There are several things that cause monsters.
The first is the glory of God.
The second, his wrath.
The third, too great a quantity of seed.
The fourth, too little a quantity.
The fifth, the imagination.

- Ambroise Pare, On Monsters and Marvels

    Once upon a time, the blank spaces of maps were labeled Terra incognita. Sailors and merchants looked at those spaces with apprehension and fear; nevertheless they traveled beyond the known world to see what was there. Many of them, when returning home, described the strangest creatures inhabiting such unexplored regions, having many heads or legs, multiple eyes distributed around their bodies, or even having combined human and animal traits. Not surprisingly, an English mapmaker would place the phrase “here be dragons” at the edges of the explored Earth. These tales were fascinating, and their fingerprint can be followed through time as cartography developed and the world was conquered. The stories became common and their details more and more compelling. And yet, as the “unknown world” shrank and Terra cognita grew, the amazing creatures living in the void of the cartographic drawings began to become rare. Eventually, they vanished and their habitats became restricted to the old maps, a unique blend of science and art.

      We might consider the loss of monsters with sorrow or happiness. It is a pity that, in the end, they were not real. They have inspired us and somehow live with us: gargoyles quietly look down on humans walking around the Gothic cathedrals of Notre Dame or Barcelona (Fig. 1). On the other hand, their disappearance from the maps is tied to the success of exploration and the rational view of nature. Over the past few centuries, the scope of scientific enquiry has been expanding.

      Today, many classical philosophical questions, including the origin of our universe and life or even the nature of consciousness, are part of the scientific, positivist program. The lands explored by science can be near, such as the insides of our heads, or as remote as the limits of our known universe. All of them are full of darkness, and we use the (sometimes weak) light of the scientific method to move through them.

       In spite of all our limitations, giant leaps have been made toward the understanding of fundamental questions. Physics shines atop all the sciences, and its success sometimes looks like magic. Indeed, there is magic in the beauty of physics.

       What makes physics so powerful? The question goes beyond the field of standard physics and pervades most scientific disciplines. Although physics has been traditionally tied to non-living systems, over the last decades of the 20th century it provided a great source of inspiration for an extremely ambitious research program: the search for the laws of complexity. In places such as the Santa Fe Institute in New Mexico, scientists coming from very diverse areas (including physics, biology, anthropology and economics) began to develop a common language and to converge toward a common picture of complex systems as entities that could also be treated in a scientific manner. The starting point in this exploration required a picture of the real world strongly departing from the analytic view of nature.

COMPLEXITY FROM SIMPLICITY: PHYSICS AND LIFE

A thing of beauty is a joy for ever:
Its loveliness increases; it will never
Pass into nothingness; but still will keep
A bower quiet for us, and a sleep
Full of sweet dreams . . .

- John Keats, Endymion

     Beyond the world of atoms and molecules is the real Terra incognita of science: the world of complexity. On the opposite side, inhabiting Terra cognita, no other field has been so successful as physics in searching for fundamental principles and laws when dealing with the smallest objects of our universe.

     The resulting theories have undeniable internal beauty: An infinite array of  phenomena can be captured by a handful of elegant equations. In a related way, biology has been able to look at nature at the scale of genes and proteins, also searching for general laws. The success of biology has been extraordinary: The finding that the genetic code is embedded inside an extremely long molecule of helical shape marked the beginning of the golden age of molecular biology. Yet the view of life as made up of genes, of genes as the ultimate explanation of everything, has been a complete failure.

     The final years of the 20th century were marked by the giant leap of human genome sequencing and by the end of the so-called analytic view of nature. Also known as reductionism, it involved the decomposition of complexity into tiny pieces. The understanding of the details, it was thought, would eventually lead to the understanding of the whole system.

     This view was claimed to be wrong by some scientists, but it was so appealing that it persisted over decades. However, complex systems exhibit properties and behaviors that cannot be reduced to the properties of their basic elements. This general observation has often been summarized as “the whole is more than the sum of its parts.” I think that a more appropriate sentence is actually “the whole is different from the sum of its parts”: When simple units interact, new phenomena emerge. Such emergent order is irreducible: It cannot be explained from the properties of the isolated units.

