A City Is a Complex Network
Faculty of Engineering and Sustainable Development, Division of GIScience
University of Gävle, SE-801 76 Gävle, Sweden
(Draft: August 2015, Revision: December 2015, January, February 2016)
A city is not a tree but a semi-lattice. To use a perhaps more familiar term, a city is a complex network.
The complex network constitutes a unique topological perspective on cities and enables us to better
understand the kind of problem a city is. The topological perspective differentiates it from the
perspectives of Euclidean geometry and Gaussian statistics that deal with essentially regular shapes
and more or less similar things. Many urban theories, such as the Central Place Theory, Zipf’s Law,
the Image of the City, and the Theory of Centers can be interpreted from the point of view of complex
networks. A livable city consists of far more small things than large ones, and their shapes tend to be
irregular and rough. This chapter illustrates the complex network view and argues that we must
abandon the kind of thinking (mis-)guided by Euclidean geometry and Gaussian statistics, and instead
adopt fractal geometry, power-law statistics, and Alexander’s living geometry to develop sustainable
Keywords: Scaling, living structure, theory of centers, objective beauty, head/tail breaks
A city is not a tree but a complex network. Implicit in Alexander’s earlier works (e.g., Alexander
1965), this insight on city is a foundation for the Theory of Centers (Alexander 2002–2005).
According to the theory, a whole consist of numerous, recursively defined centers (or sub-wholes) that
support each other. A city is a whole, as is a building, or a building complex. The centers and their
nested, intricate relationship constitute a complex network (see below for further discussion). The
complex network offers a unique perspective for better understanding the kind of problem a city is
(Jacobs 1961). Based on the premise that a whole is greater than the sum of its parts, complexity
science has developed a range of tools, such as complex networks (Newman et al. 2006) and fractal
geometry (Mandelbrot 1982), for enhancing our understanding of complex phenomena. Unlike many
other pioneers in the field, Alexander’s contribution to complexity science began with creation or
design of beautiful buildings. The Theory of Centers, or living geometry, is much more broad and
profound than fractal geometry. Living geometry aims for creation (Mehaffy and Salingaros 2015),
while fractal geometry is mainly for understanding. Creation or design is the highest status of science.
This chapter will elaborate on the network city view and how its advance significantly contributes to a
better understanding of fractal structure and nonlinear dynamics of cities. I will begin with hierarchy
within, and among, a set of cities, then illustrate beauty and images emerging from a complex network
of centers, and end up with further discussions on fractal geometry and living structure for sustainable
2. Hierarchy within, and among, cities
A city is not a complex network seen from individual street segments or junctions. This is because
both street segments and junctions have more or less similar degrees of connectivity (approximately
four), very much like a regular or random network. However, a city is a complex network seen from
individual streets. The streets are created from individual street segments with the same names or
good continuity; so-called named and natural streets (Jiang and Claramunt 2004, Jiang et al. 2008).
Unlike street segments that are more or less similar, there are all kinds of streets in terms of lengths or
degrees of connectivity. The topological view helps develop new insights into cities. To illustrate, let
us look at the street network of the historic part of the city Avignon in France. The network comprises
341 streets, which are put into six hierarchical levels based on the head/tail breaks, a classification
scheme, as well as a visualization tool, for data with a heavy-tailed distribution (Jiang 2013a, Jiang
2015a). Given the set of streets as a whole, we break it into the head for those above the mean and the
tail for those below the mean, and recursively continue the breaking process of the head until the
notion of far more less-connected streets than well-connected ones is violated; the head/tail breaks
process can be stated as a recursive function as follows.
Recursive function Head/tail Breaks:
Break a whole into the head and the tail;
// the head for those above the mean
// the tail for those below the mean
While (head <= 40%):
Head/tail Breaks (head);
Figure 1: (Color online) Hierarchy of the street network of Avignon, and its connectivity graph both
showing far more less-connected streets than well-connected ones
(Note: The hierarchy is visualized by the spectral color with blue for the least-connected streets and
red for the most-connected ones. The 341 streets and their 701 relationships become the 341 nodes
and 701 links of the connectivity graph.)
The head/tail breaks enables us to see the parts and the inherent hierarchy. The resulting hierarchy is
visualized using the spectral color, with blue for the least-connected streets and red for the
most-connected ones (Figure 1a). The 341 streets and their 701 relationships (intersections) are
converted respectively into the nodes and links of a connectivity graph (Figure 1b). The connected
graph is neither regular nor random, but a small-world network – a middle status between the regular
and random counterparts (Jiang and Claramunt 2004, Watts an Strogatz 1998). The ring-like
visualization shows the connectivity graph with a striking hierarchy of far more small nodes than
large ones, with node sizes indicating the degrees of connectivity. Networks with this scaling
hierarchy have an efficient structure, commonly known as scale-free networks (Barabási and Albert
1999). Both small world and scale free are two distinguished properties of complex networks. A
complex network is highly efficient, both locally and globally, inherited respectively from the regular
and random counterparts. How is a complex network developed? What are the underlying
mechanisms of complex networks? How do we design a complex network of high efficiency? These
questions are design oriented, with far-reaching implications for architectural design and city planning.
