**Assessing Urban Complexity**

*Salat, S., Bourdic, L. & Nowacki, N.*

International of Sustainable Building Technology and Urban Development.

SUSB Vol 1, n°2. Dec. 2010

Abstract

This paper explores the concept of urban complexity and aims at exploring it in details. Based on a scientific approach, it presents quantitative indicators and mathematical formulae to assess several types of complexity in urban fabric. These indicators will enable to compare different types of urban tissues and analyze the correlation between energy, environmental, economic and social efficiency and urban complexity.

**Introduction**

Cities around the world appear in an astounding variety and complexity. To approach these in a scientific way means to look for the simple patterns which stand behind their apparent complications. Paris and Tokyo against Vienna, Barcelona or Kyoto grew without any real overall plan, but their structure, however impervious it might be to any sort of topographic regularity, reveals a shape of complex order. It is different for both cities, giving them a seal of an irreducible identity. The formal impoverishment of modernism, reducing cities to isolated objects, leads to the loss of complexity of the compositional rules. This paper aims to understand the minimal threshold of complexity and articulation which gives these urban entities an understandable language rather than being disorders of vague parts. We have to go in search of these minimal rules of organization of these urban complexes, not to copy the past but to move towards morphologies that are vaster and more intimate at the same time. These incorporate complexes of scales never reached before including those with tens of millions of inhabitants. These rules are ones of complexity. After defining the various meanings of urban complexity, this paper aims to supply operational tools to quantify objectively the complexity of urban structure. These indicators are not so much prescriptive tools but aids in the decision-making and communication process. They allow us to compare several projects by analyzing the energy and environment consequences resulting from the choice of form. They are not intended to be used as absolute values, but rather as instruments of comparison between different types of form. Any attempt to use them as absolute values would only fragment the urban conception in a series of technical targets, forgetting thereby its character as a structured system. We hope the system we are proposing will serve as a support for a dialogue-based investigative approach to the process of form selection conducted by local stakeholders in urban development. In this spirit, instead of target values, we will propose in future papers gradients of reference values corresponding to real cities. Moreover, our indicators offer the possibility of being adapted to specific projects, notably by way of comparison to the structural objectives defined by public actors. Once we had established the broad lines of our approach, we started by examining the nature of each indicator. This meant setting down limits to define its scope. The decisive question rested on the definition of the role of each indicator. Why choose it? What information does it provide? What information, on the other hand, does it not give? These questions were indispensable for finding the most appropriate name and mathematical formula for the indicator. However, the translation into mathematical formulae requires particular care since expressing an idea or an objective by a formula, a percentage or a ratio is never an innocent decision. It can alter the sense of the indicator and lead to an erroneous interpretation of the results. The quantitative and not qualitative measurement is primal because it alone allows for comparisons between cities with different geographic horizons. This analysis can give us the distribution of the different existing urban fabrics and point to common characteristics between some cities, thereby creating types that may or may not correlate with an historical or geographic distribution, and this would bring us to raise other questions and eventually change the direction of the research. It also allows us to study the correlation between indicators, providing us with the possibility of proposing recommendations, in the form, for example, of groups of values in which it would be preferable to be situated. Speaking of groups and not of an absolute value is crucial because the urban system is of such complexity that certain indicators may impact energy or environmental performance in opposing ways. We cannot therefore seek to optimize each indicator in absolute terms, since the optimization of one may involve the degradation of another. Defining a group of values allows us to find an optimal balance between opposing indicators. However, defined groups of values are not necessarily the same for all cities in the world, if only for reasons of climate. Some quantitative objectives will need to be modulated case by case, which does not mean that the overlying principles will be forgotten. Lastly, the desire to quantify and to keep the most objective indicators possible led us to pay even more attention to making sure that these quantitative measurements do not mislead us under the guise of objectivity.

