Network Science

Network Science

Albert-László Barabási

Networks are everywhere, from the Internet, to social networks, and the genetic networks that determine our biological existence. Illustrated throughout in full colour, this pioneering textbook, spanning a wide range of topics from physics to computer science, engineering, economics and the social sciences, introduces network science to an interdisciplinary audience. From the origins of the six degrees of separation to explaining why networks are robust to random failures, the author explores how viruses like Ebola and H1N1 spread, and why it is that our friends have more friends than we do. Using numerous real-world examples, this innovatively designed text includes clear delineation between undergraduate and graduate level material. The mathematical formulas and derivations are included within Advanced Topics sections, enabling use at a range of levels. Extensive online resources, including films and software for network analysis, make this a multifaceted companion for anyone with an interest in network science.

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Bursts

Bursts

Albert-László Barabási

Bursts: is about... The Hidden Pattern Behind Everything We Do... A revolutionary new theory showing how we can predict human behavior. Can we scientifically predict our future? Scientists and pseudo scientists have been pursuing this mystery for hundreds and perhaps thousands of years. But now, astonishing new research is revealing patterns in human behavior previously thought to be purely random. Precise, orderly, predictable patterns... Albert-László Barabási, already the world's preeminent researcher on the science of networks, describes his work on this profound mystery in Bursts, a stunningly original investigation into human nature in the light of Big Data. His approach relies on the digital reality of our world, from mobile phones to the Internet and email, because it has turned society into a huge research laboratory. All those electronic trails of time stamped texts, voicemails, and internet searches add up to a previously unavailable massive data set of statistics that track our movements, our decisions, our lives. Analysis of these trails is offering deep insights into the rhythm of how we do everything. His finding? We work and fight and play in short flourishes of activity followed by next to nothing. The pattern isn't random, it's "bursty." Randomness does not rule our lives in the way scientists have assumed up until now. Illustrating this revolutionary science, Barabási artfully weaves together the story of a 16th century burst of human activity-a bloody medieval crusade launched in his homeland, Transylvania-with the modern tale of a contemporary artist hunted by the FBI through our post 9/11 surveillance society. Barabási's astonishingly wide range of examples from seemingly unrelated areas include how dollar bills move around the U.S., the pattern everyone follows in writing email, the spread of epidemics, and even the flight patterns of albatross. Bursts reveals what this amazing new research is showing us about where individual spontaneity ends and predictability in human behavior begins. The way you think about your own potential to do something truly extraordinary will never be the same.

EDITORIAL REVIEWS

"In Linked, Barabási showed us how complex networks unfold in space. In Bursts, he shows us how they unfold in time. Your life may look random to you, but everything from your visits to a web page to your visits to the doctor are predictable, and happen in bursts." -Clay Shirky, author of Here Comes Everybody
"Barabási is one of the few people in the world who understand the deep structure of empirical reality." -Nassim Nicholas Taleb, author of The Black Swan "Barabási brings a physicist's penetrating eye to a sweeping range of human activities, from migration to web browsing, from wars to billionaires, from illnesses to letter writing, from the Department of Homeland Security to the Conclave of Cardinals. Barabási shows how a pattern of bursts appears in what has long seemed a random mess. These bursts are both mathematically predictable and beautiful. What a joy it is to read him. You feel like you have emerged to see a new vista that, while it had always been there, you had just never seen." -Nicholas A. Christakis, M.D., Ph.D., coauthor of Connected: The Surprising Power of Our Social Networks and How They Shape Our Lives "Bursts is a rich, rewarding read that illuminates a cutting-edge topic: the patterns of human mobility in an era of total surveillance. The narrative structure of Barabási's provocative book mimics the very pattern of bursts, as abrupt jumps through the lives of a post-modern sculptor, a medieval Hungarian revolutionist, and Albert Einstein eventually converge on a single theme: that our unthinking behaviors are governed by a deeper meaning that can only be deciphered through the brave lens of mathematics." -Ogi Ogas, Ph.D., and Sai Gaddam, Ph.D., Boston University "Barabási, a distinguished scientist of complex networks, bravely tests his innovative theories on some historic events, including a sixteenth-century Crusade that went terribly wrong. Whether or not the concept of "burstiness" is the key to unlocking human behavior, it is nonetheless a fascinating new way to think about some very old questions." -Thomas F. Madden, Ph.D., Professor of Medieval History, Saint Louis University, author of The New Concise History of the Crusades In his first book, Linked, Barabási introduced us to the interrelatedness of the universe and to the emerging field of network science. Here, the physicist shows how to use that knowledge to predict seemingly random human behavior. Or the spread of a viral epidemic through populations. Or the convoluted trails that money follows. Like the “unexplained” erratic motion of tiny objects floating through water that fascinated Einstein at the turn of the 20th century, apparent stochasticity, says Barabási, can all be explained—and predicted—by elegant mathematical formulas. And for the first time in history, we’re beginning to have the right data to plug into such formulas. Using algorithms built in his lab, fueled by reams of data we unthinkingly create in our daily digital interactions (carrying around and communicating with mobile devices, withdrawing money from ATMs, making online purchases), Barabási demonstrates how much of human activity occurs in quantifiable patterns known as “bursts.” These bursts seem to define us: from our emailing and web-browsing patterns to how we move about the world. But in Bursts, this realization surfaces only as the sum effect of a nigh-schizoid storyteller’s account of historical and personal events. Driven by colorful characters and an experimental plot structure that jumps between ostensibly unrelated narratives, the book weaves a bloody crusade, the papacy, 9/11, and FBI surveillance into a tidy package. The effect is enthralling: less like listening to a lecture at a research conference, and more like sitting at a bar with a clever friend who charms you with his semi-implausible anecdotes. After nursing the last beer, beyond being amused, you’ll have learned something truly profound about the curious paths of human activity.
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Questions & Answers with Albert-László Barabási

