Hierarchy is an entrenched social concept. The Internet however, presents the possibility of envisioning social relations as a level or ‘flat’ configuration. The Internet fosters relationships that are networked, heterogeneous and horizontally distributed. This article contemplates the surface features of networked structures like the Internet by using topographic imagery.
Visualizing the shape of complex networks
Visualizing small worlds
Topographic network representations
Heterogeneity and complexity
Hierarchy is a concept that has disproportionate social consequences. Dominant social institutions tend to be arranged into hierarchies. The shape of such institutions may be compared to a pyramid with a few individuals on top while the majority inhabits the bottom. Websters Dictionary defines hierarchy as, a group of persons or things arranged in order of rank, grade, or class. In social relations, hierarchy is associated with the categorical separation of individuals based on an assumed understanding of their status. As an ordering system, hierarchy is prone to stereotyping and simplifying in order to keep things in their place.
If looked at from a geometric perspective, hierarchy is based on a Euclidian separation between objects, shapes and scales. Persons and objects exist within the hierarchy as contained entities lying on a hierarchic scale of top and bottom. It is not possible to inhabit many scales at once. Hierarchy is a system of fixed things that are set apart from others by law of arrangement. At best, the idea of geometric hierarchy is summarized by the adage, A place for everything, and everything in its place. Hierarchy depends on a sorting mechanism that keeps things segregated based on an assumed intrinsic nature of objects and people on that scale.
When considering the social consequences of hierarchic systems, some have argued that social theory is inherently polarizing because it starts from an assumed hierarchy and then dictates subordinate status to all else . This sets up a series of social juxtapositions, (between the macro and micro, global and local, core and periphery, higher and lower class, developed and underdeveloped world). In this manner, hierarchy produces its own opposite through the language of dominance and subordination.
By contrast, nonhierarchic social relations have been associated with greater levels of innovation. AnnaLee Saxenian (1994) identified increased productivity in the nonhierarchic practices of workers in the West coast semiconductor industry. Decentralized computer programming has proven more innovative than hierarchic control in computer programming (Raymond, 1998; Kuwabara, 2000; Iannacci and MitletonKelly, 2005) . Moreover, anthropologists have long argued that diversity in social relationships is central to human evolution and cultural adaptation (Lansing, 2000; Mooney, 1996). Nevertheless, ideas about hierarchy persist in the cultural imagination.
Every social theory carries a reinforcing logic that validates the cultural reality which is being described. Whenever a concept such as hierarchy is in play (both in our social theories and in social relations,) we must speak of a specific instrumentality at work. Carlos Delgado Díaz (2004) compares the idea of hierarchy to an optical lens for seeing reality. The emphasis on hierarchy predisposes social theory to look for, and therefore focus on, systems that resemble hierarchic structures. At the same time, those forms of sociality that do not conform to hierarchy (those that are perhaps more complex) are marginalized in social theory. The Internet however, presents an altogether different kind of social structure that diverges from the hierarchic models of social organization. The Internet requires a new paradigm for describing the way the system is configured — and by extension, the social relations it enables.
It could be said that the Internet represents a level of complexity that overflows the social theories that attempt to represent it. Furthermore, the Internet with its dazzling array of human and nonhuman intersections may provide a template for reimagining social relations in a new figuration. This article considers the Internet as a prototype of nonhierarchic social relations. It explores the shape of such relations by drawing upon topographic network maps. The article is an exercise in imagination. It argues that aesthetics is a particular way of thinking that allows us to formulate new theoretical models of sociality which are flat rather than hierarchic.
Immediately, the question arises, what does aesthetics have to do with envisioning social networks and the Internet? The topic of aesthetics has been construed in a number of different ways but it is generally not considered part of any analytic or scientific inquiry. The word aesthetics is derived from the classical Greek (aisthetikos) which means to perceive, to sense, and to have the faculty of feeling. Likewise, the word art is from the Latin (ars) meaning to fit, to join, and to make form. With these two definitions there is an appreciation of the way in which aesthetics provides an entry point for perceiving the nature of structure (Guyer, 2005).
Nevertheless, aesthetics has come to be associated primarily with beauty, decoration and the appreciation of works of art. During the Enlightenment, aesthetics was thought to represent transcendental values of beauty and truth. Such concepts continue to inform the social definition of aesthetics and are evident in notions like, Art for arts sake. 
Yet, if we get back to the original definition of aesthetics as a problem solving activity, then aesthetics provides a way of envisioning structure that allows for experimentation and the formulation of social theory. Aesthetics is a tool for articulating newness, which is why it is so frequently associated with the avantgarde. The inspiration that derives from an aesthetic sensibility provides a source inner vision that feeds the science of discovery.
Aesthetics entails imagining structure through a kind of inner visioning. Carole MacKenzie and Kim James (2004) call this kind of thinking a graphic referencing technique.  Such a visualization technique supplies a practical resource for envisioning the complex interconnections in social organization. In the case of the Internet, aesthetics allows us to imagine the nature and disposition of the system, including a number of features such as accumulation, aggregations, acquisitions, borders, connection, constellation, dispossessions, dissipation, fission, fusion, formations, fusion, edges, margins, oscillations, spread, severance and new structural dynamics. By extension, aesthetics provides a means for examining social organization that is at once flat, and yet highly complex and multidimensional.
Visualizing the shape of complex networks
An important consideration for any inquiry structure of the Internet is whether or not the system can be said to have selfsimilar characteristics? This is an aesthetic question because it focuses on the structure of the complex network and how such a shape impacts the sociality that arises from it. To explore this theme, the article borrows ideas from fractal geometry and complexity studies, not merely to describe the features of the system but to discover the possibility of a selfsimilar social surface that may be intrinsic to the network form. Therefore, the inquiry not only speculates on what kind of shape a network like the Internet adopts, but also by implication the transcendence of that shape.