     Examples of irreducible order are all around us. Ants and termites, for example, are very limited in cognitive terms, but can build very complex nests. There is no blueprint of the global nest organization in the brain of a single ant or termite: Each ignores everything about what is being constructed. The colony is a superorganism and its properties result from interactions among individuals. Inscrutable workmanship that reconciles Discordant elements, makes them cling together in one society —William Wordsworth

     Our starting point will be social interactions. Imagine a society. Imagine an individual within it. Every individual has a context, a circle of friendships and acquaintances. It is not possible to reduce friendship to an equation, nor to measure it. If we want to gain understanding of the nature of social organization from a global view, we need to ignore the details and think of social ties as simple links. Imagine that we want to draw a map of social complexity. Looking at individuals as the elementary units, our target is mapping the web connecting humans.

Fig. 1. Monsters in the city: a gargoyle of Notre Dame Cathedral, Paris. (Photo © Ricard Sole)

       We can obtain some insight into this invisible web by doing some simple experiments. One such was performed by U.S. sociologist Stanley Milgram in the 1960s. He conducted a series of experiments that revealed a surprising phenomenon: the “small world” property of social webs. Using a group of people living in Omaha, Nebraska, he provided them with a letter to be delivered to a single target person: a stockbroker living in the Boston area. No one in the initial group knew this person, and they received as information only the person’s name, broad geographic location and profession. The instructions were simple: Send the letter to an acquaintance and re-send it to someone else you know who can get the letter closer to this unknown person. How many contacts were required to reach the target? One might think that a very long chain of letter exchanges would be required. However, the answer turned out to be rather different: An average of only six steps is all that was needed to find the target. The world is, indeed, small. The same occurs with memory in the brain: It is not a property of individual neurons. Such irreducible order is not— as it might seem at first sight - bad news. Although it tells us that it might be pointless to reduce a system to its parts in search for global laws, it also means that theories and models must consider how units communicate. As we will see below, looking at the whole can be a really insightful experience.

      A search for the laws of complexity is a difficult task. We do not seem to have powerful icons such as atoms, fields or strings guiding us through the dark. The real world is far from simple and seems to have many different faces and levels of description. Can we achieve elegant, unifying metaphors, even explanatory laws, driving the emergence of complexity? Although the task seems almost impossible, there is an undeniable taste for elegance pervading science that can help us. If we have two candidate explanations for a given phenomenon, we spontaneously incline ourselves toward the simpler explanation. We have found something that seems to provide a good unifying theme: networks. Networks pervade complexity at multiple scales. They are the fabric of social interactions and sustain life from cells to ecosystems and allow thoughts to flow through our brains.

COMPLEXITY IS MADE UP OF NETWORKS

Dust as we are, the immortal spirit grows
Like harmony in music; there is a dark
Inscrutable workmanship that reconciles
Discordant elements, makes them cling
Together In one society

- William Wordsworth

    Our starting point will be social interactions. Imagine a society. Imagine an individual within it. Every individual has a context, a circle of friendships and acquaintances. It is not possible to reduce friendship to an equation, nor to measure it. If we want to gain understanding of the nature of social organization from a global view, we need to ignore the details and think of social ties as simple links. Imagine that we want to draw a map of social complexity. Looking at individuals as the elementary units, our target is mapping the web connecting humans. We can obtain some insight into this invisible web by doing some simple experiments. One such was performed by U.S. sociologist Stanley Milgram in the 1960s.

Fig. 2. The architecture of the Internet. (From http://en.wikipedia.org/wiki/internet>.) Far from a planar map, it inhabits a high-dimensional space, where topology instead of geography is the relevant word. One specific hub is indicated at the bottom right.

