Applied Fractals: Searching for the Holy Grail in Nature with Scale -- Part III

In part II, we considered the ways that biological systems and corporations scale and discovered that they are similar in many ways.  But what if we go up another level to societies  -- specifically to cities?

And what does this all mean in the grand scheme of things?  Where have we seen it before?

​Scaling Cities (in a Single Bound!)

There is a science of cities developing under the Fractal Lens and complexity theory.  Cities turn out to be fractal-like in many ways.  To wit:

Cities scale in two ways.  Like biological scaling, some of the scaling is sublinear.  This applies most directly to city infrastructure— such as the length of roads, electrical cables, water pipes, and the number of gas stations.  For every doubling of the size of a city, only 85% more infrastructure is required – a savings of 15%.  This indicates a systematic economy of scale with a power-law exponent of about 0.85.   This does not seem to be that much of a savings at first until you consider that a city 100 times the size of another only requires 50 times more roads, electrical cables, water pipe and gas stations.

On the other hand, many socioeconomic factors in cities scale at a superlinear rate.  These include wages, wealth, patents, AIDS cases, crime, and educational institutions.

"[T]hese metrics not only scale in an extremely simple fashion following classic power law behavior, but they all do it in approximately the same way with a similar exponent of approximately 1.15 regardless of the urban system. So in marked contrast to infrastructure, which scales sublinearly with population size, socioeconomic quantities— the very essence of a city— scale superlinearly, thereby manifesting systematic increasing returns to scale. The larger the city, the higher the wages, the greater the GDP, the more crime, the more cases of AIDS and flu, the more restaurants, the more patents produced, and so on, all following the “15 percent rule” on a per capita basis in urban systems across the globe.

Thus the larger the city the more innovative “social capital” is created, and consequently, the more the average citizen owns, produces, and consumes, whether it’s goods, resources, or ideas. This is the good news about cities and why they have been so attractive and seductive. On the other hand, cities also have a dark side, so here’s the bad news. To approximately the same degree as for the positive indicators, negative indicators of human social behavior also systematically increase with city size: doubling the size of a city not only increases wages, wealth, and innovation by approximately 15 percent per capita but also increases the amount of crime, pollution, and disease to the same degree. Apparently, the good, the bad, and the ugly come together hand in glove as an integrated, almost predictable package. A person may move to a bigger city drawn by more innovation, more opportunity, better wages, and a greater sense of “action,” but she can also expect to confront an equivalent increase in garbage, theft, stomach flu, and AIDS."

This type of scaling is present in all countries, although the comparisons must be made between cities within the same country.

"These extraordinary regularities open a window onto underlying mechanisms, dynamics, and structures common to all cities and strongly suggest that all of these phenomena are in fact highly correlated and interconnected, driven by the same underlying dynamics and constrained by the same set of “universal” principles.  Consequently, each of these urban characteristics, each metric— whether wages, the length of all the roads, the number of AIDS cases, or the amount of crime— is interrelated and interconnected with every other one and together they form an overarching multiscale quintessentially complex adaptive system that is continuously integrating and processing energy, resources, and information. The result is the extraordinary collective phenomenon we call a city, whose origins emerge from the underlying dynamics and organization of how people interact with one another through social networks. To repeat: cities are an emergent self-organizing phenomenon that has resulted from the interaction and communication between human beings exchanging energy, resources, and information."

"To summarize: the bigger the city, the greater the social activity, the more opportunities there are, the higher the wages, the more diversity there is, the greater the access to good restaurants, concerts, museums, and educational facilities, and the greater the sense of buzz, excitement, and engagement. These facets of larger cities have proven to be enormously attractive and seductive to people worldwide who at the same time suppress, ignore, or discount the inevitable negative aspects and the dark side of increased crime, pollution, and disease. Human beings are pretty good at “accentuating the positive and eliminating the negative,” especially when it comes to money and material well-being.