Marco Visscher, Ode: What exactly is a burst? ALB: A burst is a sudden escalation in our activity pattern, characterized by an excessive focus on a certain type of task at the exclusion of all other responsibilities. It is like the thunder of drums in a Beethoven masterpiece, punctuated by the pleasing sound of the violins that preceded and follow them.
Marco Visscher, Ode: What is a good example of a burst? ALB: We may not even be aware of it, but each day we participate in many small bursts. We originally discovered bursts in the email pattern of individuals. Indeed, if we follow the sequence of emails sent by any person, we will not see a random and uniform stream of messages that most previous theories of human communication predict--we will witness instead short periods of intensive email activity, when the users fire out several, occasionally dozens of emails, followed by long periods of e-science. Soon we started to see similar bursts in all human activities that we could collect data for, from phone calls to the financial transactions of stock brokers; from Wikipedia edits to visits to the library. What surprised us most, however, was that all these bursty patterns followed the same precise mathematical law. We were seeing a peculiar rhythm of life that none of the individuals generating these bursts were aware of.
Marco Visscher, Ode: How do bursts occur? ALB: At first it appeared that burst occur randomly but we soon learned that they have a simple origin: prioritizing. Indeed, whether we do it consciously or subconsciously, we prioritize, often many times each day.As I show in Bursts, each time people prioritize their tasks, their behavior becomes bursty. If, however, we let a dice run our life, all signatures of burstiness disappear.
Marco Visscher, Ode: Can we predict the next burst (when, what)? ALB: Yes and no. To be sure, our daily activity is far more predictable than we are often comfortable of acknowledging, a major topic of the book. Consequently, we can in principle predict quite a number of things, from our whereabouts to potentially the timing of our email messages. While we know exactly our predictability when it comes to where we are (and it is frighteningly high), we have not yet tried to predict bursts. It may not be impossible.
Marco Visscher, Ode: How does the idea of bursts change the way we look at society? ALB: We often think of the society as a smooth machinery with its internal clock, where events proceed more or less seamlessly along their tracks. In reality, most events follow a bursty pattern, which, if understood, will change the way we approach them, and the way we get things done. Bursts is not a self-help book, but I believe that if we understand the patterns behind the rhythm of our daily activity, we are in much better position to be in tune with them and eventually exercise control over them. It has certainly changed they way I deal with the people I work with-- if we are stuck, the most remedy comes through revisiting our priorities, rather than placing blame. Marco Visscher Managing Editor
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Linked