Complex systems that are networks are represented as graphs in mathematics. Technically speaking, a network is considered a set of fundamental units that are called vertices or nodes with connections between them. A node is the point at which a curve intersects itself. The line connecting two vertices is called either an edge, or bond (physics) link (computer science), or tie (sociology). An edge is directed if it runs in one direction or undirected if it runs in both directions. Directed edges may also be called arcs. In the scientific literature, there are numerous examples of networked systems including the Internet, the World Wide Web and a number of variously connected social and biological systems, (for instance a network of friendships) .
Figure 1: A diagram of a simple network in which a curve eventually intersects itself.
The importance of a network structure lies not with its inventory of individuals, but rather how they assemble together. The connections affect the vitality of a network. The connections run on a recursive logic. Recursivity is an inherent feature of complex networks and refers to objects or actions that appear at intervals, overlap, reappear and come back again. Recursive systems often function on a cycle of feedback loops. When accumulated, the feedback in a recursive system may display an underlying chaotic trajectory because of the build up of disturbances in the feedback cycle (McNeil, 2004; Castells, 1996).
A network like the Internet is comprised of looping circuits that continually send feedback messages to each other. The distribution of feedback in a network may result from the connections between individuals. Or it may result from internal social disturbances, or changing external and environmental conditions within the network. Negative and positive feedback is central to the path a network takes.
Often, negative feedback tends to keep a system in check. It is associated with the maintenance and stability of a complex system. Negative feedback is also associated with a networks regulation. Positive feedback however, may lead toward a rich, gets richer phenomenon. This feature of positive feedback is alternatively referred to as, preferential attachment. Positive feedback relates to an amplification of tendencies in a system because it reinforces the trajectory a system is taking (Bateson and Bateson, 1987) .
Feedback relates to the distribution of communication signals within a social organization. The constant streaming of information in a social network amounts to a feedback cycle that has cumulative results over time. Positive, local feedback may also be associated with the process of inducing larger, emergent changes in a system. Within a social network, recursive change takes on a chain reactionlike sequencing of behaviors and events. Each action produces a reaction which may then influence another action and so on. Social feedback may oscillate between negative and positive signals to determine the characteristics of events. At the same time, what happens in an organized complex network is not matter of simple cause and effect because the deviations contribute to novel evolutions within that system.
Networks can also be described in a general formulation way when referring to social organization. At the most basic level, a network is a group of individuals who know one another, and perhaps share certain affiliations, goals or resources. Networks usually lie embedded within or alongside of each other, and in this way can be said to be nested. The size of a network may be difficult to determine as a consequence of the varying scales that networks adopt. For instance, a church network of friends may connect to larger regional, national and even international networks of religious organizations. All of the cities of the world may be envisioned as networked together. The Internet may be considered the greatest humanlymade network since it is a network of other human networks. Thus, networks, both in their human and non human manifestations tend to exhibit scaling behavior that is synonymous with the small world phenomenon.
Visualizing small worlds
Most social networks carry the ability to scale upward to include larger and larger groups of people. However, the size of a social network remains a difficult concept to visualize. An examination of networks involves a reflection on the nature of social organization as a varied, overlapping and dynamic prospect. This is as opposed to conceiving of the world in terms of established vertical entities like a company, an institution or the state. The exercise entails examining sociality in its micro components and investing these with power, movement and agency.
The problem of visualizing a social network illustrates an intriguing phenomenon that was identified by Stanley Milgram (1967) in a paper entitled The Small World Problem, Milgrams hypothesis was that any person on any continent may be traced to any other by a brief chain of acquaintances. He argued that a fundamental feature of social networks is that they are grouped into small clusters each distinguished by a chain of friends and acquaintances (Watts and Strogatz, 1998; Milgram, 1967).
When Milgram designed his sociological protocol he called it the Small World Method in order to emphasis his hypothesis about connection. The object of the experiment was to see whether or not a package might be delivered to a chosen person through a series of acquaintances who did not know one another. To make the experiment more difficult, Milgram chose Omaha, Nebraska as the starting point for the experiment and Boston, Mass., as the target destination. Nebraska was chosen rather than a geographically more distant location, such as San Francisco, since it was deduced that the social distance from Omaha to Boston could not be greater. Three hundred packages were forwarded in a chain maillike fashion. Each sender was asked to forward the packet to a friend closer to the target person (Watts, 2004).
Milgrams original experiment was not as successful as he anticipated. Of all the chains of acquaintances he researched, only a fraction were completely mapped. A number of the packages were lost. In addition, the experiment collided with the computational limitations of the day. Given the technology of the 1960s, there was simply no easy way to calculate all of the connections involved in the Omaha to Boston network. Nevertheless, of the letters that reached their target the typical amount of exchanges between people amounted to six. Hence the experiment is alternately referred to as six degrees of separation (Milgram, 1967; Newman, 2003; Watts, 2004) .
Figure 2: A represention of Milgrams Small World Experiment based on my extrapolation of shape.
Most diagrams of the Small World Problem like Milgrams represent the acquaintances in a typical social network as a circle or a vertex. In a social network diagram, the lines represent the connections between the circles (vertices). A chain of connections between the social circles are represented as circles strung together like beads on a necklace. This is the way Milgram conceptualized his network model based on the juxtapositions of people nested side by side. In this sense, the points on the network are external to each other. The model exhibits a Euclidean separation between objects and individuals in geographic space.
In practice, Milgrams model implies how social networks operate in a radiating fashion. While a certain New Yorker may not personally know anyone living in Africa; a neighbor who works for an NGO might. Thus, the two hemispheres stand connected by a simple degree. In theory, each person need only possess a handful of acquaintances for such degrees to avalanche to include the billions of the worlds inhabitants. In some way, Milgrams experiment could be said to replicate a human version of mathematical recursion. In the experiment, each social circle carried out the same task which was a subset of the larger task of delivering the package to the designated location.
