# NetworkTheory

### From WikiWorld

2005-03-15: [Adjustment to Metcalf's Law]: Some researchers have refined Metcalfe's Law. Metcalfe's Law states that the more of something in the network, the better, and predicts how much better/value a joining of two networks would make. Just another link/story for Jim. ;) Jim, read, and put this link wherever you like. A link section, file 13, etc etc etc. --StarPilot

2005-02-08: [How SocialNetworks affect Animals, and animal movement]: Fun little story for JimScarver, about how social animals, from bees to humans, are affected by a few knowledgable individuals in a mass of uninformed. Happy Unbirthday --StarPilot

2003-01-05: LatticeNetworks and TransportationNetworks are useful models for SocialNetworks, NeuralNetworks, CollectiveIntelligence, InformationPhysics, PubWan, etc..

The new laws of network theory seem somewhat counterintuitive. Introducing a tiny number of random links into a network may easily cause radical changes in the overall network behavior for good or ill. Introducing error can actually improve speed and reliability. Practical application of these new found laws of information networks is being found in many diverse areas from making semiconductors faster and more reliable to combating the AdministratiumAtom and the understanding of SmallWorlds and SmartMobs.

http://pup.princeton.edu/catalogs/series/psc.html

## Contents |

## Small Worlds and Networks

Small Worlds:Network theory's new math

Everyone knows the SmallWorld phenomenon: soon after meeting a stranger, we are surprised to discover that we have a mutual friend, or we are connected through a short chain of acquaintances. In his book, Duncan Watts uses this intriguing phenomenon--colloquially called "six degrees of separation"--as a prelude to a more general exploration: under what conditions can a small world arise in any kind of network?

The networks of this story are everywhere: the brain is a network of neurons; organizations are people networks; the global economy is a network of national economies, which are networks of markets, which are in turn networks of interacting producers and consumers. Food webs, ecosystems, and the Internet can all be represented as networks, as can strategies for solving a problem, topics in a conversation, and even words in a language. Many of these networks, the author claims, will turn out to be small worlds.

How do such networks matter? Simply put, local actions can have global consequences, and the relationship between local and global dynamics depends critically on the network's structure. Watts illustrates the subtleties of this relationship using a variety of simple models---the spread of infectious disease through a structured population; the evolution of cooperation in game theory; the computational capacity of cellular automata; and the synchronization of coupled phase-oscillators.

See SmallWorlds.

### Network Theory's New Math

Some are born connected, others achieve connection, still others have connectedness thrust upon them. Everyone is networked. Everyone is either a node or a hub in someone else's network. Much as the quality of life is influenced by the quality of our networks, our standard of living is increasingly determined by network standards. To paraphrase Marshall McLuhan, we shape our networks and then our networks shape us.

The notion of networks as a dominant organizing principle to explain how the world really works has attracted enormous interdisciplinary interest. Physicists are talking to mathematicians who are talking to sociologists and economists who are talking to physicists. In barely a decade, networks of researchers have sprung up to research networks. Executives are beginning to turn to these experts for usable insights into the network dynamics shaping both threats and opportunities in business.

This is no surprise. Transportation networks have striking similarities to telecommunications networks. The Internet's technological behaviors map well onto the ecological behaviors of the biosphere. The complex interconnections between people in research laboratories around the world can be cost-effectively etched onto the design of silicon chips. Similarly, the myriad networks that define corporate connectedness are alike. Economies aren't merely marketplaces; they're networks. Executives need to understand network forces, not just market forces.

As Albert-Laszlo Barabasi, author of "Linked: The New Science of Networks" (Perseus Publishing, 2002), writes, "The diversity of networks in business and the economy is mind-boggling. There are policy networks, ownership networks, collaboration networks, organizational networks, network marketing--you name it. It would be impossible to integrate these diverse interactions into a single all-encompassing web. Yet no matter what organizational level we look at, the same robust and universal laws that govern nature's webs seem to greet us."

These laws of networks may prove as robust and universal as Newton's laws of motion. But making network laws, which like Newtonian laws are steeped in mathematics and metaphor, comprehensible to the layperson is hard work. The New Yorker's Malcolm Gladwell took a successful first cut with his best-selling "The Tipping Point: How Little Things Can Make a Big Difference" (Little, Brown and Company, 2000). Three new books published this year go far beyond tipping points to present to the conceptually curious reader important theories that reveal the hidden order of complex networks.