Linked

Albert-László Barabási

In the 1980's, James Gleick's Chaos introduced the world to complexity. Albert-László Barabási's Linked reveals the next major scientific leap: the study of networks. We've long suspected that we live in a small world, where everything is connected to everything else. Indeed, networks are pervasive--from the human brain to the Internet to the economy to our group of friends. These linkages, it turns out, aren't random. All networks have an underlying order and follow simple laws. Understanding the structure and behavior of these networks will help us do some amazing things, from designing the optimal organization of a firm to stopping a disease outbreak before it spreads catastrophically.In Linked, Barabási, a physicist whose work has revolutionized the study of networks, traces the development of this rapidly unfolding science and introduces us to the scientists carrying out this pioneering work. These "new cartographers" are mapping networks in a wide range of scientific disciplines, proving that social networks, corporations, and cells are more similar than they are different, and providing important new insights into the interconnected world around us. This knowledge, says Barabási, can shed light on the robustness of the Internet, the spread of fads and viruses, even the future of democracy. Engaging and authoritative, Linked provides an exciting preview of the next century in science, guaranteed to be transformed by these amazing discoveries.

Table of Contents

The First Link: Introduction The Second Link: The Random Universe The Third Link: Six Degrees of Separation The Fourth Link: Small Worlds The Fifth Link: Hubs and Connectors The Six Link: The 80/20 Rule The Seventh Link: Rich Get Richer The Eigth Link: Einstein's Legacy The Nineth Link: Achilles' Heel The Tenth Link: Viruses and Fads The Eleventh Link: The Awakening Internet The Twelfth Link: The Fragmented Web The Thirteenth Link: The Map of Life The Fourteenth Link: Network Economy .. and The Last Link, by the time you it, this book could alter the way you think about all of the networks that affect our lives.

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Questions & Answers with Albert-László Barabási

In your book, you say “we have taken apart the universe and have no idea how to put it back together again.” What do you mean by that?
Science is based on the assumption that the devil is in the details. The jargon for this assumption is reductionism. When trying to understand anything from the cell to the ecosystem, we first seek to understand what it is made of, breaking it into pieces. Understanding cancer has really been a quest to find the molecules that cause it. Understanding the origins of the Universe was a race to reach down to quarks and superstrings. To be sure, reductionism has been hugely successful. We have drugs because we understood the molecular basis of some illnesses. We have the Internet because we understood how the electrons move about in a semiconductor. We always hoped, however, that once we understood the details, it would be easy to reassemble the pieces, and understand the global behavior of complex systems. Lately, scientists increasingly realize that this dream has failed, because most systems are not simple. They are made of so many and such diverse components that understanding how they all work together is very difficult. We are indeed in a situation of the crying child, who has taken apart his favorite toy, but has no idea how to put it back together.

Why should we care about networks? You call them “the next scientific revolution.” Are they really that big a deal?

Behind most complex systems there is an intricate network. Life is encoded by a complex network of molecules hidden within the cell. The Internet is a complex network of computers connected by wires. The economy is a complex network of companies, consumers, and regulatory agencies. Society is a complex network of people connected by friendship, family, and professional ties. It has only been in the past few years that we realized how important a role these networks play in shaping the behavior of most complex systems. We learned that understanding networks is the crucial prerequisite to comprehending complexity. Therefore, many scientists from very different disciplines have started a frontal attack to understand the webs with which nature surrounds us. One of the most surprising findings is that most networks in nature are very similar to each other. The social network is not that different from the four billion year old chemical network within our cells or the decade old World Wide Web.

What are these similarities?

For several decades, networks were believed to be fundamentally random, i.e. it was assumed that the nodes, such as the pages of the World Wide Web, the people on the society, or the chemicals of the cell, are randomly wired together. Yet, as we started looking at real networks, we noticed some reoccurring elements in all of them that increasingly undermine the random hypothesis. On the World Wide Web my research group documented the existence of a few websites, such as Yahoo.com, that have an extraordinary number of links pointing to them. In society sociologists have noticed the existence of connectors, a few individuals with an extraordinarily large number of acquaintances. In the cell my group and others noticed the existence of a few molecules that participate in just about all chemical reactions. These hubs, as they came to be called, could simply not be explained using the random network hypothesis. They were telling us that some common laws must exist, shared by all networks, which are responsible for the hubs.
What role do these hubs play in networks? Can you give us some examples?