**2. Complexity, information and fractals**

The aim of this section is to specify essential concepts used to describe urban fabrics. In the case of complexity and diversity, concepts that are usually treated in an imprecise way in publications in the field, we will briefly detail the theoretical aspects that support the indicators we are proposing.

**2.1. Information theory**

Complexity is one of the more essential aspects of the sustainable city. Complexity generates a rich urban fabric by maximizing, for instance, the points of contact, exchange and interface. But complexity of fabrics is also a component in the energy efficiency of cities. Conceptually complexity is a more difficult parameter to define than more intuitive parameters such as intensity or proximity. Information theory, as formulated by Shannon starting in 1949[1], sheds some light on this point. The theory was developed for the field of communication in an attempt to quantify the information transmitted in a message. Since, the applications have extended far beyond the boundaries of communication sciences alone. In the following section we will endeavor not so much to present what Shannon’s original theory (with its vocabulary strongly colored by communication sciences: receptor, transmitter, emitter…) brings to us, but rather to adapt our formulation to the field of application that interests us here: the complexity of urban fabrics.

We will ground our formulation in the example of a distribution of building types: housing/offices/shops. When we speak of the quantity of information present in a distribution, we are actually looking at the proportion of each category in the whole. The city can contain housing, offices and shops in a proportion of, say, (0.33; 0.33; 0.33) or (0.8; 0.1; 0.1). In the first case, if you take a random building, the probability is one out of three that you will find the category to which it belongs. In the second case, the probability is 8 out of 10 that it will be housing, and one out of ten for the other two categories. Intuitively we recognize that eliminating the uncertainty in the first case provides more information than doing so in the second case: supposing that the building randomly picked is housing, there is little chance of making a mistake in the second distribution, whereas in the first there is one chance in three. Quantifying the information of a given distribution (1/3 housing, 1/3 offices, 1/3 shops) involves quantifying the uncertainty that is eliminated when the real situation is given (building 1 is an office, 2 is a shop, and 3 housing). By quantifying the information present in a distribution, Shannon’s formula translates this point in quantitative terms. For a distribution (p1, p2, …, pn) denoted (pi)i=1…n, the associated quantity of information is, in denoting log2 the logarithm to the base two, given by the Shannon entropy formula: By quantifying the information contained in a distribution, this formula quantifies the complexity of the distribution. When we push the reasoning farther, in the field of statistical physics and thermodynamics, this definition corresponds in fact (except for one constant) to statistical entropy as formulated by Boltzmann, which can be interpreted as the measure of the degree of disorder of a system on the microscopic level.

**2.2. The fractal city**

The link between entropy and information quantity provides the link between information and energy in the systems studied. The maximization of the quantity of information of a system (and hence its complexity) allows us to minimize the associated energy. Consequently, the complexity of urban fabrics plays an essential role in minimizing the energy consumed.

The theory of fractals [2], [3], [4], [5] brings an interesting perspective on the complexity of systems. The theory of the fractal structure of urban fabrics was developed in detail by Nikos Salingaros [6],[7],[8] and Christopher Alexander[9],[10],[11]. As it turns out, fractal structures tend to provide an intense information field at any distance [12] (in the sense of Shannon’s theory), and lead to local equilibriums [13]. This result is echoed in many natural structures: lungs, blood vessel system (arteries, arterioles, capillaries), snowflakes, trees, ferns and so on. The fractal structure of urban fabrics, by maximizing the complexity, work to minimize the structural needs in energy. The inverse power scaling law [14] proposes and quantifies a fractal distribution of objects of different scales. Three areas of application of fractals for the city are highlighted by Salingaros in his publications.

