Transportation Geography and Network Science/Scaling and size

The use of a transportation network is highly dependent on the size and scale of the area it services. While we can make on the surface observations about relationships between size and scale of cities to certain demographics, such as all towns above 10,000 are within driving distance of a super market, basic demographic comparisons such as this fall short of describing both generalized phenomena that occur on a national (or even international) level and fail to describe intrinsic, unique qualities that deviate from the norm. The goal of this chapter is to introduce the reader to a few methods for determining and analyzing the size and scale of communities, explore one method to compare cities on a meaningful level, and to relate the size of a community to several demographic features.

Introduction
The growth and expansion of cities is one of the most important occurrences in human history. Living within close proximity to one another allows humans to exchange goods, services and ideas in ways that cannot just be explained through direct human communication. Even with modern communication, dynamics form within urban populations that aren't observed in rural or less densely populated communities. As the number of people increases, so does the amount of interaction between people, begetting both positive and negative consequences. As human populations continue to grow and a greater percentage of population start to move into urban centers, as we are witnessing across the globe, these effects are becoming more prevalent than any other point in history. While the United States has begun to produce a wide array of data on cities covering all aspects of society, it is difficult to make meaningful comparisons between cities due to the complexities of cities. In order to find meaningful (missing part of sentence?)

This chapter will be broken up into three sections: one, how does scale pervade within our society and how does this affect transportation systems?; two, how do we remove scale from our measurements of urban populations and how can we make meaningful comparisons between cities of different sizes?; and three, how do cities grow and how do we categorize its growth?

Power Law
There are occurrences throughout nature that have relationships that can be dictated by the power law. In relationships that can be described through the power law, the multiplicity of some element, whether it is in terms of number or size, decreases in a powered scale. The relationship can be expressed in the following equation:


 * $$px^m = C$$

Where p is the multiplicity of a given element, x is a characteristic scale size, m is the scaling exponent, and C is a normalized constant. In general, relationships that can be described by the power law will have values of m ranging from 1 to 2. If the log is taken of this function, it reveals an important relationship:


 * $$log(p) = -mlog(x) + C$$

On a log-log scale, this forms a linear relationship, with m denoting the negative slope of the function. Observation show that the slope of the function changes depending on the characteristics being observed.

Examples
The following are some examples of where the power law can be observed in nature, and its associated scaling exponent, m:


 * If P is the probability to have an income of x, 1.5 < m < 2.0 (also known as a Pareto distribution)
 * If P is the relative frequency in the number of authors of a given number of papers, x, published in a year, m = 2
 * If P is the number of cities with population x, m = 1
 * If P is the number of animals and x is the mass weight of the animal, m = 1
 * If P is the relative frequency of the usage of a word in a language and x is the rank of the word in the language, 1 < m < 1.5 (also known as Zipf's Law)

The Multiplicity Rule
In general, the number of instances where the power law can be used to describe is far greater than the number of observations we have made on the world we live in. Therefore, as stated in Salingaros & West's 1999 paper, nature follows what is called the multiplicity rule. Simply put, our world is comprised of many scales, exhibiting a spectrum of scales with components relatively numbered in accordance to the power law. When humans perceive objects with a "full spectrum" of scale, these seem aesthetically pleasing and natural. When shown objects with quantized scaling, where some sizes are favored over others, these are found to be less aesthetically pleasing and artificial.

One can look no further than their Lego's to see an illustration of this rule. Imagine you were trying to build your home, whether it was an apartment building or a house. First, imagine using only 50 uniform blocks of the same color. How much detail could get with those fifty blocks? Can you differentiate easily between your doors, windows, roofing material, etc.? Most likely, the answer is that most of the elements of your home look roughly the same: it is difficult to distinguish between windows and doors, you cannot tell where there are medium changes, and it doesn't much feel like a place a human could live. This home not only doesn't look very realistic, but more importantly artificial and fake. Now imagine building your home with 1,000 blocks, with an unlimited number of colors to choose from. Assuming you played with Lego's as a kid, you probably built your home structure that closely resembles your building. Not only were you able to re-size your windows and doors so that they were scaled appropriately, but the range of sizes could also be seen in the flower pots on the window sill to the shrubs in front yard to the ventilation system on the side. While still a man-made structure (made with man-made blocks), the details and range of scales that can be seen make the structure more life like and natural.

The multiplicity rule can be applied to real life city applications. While an analysis can be conducted to see the distribution of roads, buildings, project funding, etc., there are many other factors that could be identified as subjective qualities. Furthermore, to compare the distribution funding of projects for a small city such as Mankato, MN to a large city such as New York City will not be a valid comparison: a "small" project for New York City will most likely not be a "small" project for Mankato. Nonetheless, the multiplicity rule proves to be a powerful tool in determining the best course for policy: balanced cities that follow the multiplicity rule are those that follow the laws of nature that can be seen across numerous real life applications. As it pertains to transportation, there are three circumstances highlighted by Salingaros and West that are in direct contradiction to the multiplicity rule, which create unbalanced systems within our urban transportation systems.