. . . Cities provide a natural mechanism for reaping the benefits of high social connectivity between people conceiving and solving problems in a diversity of ways. The resulting positive feedback loops act as drivers of continuous multiplicative innovation and wealth creation, leading to superlinear scaling and increasing returns to scale. Universal scaling is a manifestation of an essential trait resulting from our evolutionary history as social animals common to all peoples worldwide, transcending geography, history, and culture. It arises from the integration of the structure and dynamics of social networks with the physical infrastructural networks that are the platform upon which the panoply of urban life is played out. Although this is a dynamic beyond biology, it shares a similar conceptual framework and mathematical structure as exemplified by fractal-like network geometries . . . ."

The health of a city can also be measured by its fractal dimensions.  Extensive research shows that citeis typically have fractal dimensions in the range of 1.2 but with a large variance up to values close to 1.8. 

" [P]erhaps one of the more interesting uses of a fractal dimension is as a diagnostic barometer of the health of a city. Typically, the fractal dimension of a healthy robust city steadily increases as it grows and develops, reflecting a greater complexity as more and more infrastructure is built to accommodate an expanding population engaging in more and more diverse and intricate activities. But conversely, its fractal dimension decreases when it goes through difficult economic times or when it temporarily contracts."

A Detour to Dunbar

One of the other things laid out under the Fractal Lens in Scale is the work of Robin Dunbar, who is famous for “Dunbar numbers” of social relationships.  These, too, turn out to be organized on fractal dimensions.

"Dunbar and his collaborators, who proposed that an average individual’s entire social network can be deconstructed into a hierarchical sequence of discrete nested clusters whose sizes follow a surprisingly regular pattern. . . .

He and his collaborators found that at the lowest level of the hierarchy the number of people with whom the average individual has his or her strongest relationships is, at any one time, only about five. These are the people we are closest to and care most deeply about; they are usually family members— parents, children, or spouses— but they could be extremely close friends or partners. . . . 

The next level up contains those you usually refer to as close friends with whom you enjoy spending meaningful time and might still turn to in time of need even if they are not on as intimate terms with you as your inner circle. This typically comprises around fifteen people. . . .

The next level pretty much defines the limit of your social horizon as far as personal interactions are concerned and consists of people you might refer to as “casual friends”— you know their names and remain in social contact with them. This group typically comprises about 150 people. It is this number that is usually referred to as the Dunbar number that has gained a certain degree of attention in the popular media.

 . . . You will notice that the sequence of numbers that quantify the magnitude of these successive levels of the group hierarchy— 5, 15, 50, 150— are sequentially related to each other by an approximately constant scaling factor of about three. This regularity is the familiar fractal-like pattern we saw not only in the network hierarchy of our own circulatory and respiratory systems but also in transport patterns in cities. In addition to the actual flows in these networks, the major geometric difference between them is the value of the branching ratio— the number of units, people in this case, at one level of the hierarchy relative to the next. . . .

The important point for our purposes is that viewed through a coarse-grained lens, social networks exhibit an approximate fractal-like pattern and that this seems to hold true across a broad spectrum of different social organizations."

Dunbar thought that the apparent universality of the Dunbar number has its origins in the evolution of the cognitive structure of the brain: "we simply do not have the computational capacity to manage social relationships effectively beyond this size. This suggests that increasing the group size beyond this number will result in significantly less social stability, coherence, and connectivity, ultimately leading to its disintegration."

This makes sense and takes us back to biological systems.  "[B]ecause the geometry of white and gray matter in our brains, which forms the neural circuitry responsible for all of our cognitive functions, is itself a fractal-like hierarchical network, this suggests that the hidden fractal nature of social networks is actually a representation of the physical structure of our brains.
This speculation can be taken one step further by invoking the idea that the structure and organization of cities are determined by the structure and dynamics of social networks, in which case the universal fractality of cities can be viewed as a projection of the universal fractality of social networks. Putting all of this together we are led to the outrageous speculation that cities are effectively a scaled representation of the structure of the human brain.  It’s a pretty wild conjecture, but it graphically incorporates the idea that there is a universal character to cities. In a nutshell: cities are a representation of how people interact with one another and this is encoded in our neural networks and therefore in the structure and organization of our brains."

Indeed, we have now come full circle from cities to brains, as consciousness is an emergent systemic phenomenon and not a consequence of just the sum of all the “nerve-paths and neurons” in the brain.  We explored this earlier with Michael Gazzaniga and This is Your Brain on Fractals.