Milgrams initial hypothesis about connectivity still remains a pertinent way of envisioning circular recursive social networks. The problem also considers the question of shape since Milgrams experiment was based on a conjecture about the interwoveness of social circles. Yet, for over thirty years, research on small world networks remained sparse because of the complexity of the problem and the difficulty of calculating the amount of data. Then in 1998, Duncan Watts and Steve Strogatz wrote a short but groundbreaking paper in Nature entitled Collective dynamics of smallworld networks.
Watts and Strogatz approached the topic from a different perspective. They applied their backgrounds in physics to measure networks in terms of their order and randomness. They were also able to take advantage of faster computers. In the initial investigation, Milgram was concerned with the question of how large a population a small world might encompass in other words, how small is the actual world? Watts and Strogatz did not believe such an inquiry was answerable. Instead, they asked what would it take for any world to be small? (Watts, 2004; Watts and Strogatz, 1998)
Watts and Strogatz devised an experiment using the Internet. They used an email version of a message sender which simulated the connectedness of a network. First, they designed an ordered example and then increasingly introduced randomness into it. They called the approach random rewiring. The model aimed to replicate the connectivity of a friendship network. By taking a ring of nodes that connected neighboring edges to each other, the experiment sought to replace a few of the neighboring edges with randomly selected nodes.
Of the 48,000 senders in their experiment, and among the 19 targets in 157 countries, the scientists discovered that the average number of separations was actually six confirming Milgrams initial hypothesis (Watts, 2004; Watts and Strogatz, 1998).
Duncan Watts and Steve Strogatz (1998) drew their network diagram a bit differently. They connected their circles along a central ring which retained a necklace shape but was closed at the ends. The interaction between clusters was represented through a dynamic short cut route.
Figure 3: 1. Random network, 2. Ordered and 3. Locally ordered networks exhibiting longrange connections.
The drawing is adapted from Watts and Strogatz (1998).
The most intriguing result of Watts and Strogatzs research came from the networks that were neither totally ordered nor random. This occurred in the tight, locally ordered networks that carried a few long range, random connections. What happened was that the path length was abruptly decreased resulting in a shortcut route between clusters. While the random rewiring had an impact on path length for the network, the clustering remained very high and almost unchanged. The authors chose to refer to the outcome as the small world network because it exemplified the earlier experiment of the same name by Milgram. The network exhibited short cut global separations at the same time, as maintaining the density of clustering that is typified in the majority of social networks (Watts and Strogatz, 1998).
Watts and Strogatz found that a significant property of networks is that they form into clusters of connections. These clusters tend to form into triangles of relationships. Clusters, when illustrated on a low dimensional graph take the form of geodesic triangles (they look much like Buckminster Fullers architectural creations). A network is likely to be made up of vertex A connected to vertex B and to vertex C and so on. The tendency indicates that there are a series of triangles in a network equaling the sum of threeset vertices. Yet, each vertex is likely connected to each of the others through the clustering of ties. In sociological literature, this form of local clustering illustrates the notion of network density. Thus, the probability increases that vertex A will also be connected to vertex C. In the language of social networks, you might remark The friend of your friend is likely also to be your friend. 
Figure 4: A social network comprised of threeset vertices.
The level of connection identified by clustering behavior has also been observed in the empirical, realworld studies of networks. Both in terms of connectivity and communication, clustering in a network is central to its function (Newman, 2003; White and Harary, 2001).
The work of Watts and Strogatz remains premised on finding the connections between individuals within a social network. At the same time, the problem is one of scale. That is, because each network carries ties that connect people to each other, which in turn may connect to additional networks and so on. In terms of examining a social networks reach, the problem proves quite extensive.
On a flat plane, such ties may be envisioned as radiating in all directions. At certain points, the connections increase in density to form a tight knot. Such knots serve as the nodes between clusters that are tied into the overall network through the communication ties. A few short linkages serve to act as communication superhighways and expedite the delivery of information throughout the network. Both the clustering and the presence of communication highways give the network its small world distribution.
Since Watts and Strogatzs experiment, an increasing amount of research has been conducted on the topic of small world networks. The research on complex networks points to a similar conclusion social networks tend to carry short cut paths between apparently distant individuals. The term small world derives from this feature of networks. Yet these short cuts would matter little if people were not able to discover them. An apparent trait of the small world phenomena is the way in which individuals are able to effectively navigate a network with very little information to go on.
This facet of networks was pointed out by J.M. Kleinberg (2000) who surmised that the people in Milgrams original experiment possessed no extraordinary knowledge of the network that connected them to the targeted individual in Massachusetts. Most knew only their friends and likely a few of their friends friends. In Kleinbergs random graph simulation of the problem, it was revealed that the kind of information that people possess in realistic situations made it very difficult to locate all of the short cut routes. Despite the difficulty, Milgrams participants were able to deliver the message to the targeted person in only a few steps. The result implies that there is something extraordinary about the very structure of a network (Newman, 2003; Kleinberg, 2000).
Topographic network representations
What makes the Small World Problem an intriguing research question is its generality and the way it implies a global level of connection. The Small World Problem is focused on a very broadly construed notion of abstract shape. This includes the overarching relationships that knit the world together and how human beings are connected to one another as a generic condition. The abstractness of the problem has made it particularly amenable to mathematical graphing techniques and topology but not necessarily traditional social science methods (Kelty, 2004; Wobst, 1974) .
By employing aesthetics as a visualizing technique, it is possible to imagine a number of possibilities for rendering the network shape. The chains of relationships within a network may orchestrated in any number of ways to indicate shape and connectivity.