- - Networks: distributing lengths and widths of roads as well as travel speeds by drawing on fractal theory (by analogy with the blood vessel system, for example)
- - Resources: allocate resources for big and small projects according to the inverse power law
- - Sizes: distribute the sizes of urban elements according to the inverse power law

We could have chosen to quantify the complexity of urban structures based on this indicator alone. But it covers different senses of complexity, and proves to be particularly difficult to use as an indicator on both a practical and an operational level. We will go into greater detail hereafter about two types of indicators of complexity. The first are used to quantify the diversity, spatial distribution, or mix present within urban fabrics. The second indicator type, that will assume different forms, will consist in a relationship of complexity to the energy consumed by the system. It will in fact translate the quantity of energy needed to create complexity.

**3. Defining urban complexity**

The preceding section led us to specify certain essential aspects of urban fabrics that derive from complexity: diversity, spatial distribution, or mixing. These essential concepts are rarely employed rigorously and appropriately and often cover very different objectives. By way of example, diversity can refer to jobs, land use (distribution of housing, offices, roads, etc.), or building size. In this case, it is hard to differentiate from the social or functional mix: the former is usually measured by percentage of social housing; the latter will be based on legal entities or the destinations of premises and buildings. These notions, which are close yet subtly different, are essential points in the sustainable city. To grasp the meaning of the results obtained based on the indicators we propose, it is therefore indispensable to define precisely what we mean by these terms.

**3.1. Diversity among similar objects**

The first meaning of diversity corresponds to bringing together several similar objects – that is, objects belonging to the same family but offering different configurations. By social mix, for example, we mean a bringing together in a single place of different populations having diverse income levels, ages or origins. The family is the same, since we are dealing with people, but the age configurations, for example, are different. This difference within the same family of objects is what constitutes diversity.

But diversity usually also implies an equity between the representative of these different configurations, meaning an even distribution. Indeed, instinctively, when we speak as simply as can be of a “good” diversity we are referring to an equity between configurations: as many under 20s, as 20s to 40s, as 40s to 60s, as over 60. In the case of functional diversity, there would be as many housings units as offices and as shops on a given scale.

Indicators concerning even distributions are the most commonly used these days. But this prejudice does not resist a close study of the question, and moreover it does not lend itself well to most of the problems that the indicators are expected to address. How, for instance, can there be an equal number of representatives of each age group in a district, given the current age pyramid?

The reasoning can be improved to begin with by taking socio-economic factors into account, to correspond to the reality of the distribution of populations in the city or to the needs for housing as compared to offices, for example. However, optimal distribution is a complex thing to establish: it is difficult not to say dangerous to try to program the distribution of housing/offices/shops. On the other hand, public authorities can elaborate distribution objectives based on surveys of what is lacking or imbalanced in the city or district. To counter a lack of housing and a profusion of offices in a district that work together to cause an exponential rise in real estate prices, which in turn destroys even more housing, public authorities can set down regulations to correct the imbalance. In this case, a structural objective takes priority over an even distribution.

**3.2. Diversity in space or spatial distribution**

The distribution of objects in space must also respond to an objective of “diversity”. In this case, it is more appropriate to speak of spatial distribution than simply diversity. What is sought here is a mix, meaning an equitable distribution of different activities in the district and city. We are no longer dealing with diversity on the scale of the city but rather with an equidistribution. This is often applied to the distribution of public buildings or cultural facilities on the city scale. An equidistribution reduces the distances that residents have to travel for their common everyday needs and activities, and avoids the unidirectional transport flows created by mono-functional spaces. Yet, one must be careful about the object to which equidistribution is applied and to the aims. An equidistribution of green spaces can be counterproductive if these spaces are too small for the population to really benefit from them and it can work to prevent the creation of a big green space capable of sustaining greater biodiversity. Finally, the equidistribution should not necessarily stand in the way of the formation of specific centers – environmental, in the preceding example, or economic when we think of finance or industrial centers – which can also be useful.

**3.3. Diversity of objects of different scales**

Diversity is also created by a range of different scales. This notion is rarely discussed even though it is fundamental. What makes the city varied for people walking through it derives to a great extent from the diversity of the scale of objects encountered. This hierarchy of scales plays a fundamental role in the field of street networks[10,14],[15]. Salingaros distribution[14] provides a very good description of this phenomenon of diversity that corresponds in fact to the right proportion of objects of different sizes.