Distribution of Path Lengths/Widths
When planning for a city, one must consider both the hierarchy of roads and the distribution in accordance to the multiplicity rule. While you couldn't imagine a city with only pedestrian walkways, the multiplicity rule is also violated when too much preference is given towards automobile arteries. Balanced cities are able to give preference to bipedal, bicycle and low-flow streets in terms of millage, but also enable car travel to occur through both middle to high flow streets to enter the city.

The most notable violators of the multiplicity rule are suburban housing projects. In many cases, the only way to enter these developments are through medium to high flow roads. Ultimately, this leads to a disconnect between the suburb and the city it is attached to, since it doesn't include the lower spectrum paths for human travel. Conversely, some inner city neighborhoods lack the major arterial to connect to and only have lower function roads, ultimately disconnecting them from the greater part of the city.

Distribution of Project Funding
If given a choice between large projects that make a substantial change to several neighborhoods or a city versus many small changes over the course of a decade to slowly changing the make up of the city, more often or not the large project will receive funding over the smaller ones. Unfortunately, studies show that the optimal way to distribute funding for projects is equally over all sizes of projects. How this relates with the multiplicity rule is to have numerous small projects that do minor changes, such as road maintenance or small facility restoration, fewer medium size projects, then only a couple of large projects. When public policies choose to fund more larger projects that reinvent the city and pull funding from smaller projects, infrastructure can become dilapidated and the city can begin to decay in places that do not see the effects of the large project.

Distribution of Urban Elements
As described earlier through the Lego's example, there must be an even distribution of sizes and scales throughout the cities fabric. This can be related to either the size of buildings, where we would hope to see a few sky scrapers, several medium size office and commercial buildings, and numerous smaller residential buildings. Not only does this apply to buildings, but city elements like parks, where there should be both a few large fields, a fair number of trees, but many human sized scale elements such as benches. Unlike the previous two examples that apply towards functionality, this application of the multiplicity principle applies towards the human perception of scale and what is determined to be "natural."

One way that suburban sprawl is sometimes deemed unnatural is its preference to using cookie cutter design: where all the buildings are approximately the same size, their lot sizes are uniform, and the roads are of the same design, creating a gap in the spectrum in scale that is required from the multiplicity rule. Conversely, with the newly developing cities in China, there are many cities opting to build high rise buildings instead of lower density, human size scaled buildings, putting too much emphasis on the other end of the scale.

Measurements Independent of Size
As described earlier, a difficult problem arises when we try to compare cities of drastically different sizes such as Mankato, MN and New York City: the size and scale of the cities are just so drastically different, on the onset it seems to be finding similarities to a golf ball to a beach ball. At the root of the problem, there are two main causes. The first is attempting to find scale free indicators. A common practice is to compare cities on a "per capita" basis, where each individual is assigned a contribution to the aggregated city value. While this does allow us to compare very small places to very large ones in terms of assigning a value to every individual, this doesn't tell us the whole story, as illustrated by the second problem. As a city grows and becomes more densely populated, an interesting phenomena occurs within the city. People begin to interact more frequently with others, causing each person to become more productive within the place they live. Not only can this be seen in conventional figures such as GDP, but in other socioeconomic factors such as wages, number of patents, even crime and the spread of certain diseases. So even if we were to see a larger GDP per capita in a certain city, it only scratches the surface as to what makes the city more or less productive than the average city. We will explore the additive effect of these interactions, designated the Fifteen Percent Rule, then compare two scale free indicators: per capita and the Scale Adjusted Metropolitan Indicator developed by Bettencourt et al. in their 2010 paper 'Urban Scaling and Its Deviations.'

The Fifteen Percent Rule
As one can imagine, when organisms work together, they can do more work than they could have individually. When applies to larger scales, it is observed that growth in many socioeconomic factors does not grow linearly with population: when a population doubles, it is able to produce more than double the goods, services and ideas. For every doubling in population, roughly a 115% increase in socioeconomic activities (GDP, wages, crime, etc) is observed. Furthermore, less resources are required as a population grows more dense: for every doubling in population, there needs to be a 85% increase in infrastructure development as the effects of economies of scale. The following is a short list of some advantageous and disadvantageous factors that this rule is seen to apply to:

Advantageous:
 * GDP (or Gross Metropolitan Product (GMP))
 * Increase number of patents
 * Increase number of research institutions
 * Increase number of education institutions
 * Decrease in energy needs (electricity, gasoline, etc.)
 * Decrease in infrastructure needs

Disadvantageous:
 * Increase in crime
 * Increase in traffic congestion
 * Increase in number of illnesses

Not only does this rule apply to large cities, it applies to all communities and shows similar increases to those seen in cities with similar cultural and/or geographical characteristics. This principle can be expressed by the following equation:


 * $$ Y(t)= Y_0 N(t)^{\beta}$$

Where,
 * $$Y(t)$$ is a quantity of a socioeconomic urban quality (i.e. wages, GMP, crime, etc.)
 * $$Y_0$$ is normalized constant
 * $$t$$ is the time at which we are observing the quality
 * $$N(t)$$ is the population at time t
 * β is the scaling factor