Putting Cities and Brains Back Together

"The two dominant components that constitute a city, its physical infrastructure and its socioeconomic activity, can both be conceptualized as approximately self-similar fractal-like network structures. Fractals are often the result of an evolutionary process that tends toward optimizing specific features, such as ensuring that all cells in an organism or all people in a city are supplied by energy and information, or maximizing efficiency by minimizing transportation times or times for accomplishing tasks with minimal energy.

If it were possible for everyone to interact meaningfully with everyone else as in one great big happy family, then the above argument would imply that all socioeconomic metrics should scale with the square of the population size. This would mean an exponent of 2, which is certainly superlinear (it’s bigger than 1), but significantly larger than 1.15. However, this represents the extreme and totally unrealistic case where the entire population is in a frenzied state of continuous and complete interaction with itself . . .

This exercise shows that there is a natural explanation for why social connectivity and therefore socioeconomic quantities scale superlinearly with population size.  Socioeconomic quantities are the sum of the interactions or links between people and therefore depend on how correlated they are. In the extreme case when everyone is interacting with everyone else we saw that this leads to a superlinear power law whose exponent is 2. However, in reality there are significant constraints on the intensity and magnitude of how many people an individual can interact with, and this drastically reduces the value of the exponent to be less than 2."

If we consider individuals to be like the “invariant terminal units” of social networks, this means that on average each person operates in roughly the same amount of social and physical space in a city. 

"This is in keeping with the implications of a “universal” Dunbar number and the space-time limitations on mobile activity in cities that we just discussed. Recall that the physical space in which we operate is spanned by space-filling fractal networks, such as roads and utility lines that service infrastructural terminal units such as houses, stores, and office buildings where we reside, work, and interact, and between which we also have to move. The integration of these two kinds of networks, namely, the requirement that socioeconomic interaction represented by space-filling fractal-like social networks must be anchored to the physicality of a city as represented by space-filling fractal-like fractal-like infrastructural networks, determines the number of interactions an average urban dweller can sustain in a city.

Consequently, the exponent controlling social interactions, and therefore all socioeconomic metrics— the universal 15 percent rule for how the good, the bad, and the ugly scale with city size— is bigger than 1 (1.15) to the same degree that the exponent controlling infrastructure and flows of energy and resources is less than 1 (0.85), as observed in the data.

The approximate 15 percent increase in social interactions and therefore in socioeconomic metrics such as income, patents, and crime generated with every doubling of city size can be interpreted as a bonus, or payoff, arising from the 15 percent savings in physical infrastructure and energy use. The systematic increase in social interaction is the essential driver of socioeconomic activity in cities: wealth creation, innovation, violent crime, and a greater sense of buzz and opportunity are all propagated and enhanced through social networks and greater interpersonal interaction.
It is perhaps not so surprising that there is a correlation between increased social interaction, socioeconomic activity, and greater economies of scale.

What is surprising, however, is that this pivotal interrelationship follows such simple mathematical rules that can be expressed in an elegant universal form: the sublinearity of infrastructure and energy use is the exact inverse of the superlinearity of socioeconomic activity. Consequently, to the same 15 percent degree, the bigger the city the more each person earns, creates, innovates, and interacts— and the more each person experiences crime, disease, entertainment, and opportunity— and all of this at a cost that requires less infrastructure and energy for each of them. This is the genius of the city. No wonder so many people are drawn to them."

What are Some of the Emergent Properties of Cities?

An emergent property of urbanization is that the pace of life moves faster in larger urban environments:  "[T]he superlinear dynamics of social networks leads to a systematic increase in the pace of life: diseases spread faster, businesses are born and die more often, commerce is transacted more rapidly, and people even walk faster, all following the 15 percent rule. This is the underlying scientific reason why we all sense that life is faster in a New York City than in a Santa Fe and that it has ubiquitously accelerated during our lifetimes as cities and their economies grew. This effective speeding up of time is an emergent phenomenon generated by the continuous positive feedback mechanisms inherent in social networks in which social interactions beget ever more interactions, ideas stimulate yet more ideas, and wealth creates more wealth as size increases."