Inspired by Alexander’s works (Alexander 2002–2005), a theory of network city (Salingaros 2005)
has already been developed for dealing with various urban-design issues.
Not only a city but also a set of cities (or human settlements, to be more precise) is a complex network.
All cities in a large country tend to constitute a whole, as formulated by Zipf’s Law (Zipf 1949) and in
the Central Place Theory (Christaller 1933, 1966). According to Zipf’s Law, city sizes are inversely
proportional to their rank. Statistically, the first largest city is twice as big as the second largest, three
times as big as the third largest, and so on. Zipf’s Law is a statistical law on city-size distribution, and
it does not say anything about how the cities are geographically distributed. The geographical
distribution of cities is captured by the Central Place Theory. Cities in a country or region tend to be
distributed in a nested manner, i.e., each city acts as a central place, providing services to the
surrounding areas. Conversely, small cities tend to support large ones, which further support even
larger ones in a nested manner. The Central Place Theory is about a network of cities or human
settlements that constitute a scaling hierarchy. The underlying network structure formulated by the
Central Place Theory resembles the structure of a whole, in which recursively defined centers tend to
support each other (Alexander 2002–2005, Jiang 2015b). In this regard, cities in a country or region
can be considered to be a living structure.
3. Beauty and image out of complex networks
Alexandrian living structure is a de facto complex network of numerous centers. The centers are
recursively defined, which means that a center contains smaller centers and is contained within larger
centers. Besides the nested, intriciate relationships among the numerous centers, they tend to support
each other to constitute a whole. In this context, wholeness, as defined by Alexander (2002–2005), is
considered to be a global structure or life-giving order emerging from the whole as a complex network
of the centers. This complex-network view of whole captures the mathematical model of wholeness as
part of the Theory of Centers, and enables us to compute the degrees of wholeness or beauty (Jiang
2015b). Using Google’s PageRank algorithm, beautiful centers are defined as those to which many
beautiful centers point. This definition of beautiful centers is recursive, and computation of the degree
of beauty is achieved through an iterative process until a convergence is reached. Eventually, each
center is assgined to a degeee of beauty between 0 and 1. The degree of beauty of the whole can be
measured by the ht-index, a head/tail breaks-induced index; the higher the ht-index, the more
beautiful the whole. Let us illustrate the computation using the Alhambra plan as a working example
at a building complex scale.
Figure 2: (Color online) The complex network of the centers with the plan of Alhambra
(Note: The degrees of beauty are calculated using Google’s PageRank algorithm; the bigger the dots,
the more beautiful the centers.)
The Alhambra is probably the most beautiful building complex in the world. It posseses many of the
15 geometric properies such as levels of scale, strong centers, thick boundaries, and local symmetries.
Seen from its plan, the most distinguished property is local symmetries. The plan does not look
globally symmetric, but the numerous local symmetries make it unique and beautiful. Let us focus on
the Alhambra plan that is partitioned into 725 convex spaces, each of which acts as a center. Most of
the centers are related to surrounding centers, as long as there is no barrier between them. This makes
880 relationships in total. There are a few isolated centers that do not contribute to the whole. The 880
relationships are directed from the peripheral small spaces to the central large spaces. Figure 4 shows
the result, in which the dots indicate the degrees of beauty; the bigger the dots, the more beautiful the
centers. It should be noted that there are 13 centers hidden or embedded in the network: One as the
whole, three subwholes, and nine subwholes of the three subwholes.
The living structure has deep implications for understanding the city structure from a cogntive
perspective. In this connection, the image of the city (Lynch 1960) is another classic in the field of
urban design. A large body of literature has been produced over the past 50 years. Much of of the
literature focuses on human internal representation, or how do mental images of a city vary from
person to person? In fact, it is the city’s external representation, or the city itself, or the living
structure, that makes a city imageable or legible (Jiang 2013b). To be more precise, the largest, the
most-connected, or the most meaningful constitute part of a mental image of the city. Among the five
city elements (paths, edges, districts, nodes, and landmarks), only landmarks capture the true sense of
scaling or living structure. The image of the Alhambra plan consists of three subwholes: The left,
middle, and right. Each of these comprises three further subwholes. Among the many other centers,
the most beautiful one, or the one with the most dense local symmetries, tends to shape our image of
the building complex.