In my renditions below, the overlapping social circles may be combined to form a flowerlike cluster representing the initial six separations between social groups. Multiplied by two the figure shows increased density that is reminiscent of images generated by a spirograph, a drawing toy for children. Multiplied even further by four, the composition resembles the dizzying optics of moiré patterns. The Small World Problem may also be laid flat and replicated in a wallpaperlike fashion. In this way, the structure shares a patterned structure similar to M.C. Eschers changeable and interlocking designs. In these characterizations, the prospect of interpenetration is not necessarily apparent.
Figure 5: Using aesthetic visualization to hypothesize about four possibilities for the network shape.
The depictions of shape in this case, are still two dimensional which corresponds with the networks inclination toward flatness. However, when looked at from any number of different vantage points, image 4 appears to shapeshift. The image also has a fractal quality. So, while laid on a flat surface, the image appears to defy its two dimensionality. Additionally, the images may be combined with one another, for instance; the combination of images 3 and 4 would form an incredibly dense and complicated structure. Moreover, the patterns need not be limited to two dimensions but may also be articulated in threedimensional space. When visualized graphically in any one of these ways, the six degrees/small world dynamic takes on a dazzling complexity.
The above diagrams represent some of the ways of exploring the Small World Problem as a chain of circular social relationships. One of the problems with using a circle however, is that it is not possible for a number of circles to nest alongside one another without leaving diamond shaped gaps between them. Even if the circles are overlapped, as in diagram number four, the diamond shaped gaps prevail.
Figure 6: Diagram of two interlocking hexagons representing social networking.
A way of circumventing the problem of gaps is to use hexagons instead of circles to represent a given set of social acquaintances. The hexagons have the advantage of having six sides and are also stackable. Hexagons nested alongside one another in a honeycomb sequence form a geometrically efficient surface. They leave no gaps. This leaves the shape open to recursive possibilities. At the same time, all of the hexagon shapes may be combined into one large hexagon.
One of the problems with the hexagon model is that the boundaries between shapes are more artificial than what would be encountered in the actual world. Rarely are the divisions between groups so neatly delineated. Social relations tend to overflow into each other blending one social network with the rest, and this shape shifting is exactly one of the most significant features of a social network worth exploring. In this sense, the hexagon model is incapable of indicating movement and association. The shapes identify geographic distinctions without representing the interchange between them.
Part of the drawback stems from the twodimensional limits of representation. But it also results from the boundedness of the hexagon peripheries themselves. In this sense, the model replicates the problem of certain Euclidian representations of space. Each shape is comprised of an individuated entirety lodged next to another individual shape. The model is limited in its ability to reproduce interaction.
Figure 7: Diagram of hexagons with intersecting lines dividing the shapes into triangles.
Large hexagon comprised of smaller hexagons becomes a giant component of all composite shapes.
There is however, a way of approaching the problem of shape that aims to indicate some of the interaction taking place in a social network. Starting with an individual hexagon, the phenomenon of connection may be represented by indicating the six paths that exit from the center of the hexagon by taking three lines and subdividing the shape so that it resembles the spokes on a wheel.
Divided in such a way the hexagon is broken into six triangles the radiating lines reflect the six distribution points to the periphery of the shape. And, as each distribution point connects to the center of the next neighboring hexagon, all hexagons are connected on all sides to their neighbors. Thus by extension, every neighbor is attached to the whole.
Perceived from a global perspective, the individual shapes converge into a blended entirety. This feature of selfsimilarity is referred to as creating a giant component because all of the separate hexagons combined together equal one large hexagon which is an identical copy of all the separate hexagons . It is no longer easy to distinguish between separate shapes. The model appears composed of a combination of honeycomb hexagons which are also triangles. The combination of shapes becomes a collective networked shape, the smallest unit of which is the equilateral triangle. But not exactly.
The equilateral triangle is infinitely recursive. Named after mathematician Waclaw Sierpinski (18821969), one of the most purposeful examples of a network includes the twodimensional pattern called the Sierpinski Gasket (also known as Sierpinski Triangle). The gasket combines the principle of uniformity with rotation to form a repetitive network based on the measurement of a triangle. The arrangement consists of an equilateral triangle divided by an upside down equilateral triangle in its center. The resulting pattern may be indefinitely repeated. In this way, the model becomes a two dimensional fractal.
The Sierpinski Gasket may be formed into any number of shapes including three dimensional models and patterns that resemble frost on a windowpane. When iterated in such a way, the resulting designs are said to form a carpet. The Sierpinski fractal has the novel quality of being a flat shape that is nonetheless multidimensional. It therefore, serves as an useful representation of a social network since it displays all of the organizational complexity of social organizations that may be nested but are without hierarchy (Schroeder, 1991; Stewart and Golubitsky, 1992).
Figure 8: Rendition of the Sierpinski Gasket as a useful construct for exploring human/Internet connections.
Both the Sierpinski Gasket and the hexagon topographic maps represent the mathematical properties of tessellation and rotation. Tessellation refers to a pattern that repeats itself indefinitely. Each shape is an identical copy of the shape nested along side it. Like the honeycomb hexagon pattern, there are no overlaps and no gaps between shapes. Triangular tessellations like the Sierpinski Gasket have an added feature of rotation. The interior angle of each equilateral triangle equals 60 degrees. When the connections are multiplied by six, the resulting sum equals 360 degrees of perfect rotation. By extrapolation, the Gasket, as a model of the Internet, displays an efficient circle of communication that constantly cycles and recycles information back to itself.
In three dimensions, the Sierpinski Gasket looks like any number of networked structures, such as an electric grid and the Internet. It also illustrates a principle familiar in architecture and displayed in such examples as the lacy metalwork of the Eiffel Tower or the igloo-like domed inventions of Buckminster Fuller. These geodesic structures take advantage of the solidity of the triangle. Sierpinskis Gasket, the Eiffel Tower and Buckminster Fullers geodesic domes illuminate a basic, but seemingly paradoxical truth about networks whose strength lies not in mass but in branching points. The connections radiate tension throughout the network to form a durable structure. The knowledge seems in some way counterintuitive since constructions like the Fuller domes are based on nothing but branch points. The same principle holds for the structure of the Internet in which it is not mass, but rather the flexible connections that determine the resilience on the system (Schroeder, 1991) .