Thus, connectivity is possible only when there is a variety of road sizes and the right proportion of roads of each size. Nature gives us very good examples of this phenomenon in plants and animals: irrigation systems, be it the blood vessel system or sap circulation, are based on a few big arteries, that lead to narrower but more numerous vessels, that lead to a multitude of capillaries. In the case of the city, Salingaros found that this distribution was more efficient when it followed a power law, with a great number of smaller elements, a smaller number of mid-size elements, and a very small number of big elements. This rule applies extremely well to communication roads but also to buildings and other components of the city. The multiplicity of small objects recreates a district that is on a human scale and is reassuring for people walking through it. The biggest elements are landmarks that act as such precisely because of the overlapping of scales and the right proportion of objects in each category. Here diversity stands in contrast to the concept of even distribution discussed above.

**4. Quantifying diversity**

To the three types of diversity outlined above correspond mathematical theories and formulas that improve our comprehension and use of these categories. The use of one formula or another in calculating the indicator will automatically orient the interpretation of the result. It is consequently important to underline in which case to use which formula.

**4.1. Quantifying proximity to an equidistribution**

Simpson’s diversity index Simpson’s formula is the most commonly used formula for determining richness (understood as diversity) of ecosystems. It is also used under the name of the Herfindahl-Hirschmann index to find the concentration ratio of an economic sector (concentration of market shares in the hands of a small number of businesses). The index obtained is expressed as equal to 1 minus the sum of the proportions of each object squared, the object could be animals in the case of biodiversity or market shares in the case of concentration of economic sectors. Each object is defined by: ni the number of objects of the category under consideration (for example, the number of service activities) over the total number of objects N (for example, the total number of activities), Cat being the number of categories. The index hits its maximal value when there is an even distribution . One of its shortcomings, however, is its reliance on the number of categories under consideration, which makes the results vary considerably.

The 4th and 5th columns show that the index is maximal for an even distribution. The 2nd, 3rd, 4th and 6th columns show that the greater the number of categories, the closer the index is to one. This indicator is interesting in cases when an even distribution is the optimal distribution. But the index lacks legibility because an even distribution may be translated by different values depending on the number of categories under consideration. This means that the distance from an equidistribution cannot be determined without having previously calculated the value obtained in the case of an even distribution. Finally, the reliance of the indicator on the number of categories under consideration greatly reduces the possibilities of comparison and limits its use as an indicator.

To resolve this problem, one can slightly modify the index to make the distance from an equidistribution visible without another calculation.

The closer the indicator is to 1, the closer the distribution is to an even distribution. The performance of the object under consideration and hence comparisons become simpler. For the same number of categories, the object with the highest performance in relation to the goal of even distribution will be the one whose indicator is closest to 1.

**4.2. Quantifying a spatial distribution**

The aim associated with an indicator of spatial distribution is the most “equitable” distribution possible of an object. The objective then is an even distribution but on a specific scale. In fact, it is a matter of choosing a couple of scales because what is studied is the imbrication of smaller scales within a bigger scale, which means the distribution of a quantity of objects present on scale n°1, on the level of scales n°2. The prime couples will be: city/districts, or district/blocks. We can, for example, apply this reasoning to the distribution of parks in the city. If a city has, say, 2000 ha of parks, it may be appropriate for them to be equitably distributed: 100 ha of parks in each of the 20 districts (of the same size) in the city so that all residents have green spaces nearby. This same analysis can be applied to the district/block tandem. The results associated with the indicator allow us to measure the distance from an even distribution and hence from an equal distribution between the small scales selected.