      He conducted a series of experiments that revealed a surprising phenomenon: the “small world” property of social webs. Using a group of people living in Omaha, Nebraska, he provided them with a letter to be delivered to a single target person: a stockbroker living in the Boston area. No one in the initial group knew this person, and they received as information only the person’s name, broad geographic location and profession. The instructions were simple: Send the letter to an acquaintance and re-send it to someone else you know who can get the letter closer to this unknown person. How many contacts were required to reach the target? One might think that a very long chain of letter exchanges would be required.

     However, the answer turned out to be rather different: An average of only six steps is all that was needed to find the target. The world is, indeed, small. These findings are rather counterintuitive. The above task seems as difficult as trying to find a document in a vast library composed of millions of documents. This, however, is precisely what we all do when surfing through the Internet looking for information: When we ask for a given document to be found by our computer, we do not wait for weeks for an answer. Despite the fact that the search is done through a universe of billions of potential entries, it is resolved quickly and successfully. The reason for such success must be found in the small-world phenomenon. A picture of the Web aids in understanding its origins and efficiency. Figure 2 shows a wiring diagram corresponding to the structure of the Internet. Here links connect computers through cyberspace. This picture is actually an accurate representation of what the maps of real complex systems look fragility of complex systems. If a key element of the Internet is destroyed or damaged, the loss will reverberate through the entire web: Information transfer will rapidly deteriorate. If two or three hubs are lost, the entire network will become fragmented and collapse. Hubs have been named “the Achilles heels” of complex networks. This tells us that the fabric of reality is composed of two inseparable properties: complexity and fragility. They seem to go together and cannot be separated. They are at the heart of language architecture, cancer or brain structure.

GENOMES, WEBS AND CANCER

We will have to see that we are the natural
expressions of a deeper order. Ultimately,
we will discover in our creation
myth that we are expected after all.

- Stuart Kauffman, At Home in the Universe

     Although the above examples involve social and technological systems, the picture of complexity as made up of networks goes beyond these areas. As in physics, networks provide the unifying theme that allows description of complex systems of very different natures using a common base. This might seem an odd conclusion. Human-made systems are, after all, the result of conscious design, instead of the blind, Darwinian process of selection. As the French biologist Francois Jacob pointed out, evolution is very much like a bricoleur, a tinkerer who is forced to use the ingredients available at any time. Evolution is a tinkerer unable to foresee the future, since there is no plan, as an engineer would have. Nevertheless, the power of evolution is obvious: We need only look around to see the wonders of nature, from cells to the brain. Somehow, tinkering has been able to generate the extraordinary richness of living forms.

     In capturing the nature of this complexity and richness, our next step is the human genome or, more precisely, genome architecture. The sequencing of the human genome is a great leap, but only a first step toward understanding the logic of life. In order to get further insight into such logic, we need to look at how genes or proteins interact. When scientists obtained the first picture of cellular complexity, they found themselves confronting a familiar pattern: The way molecules interact within cells is not very different from the way the Internet is organized.

Fig. 3. Proteins are the nanomachines driving cellular hardware. (© Ricard Sole) Here a protein molecule is shown, where each ball represents an atom.

      Consider a cell in the body. A single cell among the trillions that constitute our tissues and organs is an extremely complex machine. If we were able to have a look at its internal organization, we would be astonished by the impressive molecular interactions supporting life at its basic scale. Take for example proteins (Fig. 3). These complex nanomachines move around processing information, copying DNA, maintaining cell structure or connecting the cell with the external world. These structures are the hardware of life and they seldom act in isolation: Most proteins are known to perform their functions by getting in touch with others. In their physical interaction, proteins are like the pieces of more complex machines. Now we have a well-defined network: Proteins are the elements, and we will consider two proteins as connected if they can physically interact. In this way, we obtain again a map of the cellular world. What does this map look like? The so-called protein map is a small world: If we move through its connections from one protein to another, it is very easy to find a short path connecting them. This is important, because cells are systems composed by many interacting units working coherently. The picture of life as described by the molecular biology of the 20th century was strongly reductionist, considering genes and molecules as the meaningful building blocks of cellular life.    Although much has been understood from such assumptions, they become a rather limited description once we are confronted with the whole. The existence of a small-world structure tells us that, in cells too, nothing is too far from anything else. In the society of proingly complex structures they have created and the structures themselves start changing the way designers interact with them. At this point, there is no way of finding the boundaries between the two: Designers modify the system and the system modifies designers’ behavior. Such a closed loop is part of the non-reducible order that pervades the laws of complexity.