Because the equilateral triangle is infinitely divisible it represents a structure that is both singular, and continuous. A model of the Sierpinski Gasket illustrates how each triangle leads to giant component of itself. The rendering also illustrates one of the attributes that eludes general diagrammatic models of networks depth. With the gasket, depth is reflected in the layered and multiplied shapes. Whether as an isolated shape or a large interconnected form, in singles or in multiples the shape defies specificity while exhibiting many of the complex relationships of network structure. With the gasket, each triangle passes through and is cut by another triangle, and with it, the classical conception of dimension is shattered into a prism of multidimensionality.
Objects like the Sierpinski Gasket lend insight into the contradictory dimensionality of radical recursion. These forms play with the idea of shape not only as it exists in the first and second dimension but also as having an added dimensionality. In this way, perception becomes a nuanced task; first focusing on the singular triangle and then the field. The extra dimensional sensation of depth is rendered explicit and open the shape exists as a positive extension of itself. When observing the Sierpinski Gasket, the divisibility of the triangle in this sense identifies a recursive continuum. The result is striking since the continuum derives from the unbroken divisibility of the continuum itself. The brokenness gives way to wholeness (Rosen, 2004).
The gasket dimension of depth is not a trivial example of recursivity. Rosen identifies recursion as part of the contradictory quality of selfreference. In this sense, the shape refers back to itself, but it also refers to its extension and the boundaries that are crossed. The boundaries are not merely the exterior boundaries of a shape but rather represent a paradoxical threshold. Stepping into the distance the shape takes on one form. But when looking from close up it assumes another. The structure eludes closure. Thus, distinctions between objects and space, subjects and perception are not just dissolved, but rather the supremacy of the distinctions are overcome. The categorical separateness of Euclidian space is supplanted by a nonessentialized understanding of the basis of structure (Rosen, 2004) .
Complex networks like the Internet tend to swing between chaotic and antichaotic states with many indeterminate periods. Much of this variance has to do with the amount of feedback operating in the system. The manner in which the system changes over time displays nonlinear features and therefore, it rarely obtains any longterm equilibrium. Small changes may have radical effects on the systems functioning.
In a complex network, it is rare to observe simple cause and effect relationships between all of the elements. As in the Butterfly Effect, a small stimulus may have a great effect on the system. Another way to refer to the dynamic nature of a complex network is to invoke its history. The history of a complex network like the Internet explains its trajectory. Even small changes within this history may lead to large deviations which lead to more divergences in the future (Lorenz, 1980) 
Complex networks often go through transitions in the life of the system. Marc Newman suggests that the most important property of a network graph like the Sierpinski Gasket is it ability to demonstration a phase transition from one state to an entirely new and different state:... the most important property of the random graph, that it possesses what we would now call a phase transition, from a lowdensity, lowp state in which there are few edges and all components are small, having an exponential size distribution and finite mean size, to a highdensity, highp state in which an extensive (i.e., O (n) ) fraction of all vertices are joined together in a single giant component, the remainder of the vertices occupying smaller components with again an exponential size distribution and finite mean size. 
What this means is that in mathematical simulations of phase transition the vertices on a network graph begin behaving in a remarkable way by moving from a state of low density to one of high density. Typically, this density displays a powerlaw distribution. All of the separate vertices that are connected begin functioning as one larger system or, as a giant component. Mathematically, this phenomena also reproduces the Small World Dynamic that is observed in the study of social networks. All of the elements combine to form a single entirety with properties all its own.
Heterogeneity and complexity
The social structure of the Internet could be compared to a loose collectivity. Individuals connect to the medium autonomously, yet assemble into groups of their own accord. Combined, the actions represent an extraordinary level of complexity. This is a markedly different vision of behavioral communication than the hierarchic model, which dominates the literature on social organization. The Internet makes it possible to reenvision the way in which social organization is constituted; as an assembly of networks within networks teeming, cellular and much like the varied human ecologies from whence they spring.
The word that best describes the network form of sociality is heterogeneous. Heterogeneity implies radical difference both within and among structure. It applies to the dissimilar elements that make up the composition of networks and the differences between networks. Heterogeneous networks do not assemble into hierarchies. Rather, they form into concentrations governed by the laws of preferential attachment. These concentration points can be considered the highly connected nodes in a network (think of Amsterdams Schiphol Airport or the New York Stock Exchange). However, all around are billions of smaller networks that are also connected laterally. Small networks of low connectivity are not preferentially attached. However, they move information quite efficiently and are at low of a risk of disconnection and failure.
The Internet is an evolving global network. It consists not only of the individual computers, routers, cables, but also a number of communication gadgets. Many technologies are connected to the Internet including but not limited to cell phones, palm devices, video equipment and text messengers; all of which expand the entire communicative architecture. Users are adept at simultaneity, data sharing and using open software. These technologies highlight mutuality. For this reason the adoption of socially decentralized practices makes sense. Virtual communication favors small, local inputs, simultaneous organizing, openness, and sharing but with local autonomy. The practice is individuated yet collective. Virtual communication puts the means of communication in the palm of ones hands, so to speak. The logic of electronic networking conveys the transaction of power downward.
The Internet can be pictured as an electronic membrane that stretches across the planet. It is comprised of all of the individual computer technologies that are connected together. The landscape is continuous even if the technologies are not equally allocated to all peoples and places. The Internet is like the human environment is made up of different social situations (households, neighborhoods, companies, towns, cities, nations and supranational institutions) (Wobst, 2006). These social arrangements exist simultaneously. Like the Internet, human relationships depend on distribution, movement and the frequency of interaction between places.