If you inspect the flight diagrams shown in glossy flight magazines, telling you which airports are served by the airline, you will notice that most airports have only a few links, while a few hubs are connected to just about all airports. Similar hubs are present in most complex systems. A few hub molecules in the cell, or a few highly connected individuals in society are the bridges of their respective networks. These hubs have a dramatic impact on the behavior of all networks. For example, on the Internet, hundreds of routers are broken in any moment. Despite this, the Internet does not crash—a robustness attributed to the hubs. Also, one of the reasons you do not drop dead every time a molecule goes crazy in your body—and believe me, there are millions of such misbehaving molecules—is that the network within the cell is dominated by such hubs. On the other hand, if you know which are the hubs, by taking them out you can easily destroy the whole system. Think of simultaneously closing down the airports in Chicago, Dallas, New York, Detroit and Denver—in hours all U.S. air travel would be halted. In the past few years we learned that cells, the society, the Internet, and the economy all have their O’Hare, offering them a high degree of robustness against random failures, but making them fragile against attacks.

How do networks emerge, and how do they evolve?

Real networks always start out with a few nodes, to which new nodes are added gradually. Think about the World Wide Web: in 1991 it had only one webpage created by Tim Berners Lee. Then people started adding more pages, connecting to each other. Node by node the Web has grown to over a billion webpages. Life also emerged molecule by molecule from a soup of chemicals about four billion years ago. The economy grows company by company. These examples indicate that growth is a fundamental property of all networks. But recently we learned about another key element of network evolution: links are not placed randomly either. You are more likely to know about Yahoo.com than my webpage, and thus you link to it more often; companies are more likely to do business with large, established corporations; you will more likely meet people that have many friends. It turns out that most complex networks share two laws: growth, and a subtle preference to link to the more connected nodes. In a subtle way these two fundamental laws of network evolution are responsible for the hubs, and many other topological features of real networks, ranging from robustness to the spreading of computer viruses and fads.

What role do networks play in business?

A complex network describes the organization of each company, the nodes being the employees and the links the professional relationships between them. At the macroeconomic level there is an another network, whose nodes are the various economic institutions, from companies to government agencies, and the links represent the partnerships between them. Recent studies indicate that these economic networks developed a self-organized topology, very similar to the World Wide Web or the cell. The consequences of these findings are only now being understood. In the last few years we learned that the 1977 Asian economic crises, which took out companies and banks worldwide, could only be explained if we incorporate the various network effects. The same is true for the recent difficulties experienced by some of the leading companies of the information revolution, such as Cisco or Compaq: they suffered straggling losses by miscalculating network effects. It’s amazing how little attention has been paid until recently to network effects in the economy. But there are signs that this is about to change.

What about the role of networks in society?

Society is the most familiar network to all of us. We are the nodes, and the links are our social links. One of the most popular features of social networks is known as “six degrees of separation,” brought to the world’s attention by John Guare’s play by the same name. The concept is based on Stanley Milgram’s 1967 study indicating that people within the U.S. can be connected via six handshakes. Our understanding of the social network is a bit handicapped by the fact that we don’t have a map of it. But lately we have rather reliable submaps. We have detailed maps of how scientists collaborate with each other, and how actors work together in Hollywood. These maps captured the rise of Kevin Bacon as a central figure in Hollywood. Such a map is behind the Kevin Bacon game, which asks players to link actors to Bacon based on movies in which they appeared together. These maps also allow us to assign a ranking to all scientists based on the shortest path to Erdös via co-authorship. Most important, however, these maps indicate that society is unable to avoid the universal laws that govern the evolution of all networks around us: it’s dominated by hubs, a direct consequence of its growing self-organized topology. A Martian visiting Earth, apart from the scale, would not notice much of a difference between the social network, the World Wide Web, or the web within our cells.

What do you mean when you say, “only one link is required to form a society?”

Think about a group of people at a party that don’t know each other initially. Then pick two and introduce them to each other. Pick another two and introduce them as well. If you continue doing such random introductions, you’ll see groups of people emerging that will be connected to each other by social links. If you make only a few introductions, islands of people will know each other, but have no connections to other islands. The question is, how many introductions do you need to make in order to guarantee that most people are part of a large cluster, where you can reach from everybody to everybody else through a chain of acquaintances? In the 1950s two mathematicians, Paul Erdös and Alfred Rényi, offered the answer: when each person knows at least one other person in the group, most people will be part of a large cluster. Highly counterintuitive, but one link is enough for the society to emerge. This explains why nobody is isolated in the social network: we each have far more than one acquaintance, guaranteeing that no one is left out of the giant social cluster.

What role do networks play in technology?