The scaling factor,β, controls the effect we observe. When β = 1, there is a linear relationship between population and quality: “per capita” measurement is adequate as the quality grows at the same rate as population. When β ≠ 1, observed in most real life applications, there are changing rates due to population increases:
 * β ≈ 0.85 implies economies of scale
 * β ≈ 1.15 implies increasing rates

Per Capita
For alternative definition, please view the Wikipedia page:Per Capita

To compare cities on a per capita basis is to compare a specific quality or trait of a city on a per person basis. In terms of the above equation, β must equal 1, to which we can rearrange the equation in the following manner:


 * $$\frac{Y(t)}{N(t)} = Y_0$$

This implies that at any given time, t, some quality will have be constant value over time. Unfortunately, this goes against what is observed: either economies of scale decrease the overall number of resources needed per unit of population, or the increased interactions produce higher productivity per person as populations grow. In other words, we would not expect a quality Y to grow linearly over time with respect to population.

Scale Adjusted Metropolitan Indicator (SAMI)
Due to the limited applicability of a per capita measurement with respect to time, Bettencourt et al. (2010) developed a means of comparing cities not only for a scale free measurement, but on how they compare in growth over time. The unit of measurement they developed are Scale Adjusted Metropolitan Indicators (SAMIs). SAMI's are derived by the following equation:

$${\xi}_i = Log(\frac{Y_i}{Y_{0}N_{i}^{\beta}})$$

Where,
 * $${\xi}_i$$ is the SAMI for quality i
 * $$Y_i$$ is the average metric from a city
 * $$Y_{0}N_{i}^{\beta}$$ is the expected metric from a city with population $$N_{i}$$

As you can see, it is a unitless number that removes population from the indicator. Furthermore, it finds the deviation from the norm by taking the observed metric, $$Y_i$$, and dividing it by the expected value with a given population. SAMI's allow for more meaningful comparisons on whether a city is developing socioeconomical characteristics at an increased or decreased rate with respect to other cities. Click here for their interactive maps and their rankings for cities across the United States for four characteristics.

Size Characteristics
Across cities with similar characteristics historically, geographically and/or economically, one can use SAMI's to indicate how cities relate to one another as they grow in population. While we can expand the analysis from comparing cities within the same country, such as what Bettencourt et al. did in their 2010 paper, there are many factors that come into play beyond the scope of SAMI's due to various other factors. For example, when comparing cities across the United States, even though the variations are limited, cities can be grouped based on previous industries: the rust belt cities are very similar in growth patterns, and will vary significantly from other cities in growth geographically similar to them. Of course, expanding this on a world scale will introduce further problems; there are many fewer things that San Francisco has in common with somewhere such as Moscow than it did for a place such as Baltimore. One characteristic that makes a considerable difference when comparing two cities is its density, especially when thinking of topics previous presented in this chapter.

Density
As explained earlier, as the number of human interactions increases, the productivity increases and resources are able to be used more efficiently; economies of scale develop and infrastructure is able to be better utilized by the population. When a population grows within the metropolitan area serviced by a city, by definition the density of the population increases as well. A notable exception is when the serviced area grows with expanding suburbs and pushing the extent of the city. By expanding the city limits through expansion, this hinders economies of scale from developing and requires greater resources. One of the most notable examples of greater resource usage is gasoline consumption, as presented in Newman & Kenworthy’s article 'Gasoline Consumption and Cities' presented in 1989.

Newman and Kenworthy's Findings
In their 1989 article presented in Journal of the American Planning Association, Newman and Kenworthy made two comparisons: one, comparing United States cities to one another based on density, and secondly developing a model based on worldwide gasoline consumption and comparing United States cities to different regions of the world, determining how certain factors attributed to gasoline consumption. When comparing United States cities, cities with a developed Central Business District (CBD), high density living and relatively high public transportation (e.g. New York and Chicago) consumed considerably less gasoline than cities with low population densities (e.g. Houston and Los Angeles).

When they developed a model to predict gasoline consumption, a correlation was discovered between gasoline consumption and density not only within the CBD, but also within the outer suburban areas. This phenomenon emphasizes that it isn’t simply a strong CBD that promotes economies of scale, but increased activity within suburban areas also makes a considerable impact. One example that Newman and Kenworhty bring up is the city of Toronto, which has a similar percentage of jobs in the CBD (13% compared to 12% in American cities), has the same average commute (8.1 miles), but consumes 60% as much gasoline due to higher densities in the outlying areas from the CBD and higher transit ridership. Overall, the factor that contributed most towards gasoline usage was the average commuting distance, where the United States’ average was 8.1 miles, compared to 5.0 for European cities, and 2.5 miles in Asian cities. Unsurprisingly, American gasoline consumption (in 1980) was at 446 gallons per capita when compared to 101 for European cities, and 42 for Asian cities.