area or retail employees as the potential variable to differentiate between the various shopping mall sizes. This is done using each shopping mall as a destination j then calculating the accessibility variables for each until we have J (J=total number of destinations) values of accessibility to be summed at the end of the process.

Accessibility is expected to decline the farther the opportunities are from the origin. Much of the literature defines impedance using a negative exponential function. When we say “farther” that can be in terms of time or distance or generalized cost.

The previous equation is applied to measure accessibility using a single transportation mode m. Accessibility can be measured in the same manner for various modes of transportation then a comparison can be conducted. For example accessibility to jobs can be measured using automobiles, public transit and bicycling. The findings can then be compared to identify underserved areas or locations that need more attention in terms of accessibility using a certain mode.

Major disadvantages of this accessibility measure are the need to develop an impedance factor (though coefficients from destination choice or trip distribution models already estimated for regional transportation planning models are often used), and the appropriate weights for the destination (e.g. should retail be number of stores, number of retail jobs, or area). Combining the modes is also difficult. One might use one of the following composite measures: $$A_{i}=\sum\limits_{j}^ – {\sum\limits_{m}^ – {O_{j}M_{ijm}}f(C_{ijm})}$$

where:

$$M_{ijm}$$ = share of mode m in market ij (0-1)

But in the above the mode share in a market also depends on the cost of travel, so the analysis weights travel costs doubly. We could use

$$A_{i}=\sum\limits_{j}^ – {\sum\limits_{m}^ – {O_{j}f(C_{ijm})}}$$

Here we could introduce a new mode and instantly increase accessibility, even if the new mode was essentially identical to existing modes.

One might simply want to say something like this:

$$A_{i}=\sum\limits_{m}^ – {A_{im}M_{im}}$$

or $$A_{i}=\max \left( A_{im} \right)$$

But these equations ( use mode share at the origin, while mode share is a trip (origin and destination-based) phenomenon, so these measures lose information.

Illustration




The gravity-based measure developed by Hansen (1959) is still the most widely used general method for measuring accessibility, although it is complex in calculations and has some points of weaknesses. The Figure shows the Twin Cities metropolitan region with the gravity-based accessibility measured to jobs in the region (following equation 3b). The accessibility levels are shown in shades of color. The unit of analysis used in developing this measure is the TAZs, while using the reciprocal of the square of travel time between each TAZ as the impedance function. The attractiveness of a TAZ is calculated based on the number of jobs reported by the LEHD dataset that was previously used in the previous figure. The reciprocal of travel time squared, a common and widely used impedance function, is used as the impedance value when calculating this measure of accessibility.

Major disadvantages of this accessibility measure are the need to develop an impedance factor (though coefficients from destination choice or trip distribution models already estimated for regional transportation planning models are often used), and the appropriate weights for the destination (e.g. should retail be number of stores, number of retail jobs, or area). Combining the modes is also difficult.

First it is important to note that comparing accessibility measures should be done in a relative manner and not through comparing numbers directly. It is clear from comparing Figures 5 and 6 that similarities exist between the two measures of accessibility. TAZs with high levels of accessibility in the cumulative opportunities map tend to have high numbers of jobs within the 10 minutes travel time range in the gravity map. Both maps indicate centralization in the level of accessibility to jobs in the Twin Cities region similar to the centralization observed in the 10 minutes cumulative opportunity measure of accessibility. A statistical analysis conducted later in this report shows the relationship between these measures. It is clear that areas with high levels of accessibility to jobs are located in the area including and surrounding the two major downtowns in the region (Minneapolis and Saint Paul).

Another alternative is to change the impedance function used in generating the gravity-based measure of accessibility. Figure 7 shows the level of accessibility to jobs in the Twin Cities region using travel time and an exponential function with $$\theta= -0.1 $$ (following equation 3c) multiplied by the travel time between each TAZ of origin and destination.

The process of selecting the appropriate impedance function is complicated and requires several trials. The reciprocal of travel time squared was the first function used (following Newton’s Laws of Gravity). Some researchers generate various impedance functions and include them as part of a land value analysis to reach the most appropriate measure that is statistically most correlated with land value (and thus how people perceive the effect of transportation on land). This concept is explored later in the report.

Similarly an accessibility measure to the number of employees in a region can be developed to measure the ease of jobs reaching their potential employees.

Utility-Based Measure
The most complex and data intensive is the utility-based measure. Several researchers use this method since it adheres to travel behavior theories (Ben-Akiva & Lermand, 1977; Neuburger, 1971). The general specification of the measure is as follows:

$$A_n^i=ln[\sum_{\forall C_n} exp(V_{n(c)})]$$

Where

$$A_n^i$$ 	Accessibility measured for individual n measure at location i

$$V_{n(c)}$$ 	Observable temporal and spatial component of indirect utility of choice c for person n

$$C_n$$ 	Choice set of person n

This measure incorporates individual traveler preferences as part of the accessibility measure compared to the gravity model where the variation is not present across people living in the same zone. The gravity model implies that all people in zone i will experience the same level of accessibility. In reality people choose destination $$j$$ to maximize benefit. This is done through comparing the benefits from going to $$j_1$$ to the benefit of going to $$j_2$$.

For example suppose we are measuring accessibility to grocery stores. A person n will choose shop c based on prices and other factors like cleanliness of the store. Still other choices are available for this person, who weights going to this one as more valuable than the others. This measure imitates the human choice since the attractiveness of each destination is included. It is based on economic benefits that people derive from having access to certain activities. This measure has several advantages yet its complexity and data intensity are the main barriers to implementing it.

Constraints-Based Measure
High levels of accessibility to various activities in a city can be present, yet the amount of time available in a day that people can spend to reach these activities might not. This leads to the constraints-based measure or people-based measure of accessibility (Wu & Miller, 2002). For example if a person is at node i at time t1 while at time t2 the same individual has to return to i then the time t = t2 - t1 constrains the number of j destinations available.

Composite Accessibility Measure
A fifth measure is the composite accessibility measure. A composite measure is suggested by (Harvey Miller, 1999) where he combines space-time and utility-based measures in one measure. This approach introduces a higher level of complexity where time constraints are superimposed. The composite accessibility measure requires more data than utility-based measures and it is even more complex in terms of calculations and accordingly generalizing it for usage is not an easy task.