Ecology provides a paradigm for describing how the communicative practices evolve on the Internet. The Internet expands almost like an organic system, comprised of communities of selforganized populations; densely interconnected, diverse and proliferating. Such communities are small, plentiful and replicatable. Like the geographic, physical and climatic variation that sustains the planet, such human diversity unfolds in a continuum of landscapes.
At the same time, such a technoecology derives its vitality from the multitude of environments that embrace the Earth. If examined from close up, such communities thrive when they are grounded in their own locality and context. However they also grow and develop at the edges of similarly networked environments. This is the space where innovation lies, in the inbetween spaces and the overlap points where creativity, diversity and adaptation are prevalent (Kauffman, 1993). The Internet offers a set of spaces where different cultures meet in a globally oriented social ecology, which Gustavo Ribiero (1998) has called cyberculturalism.
The Internet contributes to an evolving form of socialization that emphasizes the production of shared knowledge that is relational, open and dialogic. Anthropologist Arturo Escobar claims that cyberculture enables social practices are highly networked like the technology they replicate. Cyberspace culture may be considered molecular, as it is engaged in the production of micro practices through organizational cells that are active at the local level. The organization however, is realized through the fluid architecture of cyberspace .
The networked architecture allows for the development of a space in which diversity dominates. On the Internet, diversity and difference are the reigning logics. Separately, the social networks craft spaces of creativity that are open, differentiated, innovative adaptive, and proliferating. At the same time, the infosphere fosters a form of collective intelligence made up of a multitude of communication profiles. Cyberspace culture consists of participating in a shared structure and momentum. In this way, global cyberculture acts as if a giant network that is emergent on many different levels.
Heterogeneity also refers to a process of reproduction in which the heterogeneous elements of the network multiply. The meaning indicates something about the proliferation capabilities that lie at the heart of the network shape. The Internet is not a singular nor hierarchic system but rather represents the emergence of many structures all blended together into a shapeshifting dynamic.
The ecological paradigms and the technological imagery are harmonious. Both rely upon networked imagery that is diverse and highly connected. The growth depends upon flows, fluidity and connection. Moreover, growth tends to be selforganized and multidirectional, rather than unidirectional, deterministic and hierarchic. Cybercultural networks exhibit clustering behavior, in the physical world and on the Internet. Traffic patterns, cellular activity and social behavior merge and become what philosophers Delueze and Guattari (1987) refer to as the concept of multiplicity .
A significant feature of the shape of complex networks is its flatness. Flatness is not the same as two dimensionality. Networks are based on a grid representation of geometric space. While depicted as a flat plane, networks have properties that could be considered as extradimensional.
In fact, there are a number of representational shapes that exist on a topographic network that do not appear in threedimensional space. For instance, the Klein bottle or the projective plane are objects which cannot exist in three dimensions. They have neither an inside nor an outside and exist solely as geometric oddities. The Sierpinski Gasket is a surface with an irregular relation to form and may represent a multidimensional structure.
Ultimately, the aesthetics of networks allows for imaging the properties of structures that have an extradimensional constitutions recursive weaves, Sierpinski nets, preferential knots, feedback spins, and chaotic divergences that comprise the irregularities of topographic emergence.
Because networks are flat they grow horizontally. Growth occurs by extending sideways, by attachments, seepages and underground runners. Networks spread along a lateral path that forms connections along a series of planes or platforms. These platforms may build over time like a sedimentary rock such as shale. Unlike the conventional understanding of the hardness of rock however, networks tend to be resilient and fluid (De Landa, 1997).
Networks also increase in density. Sometimes this leads to preferentially ordered components. But much of the growth spreads along the extensions or borderlands of dominant networks. And these borders constantly expand. They grow in unregulated directions. Moreover, since networks grow in unplanned and lateral directions, the Internet continually forms new offshoots, paths and extensions. Many times these additions develop out of inconsequential ties and are therefore difficult to anticipate. The lateral growth sustains the overall flat dimension.
Part of the flatness emanates from the community to community transactions that develop. Relations tend to sprawl rather than vertically accumulate. Additionally, since the components tend towards heterogeneous composition in what Albert Barabási called the inhomogenous topology of the Web there are no prevailing centers to the flat form of network organization. They are an ideal model for theorizing a kind of social organization that is particulate, multigenerous and prolific (Barabási, 2001).
The connections that form are not exclusively on the same terms, of the same dimension nor of the same kind. Another way of referring to the inhomogeneous nature is to recognize the diversity of structure. Networks tend to incorporate different elements through all of their communication linkages. Moreover, the connections do not vertically integrate like blocks on top of one another. Thus, instead of seeking out a pure or hierarchic basis of structure, topographic network maps make it is possible to examine the lateral connections, planes of distribution and flat fields of saturation that comprise social organization on the Internet.
Developing an aesthetic of networks allows for theorizing the complexity of social organization as a flat constitution. Thinking of the world in terms of flatness entails examining the micro practices of networking which are fragmented, multiple, heterogeneous and transformative. Flatness does not mean a smooth surface as in shallow, lacking in relief, perspective or depth. It is not the same as twodimensionality. Rather, it is a perceptual technique for conceptualizing the chains of associations between individuals, environments, material circumstances and their relationship to each other. A conceptual flatness allows for the comparison between the selforganizing prospects of sociality and their lateral growth.
Networks provide a useful means for examining social organization in a way that escapes many of the traps of Euclidian geometry (those that are centered on the morphology of structure). Instead, networks are an abstract representation of topographic space. As such, they emphasize the diverse and sometimes contradictory features of structure. Thus, networks reveal the nature of things proximate not in terms of geometric separations, hierarchy or other aspects of morphological closure. Rather, networks highlight the communication between individuals, behaviors, properties and the dynamics which arise among structures. The Internet provides a conceptual imaginary that is not as restrictive as the top/bottom, hierarchic language that we are used to. It allows us to see growth and vitality happening at the edges and to highlight the strength of the nonpreferentially organized connections. The flattened spatiality of the Internet may be seen as conducive to a flat sociality comprised of participation, nonhierarchy, and technoecological suitability. Such a sociality is not necessarily invested in the preferentially organized centers of hierarchic systems. At least a flat network could be organized in any number of directions since the space is not already directed by the logic of up and down.