The indicator we propose is simply a “spatialized” Simpson’s formula. The different categories in the case of parks will be the districts. The different proportionswill be the area of the park over the area of the district. More generally, we could write, for a city that has a number D of districts, each of an area Si , and Ai the item under consideration in this district: The closer the indicator is to 1, that is, to an even distribution, the more equitable is the spatial distribution.

4.3. Quantifying a distance from a structural distribution objective

As we have already noted, the diversity of objects of a similar nature (offices/housing/shops for example) is not always translated by an objective of even distribution. A structural objective can set a different distribution between objects. It is interesting in this case to create an indicator that yields the distance of the actual distribution from the targeted structural distribution. This distance can be expressed simply as the sum of relative distances from the objectives (squared) in each category. The factor in 1/Cat (with Cat the number of categories) allows us once again to disregard the number of categories. This formula translates the average of relative distances inside each category between the target and the current value.

The smaller the value of this indicator, the closer the distribution to the structural objective. The ideal result here is zero rather than 1 as in the preceding formula.

**4.4. Quantifying the imbrication of scales using Nikos Salingaros’s theory**

All of the approaches discussed until now are used to quantify the diversity or spatial distribution of objects of the same type (offices/housing/shops, green spaces/roads/buildings) without taking the size of the objects into consideration. Salingaros’s work suggests the importance of a distribution that varies as a function of the size of the objects examined. Unlike the indicators presented so far, this indicator integrates the scales of the compared objects. Indeed, the result obtained using the other indicators would be the same if there were 48 opera houses for every bus stop or 48 bus stops for every opera house. Whence the interest of an indicator that quantifies the distance from Salingaros’s type of distribution. The formula can seem complex but it translates a relationship that is often encountered empirically.

This distribution translates an intuitive phenomenon reminiscent of natural irrigation systems (take the example of the blood vessel system mentioned above): there are more mid-size streets than boulevards, and even more narrow streets.

As in the case of the distance from a structural distribution, we are aiming for the closest possible distribution to the one recommended by Salingaros, and hence zero (and not 1 as for Simpson’s formula).

The table below shows the differences observed for the 2 indicators. The scenario under consideration focuses on the distribution of boulevards (20m wide), mid-size streets (10m wide) and narrow streets (5m wide). The first case, titled “coherent”, displays a logical and intuitive distribution close to Salingaros’s. The value of the indicator that shows the “relative distance from Salingaros’s distribution” is 0.018, which is close to zero (the closer it comes to this distribution, the closer the indicator is to zero). The value of Simpson’s index is 0.79: the distribution is far from an even distribution, hence Simpson’s index is far from 1. If the distribution is reversed and there are a great many boulevards, an average number of mid-size street and few narrow streets, we immediately see appearing the limits of the Simpson index for objects of different scales, since it is strictly identical to the preceding case! On the other hand, the index based in the Salingaros formula explodes (9.56) – a clear expression of the incoherence of the situation. The last two columns show the difference between an even distribution (9,9,9) and an intermediary situation (6,9,12) between an even distribution and an optimal distribution. In these two cases, the results for the Simpson indices are very close (1 and 0.96), even though the intermediary situation seems intuitively to be more advisable than the even distribution. The index based on Salingaros’s formula is still high (0.57: the distribution remains poor), but lower for the even distribution (1.92: this distribution is nonetheless preferable to an even distribution).

** 4.5. What indicator for what diversity?**

As we have seen, the term diversity masks important nuances that determine the mathematical formula that will be used and consequently the result of the indicator. For each important theme, the notion of diversity has to be specified: does the indicator compare objects of the same nature or of different sizes and scales? In the case of objects similar in nature, we will need to know whether the objective is an even distribution or if different proportions have to be defined that will constitute a structural objective.

**5. Energy and Complexity**

**5.1. Indicator of complexity**

We have specified in the preceding sections the different senses that can be covered by the terms “diversity”, “spatial distribution”, “mix”, for similar objects, or for objects of different scales. This perspective allows us to tackle the concept of urban complexity with greater relevance than Shannon’s simple index, H, which is hardly operational or easy to use.