Cities are largely constrained by commuting time:  The average “commuting time” over centuries has been about one hour per day, going both ways.  Cities where walking is the primary mode of mobility are thus constrained to be no more than about 5 km or 3 miles across.  Cars expand the reasonable city size to about 40 km or 25 miles across.

People do walk faster in big cities:  The power law exponent is about .10, meaning that in a city twice as large as another, the people will walk about 10% faster.

The total number of contacts between residents of cities scales on a power law with an exponent of 1.15:  GDP, income, patent production, and crime all scale at the same rate.  This confirms "the hypothesis that social interactions do indeed underlie the universal scaling of urban characteristics. Further confirmation is provided by the observation that both the total time people spent on phone call interactions and the total volume of all of their calls systematically increase with city size in a similar fashion. These results also verify that the accelerating pace of life originates in the increasing connectivity and positive feedback enhancement in social networks as city size increases."

Cities trend towards unbounded growth, absent external constraints:  In contrast to the situation in biology, the supply of metabolic energy generated by cities as they grow increases faster than the needs and demands for its maintenance. Consequently, the amount available for growth, which is just the difference between its social metabolic rate and the requirements for maintenance, continues to increase as the city gets larger. The bigger the city gets, the faster it grows— a classic signal of open-ended exponential growth. A mathematical analysis indeed confirms that growth driven by superlinear scaling is actually faster than exponential: in fact, it’s superexponential. Even though the conceptual and mathematical structure of the growth equation is the same for organisms, social insect communities, and cities, the consequences are quite different: sublinear scaling and economies of scale that dominate biology lead to stable bounded growth and the slowing down of the pace of life, whereas superlinear scaling and increasing returns to scale that dominate socioeconomic activity lead to unbounded growth and to an accelerating pace of life.

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Moving Toward a Unified Field Theory of Complexity

We previously explored the limits of reductionism and how the nature of reality is one of emergent orders that build or stack upon one another in The Emergence of Emergence and the Limitations of Reductionism:  An Exploration of Hierarchies.

Scale puts yet another nail in the coffin of strict scientific reductionism or determinism and gives us a better and more fundamental idea to investigate:

"[T]he concept of a Theory of Everything connotes the grandest vision of all, the inspiration of all inspirations, namely that we can encapsulate and understand the entirety of the universe in a small set of precepts, in this case, a concise set of mathematical equations from which literally everything follows. Like the concept of God, however, it is potentially misleading and intellectually dangerous. Referring somewhat hyperbolically to a field of study as the Theory of Everything connotes a certain degree of intellectual arrogance.

Qualitatively, this extreme version of reductionism . . . [has something] missing. The “something” includes many of the concepts and ideas implicit in a lot of the problems and questions considered in [Scale]: concepts like information, emergence, accidents, historical contingency, adaptation, and selection, all characteristics of complex adaptive systems whether organisms, societies, ecosystems, or economies. These are composed of myriad individual constituents or agents that take on collective characteristics that are generally unpredictable in detail from their underlying components even if the dynamics of their interactions are known. Unlike the Newtonian paradigm upon which the Theory of Everything is based, the complete dynamics and structure of complex adaptive systems cannot be encoded in a small number of equations. Indeed, in most cases, probably not even in an infinite number. Furthermore, predictions to arbitrary degrees of accuracy are not possible, even in principle.
So while applauding and admiring the search for a grand unified theory of all the basic forces of nature, we should recognize that it cannot literally explain and predict everything.

Consequently, in parallel with the quest for the Theory of Everything, we need to embark on a similar quest for a grand unified theory of complexity. The challenge of developing a quantitative, analytic, principled, predictive framework for understanding complex adaptive systems is surely one of the grand challenges for twenty-first-century science."

And so there you have it.  A new age of scientific discovery is upon us, built upon the insights of Benoit Mandelbrot, the subsequent development of complexity theory and the ever-more clear and focused Fractal Lens.  The work of the Santa Fe Institute has only just begun.  We haven’t found a Holy Grail yet, but at least we know we’re on the right trail.