About the author
M.K. Sterpka is assistant professor in the Department of Communication Studies at Niagara University where she researches researches globalization, networks and social movements. She is also an awardwinning illustrator.
Email: msk [at] niagara [dot] edu
1. Marston, Jones and Woodward, 2005, p. 416.
2. Eric Raymond in his seminal article The cathedral and the bazaar, Raymond reflected on the importance of open source programing. His work focused especially on the culture that developed around Linux, the system released by Linus Torvolds. He saw this as contributing to a dynamic interchange based on collective contributions of volunteers. The system proved more successful at making improvements and removing bugs than other top down approaches to code. Others have also written on the topic, Federico Iannacci and Eve MitletonKelly on heterarchy as an element in programing that is decentralized, and showing many of the features of adaptation within other complex systems. Ko Kuwabara, has also written cogently on the topic.
3. Hegel, 1973; Guyer, 2005: Kant, 1987.
In his work The aim of art, Georg Hegel recognized that art might have utility but such usefulness was separate from the art itself. The usefulness of art was an imposed social criteria. He argued that the vocation of art was to reveal the truth and that was an end in itself. Similarly, Immanuel Kant argued that the faculty of judgment allows humans to experience beauty as part of a natural and ordered world that has an intrinsic purpose. While beauty can be purposeful in regard to judgment, nevertheless beauty need not have a definite purpose. It was assumed to be part of a universal truth about the nature of beauty.
4. Carole McKenzie and Kim James make an argument that aesthetics is an aid in understanding complexity particularly those things that are immeasurable. While not referring specifically to networks, they argue that discerning structure is a primary goal of an aesthetic mindset.
5. Newman, 2003; Castells, 1996, p. 470.
The origin of complex network studies is usually tied to Leonhard Euler (17071783), a Swiss mathematician. He developed a method for depicting a network as a series of points and bridges. Complex systems are studied through three complimentary approaches, network analysis, network theory and/or graph theory. Network analysis examines and maps the strength of relationships in a given social organization through the use of sociometric maps. Network theory is not so much theoretical in the traditional sense, but rather is a series of agreed upon methodological techniques utilized for the study of complex systems. Network theory is a subset of applied mathematics and examines the same subjects as graph theory namely using a graph to represent a series of connections. The technique is employed in computer programing but also used to model such disparate phenomena as gene networks, the diffusion of disease and the ties of social network maps. The more traditionally applied theoretical component of network studies has to do with the broad social changes related to networked technologies and informationalism. Mostly a theoretical direction in the social sciences, such research aims to consider social organization writ large, as a series of networks within networks. All of these topics fit the criteria for complexity and are thus applicable for network analysis.
6. Preferential attachment based on Barabási and Alberts model of growth (using the World Wide Web as an example) speculates on the cumulative advantage of directed networks that follow a power law distribution.
Positive feedback has its own power law dynamic which can be related to preferential attachment (the likelihood that a wellconnected node will attract further connections). Using the Web as an example, Watts and Strogatz chose an existing vertex and added a number of edges. From these, another edge was chosen and random edges were added onto it and so on. The supposition is fairly simple. Over time, the model gives way to a powerlaw distribution.
Kleinberg (2000) has proposed that the growth in some biochemical networks emanates from the ability of vertices to duplicate. The vertex copying model is somewhat applicable to models of growth on the Web. The code for protein duplication is already known to exist. The copying mechanisms of protein networks have been modeled by a number of researchers (Kim, et al., 2002: Sole, et al., 2002). Thus, the biological model can be used as the basis of extrapolation for modeling electronic communication systems. Jain and Krishna (1998) propose a network of interacting chemical species that evolve by copying through reproduction and mutation in a cycle that leads to selfsustaining autocatalytic loops, that the authors theorize as an explanation for the genesis of life on the planet. The idea bears a similarity to Maturana and Varelas (1992) theory of autopiosis. Preferential attachment as a growth mechanism does not apply to all networks. It is important to keep in mind that the biological models provide a template for theorizing the nature of communication networks like the Internet. There are however, important differences between biological and technological networks, hence the comparison therefore is not perfectly extrapolatable.
Lawrence Lessig is a strong proponent of keeping the Internet free of too much preferentiality, including the monitoring that goes along with determining preferential matching on the Web (for instance Amazon.coms system of matching consumer preferences with similar products through the monitoring of purchases) (Lessig, 2001, p. 133).
7. The idea of social network connection was actually was actually first proposed by the Hungarian author Frigyes Karinthy (18871938) in a story he wrote in 1929 called Láncszemek (Chains). In the story the protagonist, seeking to demonstrate that people on Earth are growing closer together challenged his colleagues to consider that a billion people might be connected through only five acquaintances (Watts, 2004). Milgrams hypothesis was one of the first experiments to demonstrate the six degree phenomenon about human connectivity. There are implications for the communication of information. It is also related to the Erdös number and the six degrees of Kevin Bacon trivia game. Watts and Strogatz explored the problem of connectivity in terms of the short cut route created out of seemingly inconsequential ties.