Let me focus only on the Internet, a complex network of computers connected by wires that allow them to communicate with each other. The most fascinating thing about the Internet is that it is not centrally designed. A network always evolves by the addition of new nodes. The nodes of the Internet are the routers—and there is no central blueprint telling us where can we one add a new one—any company can add its own router to the network without asking permission from anybody. Therefore, the Internet took up all the characteristics of a self-organized network, becoming very similar to other complex networks in nature. It’s dominated by a few highly connected hubs that carry most of the traffic, and to which everybody connects. This is good news, as this self-organized structure makes the Internet resilient against unavoidable random local failures. But it carries its Achilles’ Heel, as a group of well-trained hackers could bring it down in no time.

Why haven’t hackers attempted to bring down the Internet yet?

If hackers take down the Internet, they rob themselves of their favorite toy. Real hackers would never do that. Rogue nations might, however. The reliance of the United States on the Internet is overwhelming, which has lead to a very asymmetric threat: the U.S. could gain little by breaking down the Iraqi Internet, but Iraq could create significant destruction by disrupting the U.S. infrastructure. This is the reason why many fear that as our dependence on the Internet deepens, terrorism will move more and more toward cyber-warfare.

Does your network theory explain why the terrorists went for the World Trade Center?

It certainly gives us some clues as to why this choice was the most appropriate from the terrorist’s perspective. Most networks have an Achilles’ Heel: if you knock the hubs out, you can inflict serious damage. The World Trade Center is probably the most prominent single economic hub, the White House (which was supposed to be a target) and the Pentagon are the most visible political and military hubs. But network theory also helps us understand why the attacks failed to bring these networks down: knocking out a single hub, even the biggest one, is never sufficient. You have to go for the whole hierarchy of hubs. Perhaps even more important, network theory helps us understand how the terrorist networks emerged. It’s increasingly clear that Osama bin Laden took advantage of the natural laws of network formation to create his terrorist organization. The terrorist network’s non-centralized, hub dominated structure offers a resilience not seen in regular armies, allowing it to function successfully under very different conditions.

What would it take to knock “terrorist networks” down?
The answer is simple: knock the hubs out. Yet, as I mentioned earlier, capturing the biggest hubs is not sufficient. You really have to get the whole hierarchy. But there are other ways as well: recent results indicate that one can induce a natural self-destruction in a network by inducing a cascade of internal failures. We have seen such cascading failures in the Asian financial crisis and the 1996 California power failure. It was bad news for everybody affected by it. We could turn such cascading failures to our advantage, however, in the case of the terrorist networks.

What role do networks play in biology?

If there is any field where understanding networks will bring an unmistakable revolution in the next decade, biology is the one. We have the full list of the genes for humans and dozens of simpler organisms. The most important lesson we learned from genome mapping is how little we really know of how cells work. While we have a hint of the components, we don’t know how these components interact, or how the signatures of life emerge from the interactions of thousands of molecules. Therefore, biology is on an incredibly fast track to map out the network behind the cell, seen as the ultimate barrier for comprehending life. Just as I write these lines, Nature and Science have published four major studies telling us which proteins interact with which other proteins. A new field has emerged, called bioinformatics, that aims to make sense of the incredible amounts of network data collected by the experiments. The bulk of my current research is also in this area. The advances taking place at an unparalleled pace offer the potential for new drugs and a far better understanding of what life is and how cells function.

Are you saying doctors will never cure cancer if they keep approaching the disease on a gene-by-gene basis?

That is pretty clear to all involved in cancer research. Most diseases, ranging from cancer to manic depression, are not due to a single broken gene but arise from the interaction of several malfunctioning genes and gene products. A good example is cancer. The so-called p53 molecule is known to be mutated in a high percentage of cancer patients. Yet, this molecule alone is not responsible for cancer. It acts as a cellular cop—when mutated, it is unable to stop the harmful effect of other gene disorders. Only when other, cancer causing genes break down and interact with this p53 gene can we see cancer developing. It’s increasingly clear, and frequently discussed in scientific publications, that we won’t cure cancer until we fully understand these network effects.

You claim it takes only six handshakes to get from person to person in society. But you say it takes nineteen, on average, to get from website to website. Does that mean word of mouth is still more powerful than website buzz?