Here we propose to define the complexity of urban fabrics as the imbrication of two concepts of diversity. Let’s take the example of the distribution of uses of buildings: housing, offices or shops. We have seen above that only a structural analysis was capable of determining the optimal proportions of usages. Now let’s take the example of shops. A fractal structure for shops creates a lively, dynamic fabric: some big shops (supermarkets), more mid-size shops (mini-markets), and even more neighborhood convenience stores. The fabric’s complexity and richness can be expressed by mixing the two indicators: indicator of distance from the structural distribution and indicator of distance from the Salingaros distribution. We calculate the indicator of distance from the structural distribution of different categories, and within each category, the distance term is weighted by the Salingaros indicator.

This indicator represents once again the distance of the actual distribution from the structural distribution (proportion of offices, housing and shops, for example) with the term. The only difference from the indicator of distance from the structural distribution developed earlier is that this term is weighted by the term of distance from the fractal distribution within the category under consideration – for example the distance for shops between the number of supermarkets, mini-markets, and neighborhood convenience shops, on the one hand, and the theoretical distribution following the inverse power law, on the other:

The closer the overall distribution to the structural distribution, the closer the indicator to zero. Within each category, this distance decreases even more when the category follows a fractal distribution, but increases when the category is not very fractal. Finally, the closer the distribution to the structural distribution and the more fractal the categories within them, the closer the complexity indicator is to zero.

This innovative indicator will be the object of analyses on existing urban fabrics. It requires a great amount of data, which is not necessarily available. Note that we have expressed this concept in the case of a distribution of uses, but it can also serve to quantify urban complexity on other levels: buildings, networks, activities, land use, etc.

**5.2. Indicator of complexity and energy **

The analysis of the complexity and diversity of urban fabrics allows us to create a final indicator associating complexity (or diversity) and energy consumption, adapted from [16]. It involves quantifying the energy needed to create complexity, and vice versa, to quantify the impact of complexity on energy consumption. The most flagrant example is that of the street network. The more complex a network (in the sense of a distribution of scales, according to the analogy of the lungs or the blood vessel system), the more energy efficient it is. Rapid motorized transit permits a rapid irrigation of fabrics, while leaving room on smaller scales for cyclable or walkable networks. This indicator thereby serves to quantify the energy consumption that was eliminated thanks to the complexity of the fabrics. We are thus measuring the relationship between the complexity of the fabric and the energy. We have previously used the indicator of distance from the Salingaros law to quantify complexity: the more complex the city, the closer this indicator is to zero (this formulation is the easiest to use from an operational standpoint). For the following formula, it is necessary to use an indicator that correlates in the same direction as the complexity. We thus use the term 1/S. The more complex the city, the higher the term. We can then write the ratio E/(1/S) equal to ES. The higher the complexity of the urban fabric and the less the city consumes, the lower it is. For a network, for example, we can then express the ratio between the energy consumed for transports (divided by the area of the selection) and 1/S (inverse of the indicator of distance from Salingaros):

This network efficiency analysis can be extended to an economic analysis (economic gains due to the complexity), to an energy production analysis, etc…

**6. Conclusions**

This paper defines precisely the concepts which make the complexity of urban fabric which are sometimes treated in an imprecise way in the publications in this field. It also supplies mathematical formulae allowing us to quantify in an objective way the various constituents of this complexity: spatial distribution, structural distribution objectives, diversity among objects of different scales. Finally, it proposes innovative tools which will allow study of the correlation between the energy efficiency and the complexity of urban structures to be done in the future. The main barrier in the use of these tools lies in the data-gathering necessary to calculate the indicators – numerous data is indeed sometimes required. Nevertheless, our indicators offer the capacity of comparing several urban projects, according to the structural objectives defined by public actors and stakeholders of urban development, as instruments of comparison between different types of urban forms.

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