8. Newman, 2003, p. 11.
The clustering feature of networks is sometimes referred to as community structure. On a network graph, this translates into groups of vertices with a low density between groups. The finding tends to reflect commonplace knowledge about social structure people divide into groups of like interests. This finding is not that surprising from a sociocultural standpoint, however the prevalence of clustering may be effectively extrapolated to describe similar community formation on the Web or within ecosystems and so forth (Newman, 2003) Largely, the small world problem is one of distribution meaning the way that networks are dispersed geographically and the interconnections between these distributions by virtue of the long geodesicspanning linkages that connect the network members to each other. I use the metaphor of a superhighway to illustrate how pathways may link disparate regions. Watts and Strogatz examined comparative networks, and especially those with random distribution to elucidate this feature of connectivity and clustering.
9. There may be something eminently suitable about the network form of social arrangement. This may be its human scale though we currently refer to such relations as networks, they can be alternately called kinship groups or social bands that have been an enduring feature of human organization as part of our evolutionary and adaptive strategies. A similar modeling problem was approached by Martin Wobst in 1974 in his paper, Simulation of boundary conditions for Paleolithic social systems. Wobst chose the example of hexagons in order to simulate the territorial space of huntergatherer groups. While acknowledging that actual huntergatherer territories may have little in common with the modeling, Wobst proposed that polygon shape provided a spatial pattern that afforded an ideal approximation of Pleistocene conditions (Wobst, 1974, p. 154). This is because the model makes for a neat representation of territorial distribution with clear geographic distinctions between shapes. At the same time, there is a recognition that the model is an ideal type, and in practice huntergatherer groups may converge in any way they wish. According to Wobst, huntergatherer social groups on average are surrounded by six neighboring bands. This finding approximates the six degrees of separation found in modern social circles.
10. There are only 11 ways that the twodimensional planar network can be divided into convex, regular polygons; including the hexagon, equilateral triangle and square. The most density is achieved in those networks of regularly divided planes of neighboring polygons. In two dimensions, only 31 networks are possible overall (governed by the rule of most density packing). However, in three dimensions the number of possibilities has not been calculated. Glen Slack (1983) explored the regular packing of circles and spheres on a geometric plane. Before him, M.C. Escher worked on the problem of tessellation in the same way and George Pólya (1924) and F. Haag (1923) explored the phenomena as well.
11. The Sierpinski Gasket model illustrates a number of the features of fractals including, divergence and chaos. While the model has an inferential appeal, preliminary research also serves to identify the Sierpinski Gasket as a fair representation of both the structure and properties of the network of the Internet. Mohammed El Houssain EchCherif El Kattani (2002), in an estimation of long range dependence uses the Gasket to model the nature of network traffic. Similarly, NguyenDat Dao argues that the selfsimilarity of Sierpinski triangles are most suitable for describing traffic on the World Wide Web.
The Sierpinski gasket is but one of a family of geometric constellations that may be combined to form a variety of two and three dimensional patterns, including cubes, circular shapes and delicate patterns which resemble frost on a windowpane. When iterated repeatedly, the ensuing formula is said to form a carpet. Sierpinski carpets also bear obvious relation to Euler network geometries with their prominent points, bonds and vertices. In comparing the triangle with Internet traffic, Dao writes:The triangle is composed by 3 congruent figures having exactly half of the size of the original triangle. Therefore, if we magnify one of these three little replications by a factor of 2, again we get another copy. So, the Sierpinski triangle consists of 3 self similar copies of itself, each with a magnification factor of 2. In fact, the Sierpinski triangle can be decomposed in different ways: by 9 self similar copies, each with a magnification of 4, or 27 copies (magnification factor of 8) and so on. This type of self similarity at all scales proves the fractal characteristic of an image. Therefore, self similar data have a common property with fractals: when we zoom in a part of the traffic generated by the data, we observe the same structure. Many successive zooms show the same result. (Dao, 1999, p. 27)
Like the Sierpinski Gasket, the data generated from Internet traffic will hold structure irrespective of scale. But an even more novel quality emerges when running the triangles through a number of iterative successions. The combined sequence forms the phenomenon of divergence. This is because a fascinating characteristic of Sierpinski points is that they are all repeller points. When mathematically calculated, a point near a Sierpinski point will not be attracted. Its distances diverge from the Sierpinski point and the divergence is exponential. Manfred Schroeder explains the phenomenon:The reason for this exponential divergence is easy to see because our mapping corresponds to left shifts of the binary digits that encode the coordinates of the points. Thus, sooner or later, the first error bit will arrive at the binary point, which means that the initial difference, no matter how small, will have been magnified to half the height of the triangle. After that, all succeeding bits are random errors; the motion of an initially almost periodic point will become chaotic. In fact, this simple example contains the very essence of chaos and fully accords with its definition: small initial errors grow exponentially until they dominate any regular motion. (Schroeder, 1991, p. 25)
Schroeder refers to the appearance of error bits. Another way of thinking about such errors is through the phenomenon of divergence/or strange attractors. Strange attractors are nonintegral numbers that are encountered in nonlinear systems, such as biological or social systems. They reveal that while the rules governing a system may be deterministic, the results are anything but.
12. Rosen explains recursion in terms of its semiotic qualities. The author turns to topographic representations of irregular forms, such as the Klein bottle, in order to explain the concept of nonmorphological closure. The surface features of irregular shapes become representative of a semiotic argument that illuminates the dialectism and contradictory nature of identity (in this case the identity of form). The shapes represent paradoxical selfreference.
13. Edward Lorenz (1817) is generally credited with performing the first experimentation on the effects of chaos. His computer simulation in the 1970s attempted to model and predict weather patterns and in the process discovered how the slightest difference in initial conditions meant for wildly unpredictable results. The hypothesis was called the Butterfly Effect. The Internet shares many features with other complex systems, including an inherent instability and potential escalation of chaotic trajectories.
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Paper received 2 October 2006; accepted 15 August 2007.
Copyright ©2007, First Monday.
Copyright ©2007, M.K. Sterpka.
The aesthetics of networks: A conceptual approach toward visualizing the composition of the Internet by M.K. Sterpka
First Monday, volume 12, number 9 (September 2007),