THE THREAD OF LANGUAGE

Remembering speechlessly we seek the
great forgotten Language, the lost laneend
into heaven, a stone, a leaf, an unfound door.

- Thomas Wolfe, Look Homeward,Angel

      It is said that Jorge Luis Borges once claimed that one of his dreams was to connect two words that no one else would have. Here connecting words means using them together in a meaningful way. Of course language has some rules: syntax. We combine words in an extraordinarily flexible way, but not everything is allowed. Only poets and fools can make arbitrary decisions. But then, once again, we might be moving into Terra incognita. Language is the most prominent trait of humans. It is unique as an evolutionary transformation that allowed us to build societies, create symbols and invent culture.

       Nothing would be sustained in a society without language. And yet not much is known about how it emerged, how it is organized and how we cope with it in such an efficient way. In fact, we might think of language as a very big collection of words, which we combine following rules. The rules themselves are complex: We all know what syntax is, since we learned the rules when we were kids; but if requested to explain them, we would have a hard time. How do we manage to combine words properly and effectively, generating complex sentences without much thought or finding semantic relations between two words from different perspectives?

       Languages differ and yet are close in resemblance. In spite of their obvious differences, the study of their architecture reveals, once again, hidden patterns: All of them appear to be organized in common ways. For example, take a Chinese, English or German book and count how many times every word appears in the text. You will find that the great majority of words appear represented just a teins, talking to each other by linking together, constantly merging and splitting, communication at a global scale is also important. The cellular map also reveals the presence of hubs: Although most proteins talk to just a few others, some of them interact with many others. What are these proteins? It has been shown that they are very important elements of cellular machinery.

     Not surprisingly, many of them are important in the context of cancer. The most famous is p53, a protein that appears mutated in a large number of human cancers (about half of them) (Color Plate B). This protein directly interacts with DNA and is associated with key biological processes, particularly those involving the maintenance of cell integrity. Before a cell divides, several cellular checkpoints (such as the p53 protein) are used in order to guarantee that the replication process is done accurately.

     Otherwise, the absence of such checkpoints would quickly and inevitably lead to mutations and eventually to cancer. This, however, is precisely why p53 is an Achilles heel: Its failure has huge consequences for the stability of cells. Conversely, because most proteins interact with only a few others, a random failure will have weak or no consequences for cell viability. Biology seems to recapitulate technology, and some scientists have explicitly compared the set of elements interacting with p53 (the p53 box) with the Internet.

     Although the relation seems weak, it is not. Eventually, both systems must manipulate, transfer and process information. The finding that most networks, from the human genome to the Internet, share a common architecture is a far reaching discovery. It provides us a real window onto the structure of complexity.

     The picture we see is elegant and evocative. Looking at the previous figures, we perceive order, and this order is common to all these systems. It is a subtle feature that is connected to functional properties but also to weaknesses. In sharing their architecture, life and technology have similarities, indicating common organizing principles and perhaps even common evolutionary rules. In fact, it has been shown that some large-scale technological systems are also predated by tinkering: Although engineers are indeed designers, some technological designs, such as software systems, become so complicated at some point that their evolution proceeds effectively by tinkering.

     Engineers then have to cope with the constraints imposed by the increasingly complex structures they have created, and the structures themselves start changing the way designers interact with them. At this point, there is no way of finding the boundaries between the two: Designers modify the system and the system modifies designers’ behavior. Such a closed loop is part of the non-reducible order that pervades the laws of complexity.

THE THREAD OF LANGUAGE

Remembering speechlessly we seek the
great forgotten Language, the lost laneend
into heaven, a stone, a leaf, an unfound door.