Perhaps the most important aspect of this finding is not the difference between six and nineteen, but their similarity. Look at it this way: six billion people linked by six handshakes. Over a billion webpages linked by nineteen clicks. I can continue the list: close to a half million Hollywood actors, each only three links from other actors. Each molecule in the cell is only three reactions from the other molecules. A generic property of networks is that they create small words: you can navigate between a huge number of nodes with only a few links. It’s this universality of small world behavior that I find intriguing. From this perspective there is little difference between six and nineteen. With regard to word of mouth, the spread of buzz is less determined by the small world effect than by the network’s topology. One of the most incredible recent findings is that in hub-dominated networks, buzz and viruses can spread indefinitely. This means that no matter how weakly contagious a virus is, it has a fair chance to reach all of us. This finding has turned everything we know about how to protect ourselves from virus outbreaks upside down, and explains why the weakly infectious AIDS virus has reached such a significant percentage of the population. Indeed, the sexual network is hub dominated—there are a few individuals that have hundreds of sexual partners, and they have an incredible impact on our ability to stop diseases.

You say that Vernon Jordan is only three handshakes away from any Fortune 1000 corporate director. Does that make him the country’s best networker?

It certainly makes him the country’s most central director. Jordan’s case is a wonderful example of the power of networks and networking: we can follow step-by-step how his first directorship, via network effects, has landed him on more and more company boards, eventually making him the most central director, with membership on eleven boards. Once you follow his path and see how members of one board recommended him to members of other boards, you can understand why he emerged as a central node in the “six degrees of Monica Lewinsky” scandal.

What made you get interested in networks?

I lived in the Bronx at that time, and visited Manhattan daily. One day I envisioned the millions of Internet, phone, and electric cables hidden beneath the pavement that make the city functional. You don’t see them, but you know that there’s a very complex nervous system with uncountable nodes and links. I felt that there must be some organizing principles that describe the structure of this amazing construct, and I’ve been thinking about networks ever since. The project had a very long incubation period, however: it was five years before this thinking resulted in some tangible results when we predicted the diameter of the World Wide Web.

How has your knowledge of networks affected your personal life?

Well, since I started work on the book, I’ve ceased to have a personal life…but seriously, it did give me a completely different perspective on the world. Before many systems looked hopelessly entangled and complicated. After uncovering the network behind them, they became surprisingly simple and elegant. Phenomena that I was always baffled by, from market crashes to cellular robustness, now make sense. In general, I have a far better understanding of why and how things happen—from careers to financial issues, from the rise of terrorism to searching the World Wide Web—than I did before. I guess you could say this new approach changed the way I internalize the events around me.
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The Structure and Dynamics of Networks

The Structure and Dynamics of Networks

Albert-László Barabási, Mark Newman, Duncan J. Watts

From the Internet to networks of friendship, disease transmission, and even terrorism, the concept--and the reality--of networks has come to pervade modern society. But what exactly is a network? What different types of networks are there? Why are they interesting, and what can they tell us? In recent years, scientists from a range of fields--including mathematics, physics, computer science, sociology, and biology--have been pursuing these questions and building a new "science of networks." This book brings together for the first time a set of seminal articles representing research from across these disciplines. It is an ideal sourcebook for the key research in this fast-growing field. The book is organized into four sections, each preceded by an editors' introduction summarizing its contents and general theme. The first section sets the stage by discussing some of the historical antecedents of contemporary research in the area. From there the book moves to the empirical side of the science of networks before turning to the foundational modeling ideas that have been the focus of much subsequent activity. The book closes by taking the reader to the cutting edge of network science--the relationship between network structure and system dynamics. From network robustness to the spread of disease, this section offers a potpourri of topics on this rapidly expanding frontier of the new science.

Fractal Concepts In Surface Growth

Fractal Concepts In Surface Growth

Albert-László Barabási, H.E. Stanley

Fractals and surfaces are two of the most widely-studied areas of modern physics. In fact, most surfaces in nature are fractals. In this book, Drs. Barabási and Stanley explain how fractals can be successfully used to describe and predict the morphology of surface growth. The authors begin by presenting basic growth models and the principles used to develop them. They next demonstrate how models can be used to answer specific questions about surface roughness. In the second half of the book, they discuss in detail two classes of phenomena: fluid flow in porous media and molecular beam epitaxy (MBE). In each case, the authors review the model and analytical approach, and present experimental results. This book is the first attempt to unite the subjects of fractals and surfaces, and it will appeal to advanced undergraduate and graduate students in condensed matter physics and statistical mechanics.