4. Fractal and living structures
The topological perspective differentiates it fundamentally from the perspectives offered by Euclidean
geometry and Gaussian statistics. Euclidean geometry aims for measuring regular shapes, and
Gaussian statistics aims for analyzing more or less similar things. These two mathematical tools show
some constraints while dealing with complexity of the world. Instead of more or less similar things
and regular shapes, there are far more small things than large ones, and irregular shapes. To put them
in perspective, Euclidean geometry aims for measurement or scale, while fractal geometry aims for
scaling or the scaling pattern of far more small things than large ones. Gaussian statistics aims for
average things, while power-law statistics aims for outliers. Events of a small probability in Gaussian
statistics are impossible, whereas events of a small probability in power-law statistics are highly
improbable or vital. To a great extent, Euclidean and fractal geometries complement each other, and
one cannot stand without another. This is because one must measure all things under the framework of
Euclidean geometry to recognize scaling. However, our thinking in architecture and urban design is
very much dominated by Euclidean and Gaussian thinking. For example, to characterize a tree, we
tend to only measure its height, rather than all its branches. To illustrate, let us examine two patterns
shown in Figure 3.
Figure 3: Fractal versus Euclidean patterns
(Note: The left pattern appears appears metaphorically in traditional buildings, in the sense of all
scales involved rather than of precisely the same pattern, whereas the right pattern is pervasively seen
in modern buildings metaphically and in terms of precisely the same pattern.)
The square of one unit is cut into nine congruent squares, and the middle one is taken away. The same
procedure is recursively applied to the remaining eight squares again and again, until we end up with
the pattern commonly known as Sierpinski carpet (Figure 3a). This particular carpet of three iterations
comprises one square of scale 1/3, eight squares of scale 1/9, and 64 squares of scale 1/27. A
Sierpinski carpet is hardly seen in reality, but it helps illustrate some unique properties shared by the
real-world patterns, referring to not only those in nature but also those emerging in cities and
buildings. First, a pattern recurs again and again at different scales, known as self-similarity. Second,
there are multiple scales, rather than just one. It is essentially these two properties that differentiate the
left pattern from the right one in Figure 3. It is important to note that the right pattern is with nine
squares, which are disconnected each other. However, all the squares of the left pattern are connected
each other, according to Gestalt psychology (Köhler 1947). The largest square is supported by the
eight middle-sized squares, each of which is further supported by the eight smallest squares. This
support relationship is very much similar to the framework of the Central Place Theory.
Unfortunately, modern architecture has been deadly misguided by Euclidean geometry and Gaussian
thinking towards so-called geometric fundamentalism (Mehaffy and Salingaros 2006). Geometric
fundamentalism worships simple and large-scale Euclidean shapes, such as cylinders and cubes, so
removes small scales and ornament. However, scaling laws tell us that all scales ranging from the
smallest to the largest (to be more precise, many smallest, a very few largest, and some in between)
are essential for scaling hierarchy and for human beings. This scaling hierarchy appears pervasively in
traditional buildings such as temples, mosques, and churches, yet has been removed from
contemporary architecture and city planning. The life of living structure lies on the smallest scales or
fine structure (Alexander 2002–2005) as demonstrated in Figures 1 and 2. It is time to change our
mindsets toward fractal geometry, power-law statistics, and Alexandrian living geometry to develop
sustainable cities and architecture.
5. Concluding remarks
A city is not a simple network, as simple as a regular or random network. Instead, a city is a complex
network, or a middle status between the regular and random counterparts. It is highly efficient locally
and globally, inherited respectively from regular and random counterparts. Many urban theories, such
as the Central Place Theory, Zipf’s Law, the Image of the City, and the Theory of Centers can be
better understood from the perspective of complex networks. Network cities bear the scaling
hierarchy of far more small things than large ones, or living structure in general. This is the source of
structural beauty and the image of the city. The scaling hierarchy should be interpreted more broadly,
i.e., far more unpopular things than popular ones in terms of topology, or far more meaningless things
than meaningful ones in terms of semantics. In this connection, a city is indeed a tree in terms of the
The kind of complex network thinking is manifested in a series of Alexander’s works that are highly
iterative, such as Notes on the Synthesis of Form (Alexander 1964), A City Is not a Tree (Alexander
1965), The Timeless Way of Building (Alexander 1979), and The Nature of Order (Alexander 2002–
2005). The complex-network perspective implies that within a city, every element depends on every
other element, and changing one element would affect virtually every other in a design context. In this
chapter, I have shown the power of complex-network perspective in understanding city complexity, in
particular the topological view of city structure. Further work is expected towards the integration of
the Theory of Centers and network science, and of living geometry and fractal geometry, for
sustainable urban design.
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