- Thomas Wolfe, Look Homeward,Angel

      It is said that Jorge Luis Borges once claimed that one of his dreams was to connect two words that no one else would have. Here connecting words means using them together in a meaningful way. Of course language has some rules: syntax. We combine words in an extraordinarily flexible way, but not everything is allowed. Only poets and fools can make arbitrary decisions. But then, once again, we might be moving into Terra incognita.

    Language is the most prominent trait of humans. It is unique as an evolutionary transformation that allowed us to build societies, create symbols and invent culture. Nothing would be sustained in a society without language. And yet not much is known about how it emerged, how it is organized and how we cope with it in such an efficient way. In fact, we might think of language as a very bi  g collection of words, which we combine following rules. The rules themselves are complex: We all know what syntax is, since we learned the rules when we were kids; but if requested to explain them, we would have a hard time. How do we manage to combine words properly and effectively, generating complex sentences without much thought or finding semantic relations between two words from different perspectives?

Fig. 4. Language webs: Here the network of word-word interactions from the first chapter of Moby Dick is shown. (© Ricard Sole) Balls are words, and links indicate that they appear next to each other within at least one sentence.

      Languages differ and yet are close in resemblance. In spite of their obvious differences, the study of their architecture reveals, once again, hidden patterns: All of them appear to be organized in common ways. For example, take a Chinese, English or German book and count how many times every word appears in the text. You will find that the great majority

of words appear represented just a few times, and a few words are very common. All languages follow this rule. In fact, the previous observation can be formulated as a very simple mathematical law: This is known as Zipf’s law. The reasons for such universality have been a matter of discussion over decades. Is this a mere statistical artifact? Does it indicate a general principle of organization?

      Looking at the problem from a network perspective can help in finding some answers. How is it possible to build a language network? There are several ways of doing it. Let us first consider word associations. If I ask you to find words related to a given word such as “tree” you are likely to come up with “leaf” and “root,” or perhaps “shadow” or “family.” A multiplicity of relationships among words can be established, involving different ways of

defining word categories. Using a given relation, we can define a network by linking those words, which are related. This includes, for example, synonyms. We can build nets of words defining word relations and we can navigate from word to word following their links. This is a complex web in which once again we discover the small-world effect: It is extremely easy to navigate through the semantic web to find a chain of associations connecting any two arbitrary words through semantic relations. Words involving very general, sometimes fuzzy conceptual categories are the hubs, and they allow easy location of paths connecting seemingly remote words through meaningful associations.

      A different type of language network can be easily constructed using a simpler form of word-word link: precedence. Two words will be considered thus linked if one precedes the other at least once in a sentence. Consider, for example, the language network obtained from the words of the first chapter of Melville’s Moby Dick (Fig. 4). Here are shown only a small number of words and their links, but we can clearly appreciate the same regularities already discussed in the previous section. Words seem to form a society, not everyone interacting with everyone: Most words are seldom linked to others, whereas a handful of them appear connected to many others. Such difference suggests a compromise between specificity (provided by words rarely used and thus having few links) and generality

(given by commonly used items). What are those highly connected words? Interestingly, words having many links are those with the smallest semantic content, such as prepositions or articles. As occurs with the Internet, highly connected words are information chanalizers that make traffic efficient and the world small.

       For the Moby Dick web, for example, just four degrees of separation are required on average to reach one word from another. It is very easy to connect two words. The study of different languages reveals that they all share the small-world property and have the same basic architecture. What we learn from our network picture of language is that our brains organize language in a subtle, but very efficient structure. Borges dreamed of connecting two words impossible to connect. Maybe, however, the really impossible thing is to make two words totally apart from each other. THE PHYSICS OF AESTHETICS

NETWORKS OF MIND

The brain is just the weight of God,
For, lift them, pound for pound,
And they will differ, if they do,
As syllable from sound

- Emily Dickinson, The brain

     Our journey through network complexity ends with the mystery of all mysteries: the depths of human mind. No other complex system is so close and yet so unknown. You read these lines and a stream of thoughts flows within your brain. But we so routinely use our mind that we almost forget how extraordinary it is. The brain burns 25% of the total energy consumed by the body and is able to store an apparently infinite set of memories. Imagination and consciousness are two marvels of the mind, emerging from the activity of large neural masses. As happened with the genome, many scientists were traditionally attached to the view of the mind as reducible to the behavior of neurons or small, spatially located brain modules. Once again, however, complexity fails to be captured by the analytic picture of reality. Here also the whole is not reducible to the sum of its parts. At this point, one will be not surprised to learn that brain architecture also displays small-world organization. Neurons and neural assemblies communicate very efficiently, and the neural society seems to work very efficiently. Also, as with the Internet or the World Wide Web, brain networks are organized in complex ways: some neural domains appear to be related to many others, whereas most brain areas communicate mainly with the few nearest regions.

     Cognitive science has a long tradition of network analysis, and it is seldom a matter of discussion that most properties exhibited by the brain are the outcome of multiple interacting pieces of gray matter. Inside the brain, about 100 billion neurons exchange bits of information from which complex behavioral patterns emerge. The brain can be compared with a very large parallel computer performing many different tasks simultaneously. Just the way we receive and process information from our senses is a remarkable achievement, but—on top of that—we think, dream and look to our universe in search of explanations. Brains have been compared with different technological inventions as humans developed them. It started with the steam engine, a metaphor soon to be replaced by networks: first the telegraph, then the telephone network and finally the computer.

    However, although brains and parallel computers have several common traits, there is a widespread property of brains that causes them strongly to depart from the computational picture: their enormous plasticity. Every day we lose thousands of neurons without noticing. We can even be injured and fully recover from a significant loss of neural mass. More interestingly, some special case studies reveal an amazing underlying plasticity. Take for example language. Language appears to be associated to some more or less well defined areas of the left half of the brain. If these areas are damaged, we are likely to experience serious problems in language understanding or production. We might conclude that evolutionary pressures organized brain development in such a way that a specific module was placed in a given domain of the left hemisphere. An area communicates with others in such a way that the resulting network allows it to integrate the flow of symbols coming from the outside world and properly process them.  

      Now imagine that a young child loses his left-brain cortex. What might happen then? Although it sounds like a rather bizarre question, this actually happened to Nico, a boy who suffered a severe form of epileptic seizure. The story, told by neuroscientist Antonio Battro in his fascinating book Half a Brain Is Enough, illustrates how far we are from a full understanding of the logic of brain function and organization. Because of the damage caused by strong epileptic attacks and loss of consciousness, surgeons decided to use functional hemispherectomy, an extreme operation in which the left half of the brain cortex is removed. One might easily conclude that Nico would be unable to speak afterwards. But nature is often counterintuitive: Nico fully developed a normal language ability. How is that possible at all? How can a full mind be sustained by half a brain? Somehow, after the surgery, the growing brain of the little boy experienced a deep reorganization: New brain areas were recruited on the right side that were fully able to support language development. Actually, although Nico had some movement problems, he was at the top of his class in written and spoken language. This is not an isolated case, and it tells us that the potential of brain areas to develop different functionalities when required is enormous and unexplained.

    Although plasticity has great advantages, such an almost unlimited flexibility seems difficult to explain in evolutionary terms. This story is fascinating and scientifically challenging. The brain, or what Battro has called “the Brain Wide Web,” is the most complex piece of matter we know. Physicist Peter Coveney rightly named it “the cathedral of complexity.” How appropriate: although our universe is already complex and tangled, our mind has been able to provide meaning to the apparent chaos, is able to imagine alternative universes and even to dream. In our minds, the networks of Terra cognita and Terra incognita meet, and monsters and marvels live together.

Bibliography

Unedited references as provided by author.

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A.M. Battro. Half a Brain Is Enough. Cambridge U. Press, Cambridge UK (2000)

M. Buchanan, NEXUS, Norton, New York (2004)

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R.V. Sole, R. Ferrer, J. Montoya and S. Valverde. Selection,Tinkering and

          Emergence in Complex Networks. Complexity 8, 20–33 (2002)

D.J. Watts, Six Degrees: The Science of a Connected Age. Norton, New York  (2003)