Barabasi, the author of "Linked," is a physicist and leading researcher in the field who uses the Internet as his dominant research medium for analyzing the peculiar properties of networks. His book is ideal for those looking for the perspective of a network researcher and practitioner; it's even spiced with a few equations. Mark Buchanan's "Nexus: Small Worlds and the Groundbreaking Science of Networks" (W.W. Norton & Company, 2002) is the product of a physics Ph.D. who writes for the noted scientific journals Nature and New Scientist. Although Buchanan draws heavily on Barab's work, his intellectual focus is the intriguing so-called small-world networking theories of mathematicians Duncan Watts and Steve Strogatz. Small-world theories, which are derived from theoretical mathematics and practical reality, prove that seemingly distant, disconnected, and disparate populations, events, or actions can be easily linked to one another. Like many scientists turned writers, Buchanan is a bit of an ideologue who seems more comfortable discussing network ecologies than network economics. Then again, because of the transcendent nature of networks, the distinctions between ecology and economics aren't that great.

### Order and randomness

As described in "Nexus," the ideas underlying Watts and Strogatz's "small worlds" are simple, powerful and compelling. In effect, Watts and Strogatz validated the "six degrees of separation" phenomenon, the belief that any two people on earth are separated by no more than five people connected to each other in some meaningful way.

Inspired by earlier research on social networks, the two struggled to find a coherent mathematical way to describe how these networks were connected. What Watts and Strogatz found was counterintuitive and profound: By injecting just a few random connections into a complex network, they could make that network both more efficient and more effective. The right random links create small worlds from vast complexities. Randomness can dramatically improve the performance of a complex system rather than ruining it.

When Watts and Strogatz published a paper on their small-world theories in Nature in 1998, it "touched off a storm of further work across many fields of science," Buchanan writes. "A version of their small-world geometry appears to lie behind the structure of crucial proteins in our bodies, the food webs of our ecosystems, and even the grammar and structure of the language we use. It is the architectural secret of the Internet and despite its apparent simplicity is in all ways a new geometrical and architectural idea of immense importance."

This finding on randomness has already had a significant impact on the design of telecommunications networks and silicon chips. Microprocessor companies like Intel and Motorola now use elements of small-world theory to link circuits on their semiconductors to make them run faster and more efficiently. Engineers are now aggressively exploring the role of randomness in performance enhancement of their products. Purely rational design that once treated randomness as the enemy has been transformed; designers now play with randomness as a tool to create "small worlds" that exploit this power of serendipitous connection. The result is more robust networks and ever-faster silicon chips. These innovations wouldn't have occurred without the proofs outlined by Watts and Strogatz.

It's important to remember--and this theme is stressed in each of the books--that small-world theory findings are the direct result of interdisciplinary interaction and observation. Empirical observation is just as important as clever theory. The beauty of the small-world hypotheses is that they can be tested in the real world very quickly.

### Power laws

Random geometries of small worlds is just one network law that commands respect. While ambitious managers read Machiavelli to better understand the laws of social and political power, effective executives need to understand that mathematical "power laws" profoundly shape laws of personal power.

"If you are not a physicist or mathematician, most likely you have never heard of power laws," asserts Barabasi. In Linked, executives will recognize their importance, because power laws can reveal as much about marketing and finance as they do about math and physics.

The "power" in power laws is not a function of Machiavellian manipulation but the "power" found in exponential functions; numbers squared or cubed or taken to the 10th power, etc. Power laws strike at the heart of what businesspeople think they understand about playing the odds and managing risk. Why? Because power laws are the sworn enemy of a basic statistical concept: the notion that probabilities present themselves in the average distribution of bell-shaped curves. In a networked world ruled by power laws, the bell curve is a dangerous lie.

In fact, power laws describe a radically different kind of distribution. There are no peaks; no symmetries; no bell curve. Power laws look nothing like traditional school-taught statistics. Yet they do a far better job of reflecting how much of the real world behaves. The distinguishing feature of a power law, Barabasi writes, is that its distribution is wildly skewed: Numerous tiny events coexist within the few very large ones that actually matter.

from http://news.com.com/2009-1122-978596.html?tag=fd_nc_1

## Addenda

As an addenda, its worth noting that both Duncan Watts and Steve Strogatz have produced their own popularization books on networks. Watts' book 'Six Degrees' accomplishes more theoretically and mathematically than Barabasi's or Buchanan's, while citing more of the historical figures in the social sciences who paved the way for network analysis. Unlike the other two, there is also an excited spirit of discovery that tinges the book with wonder known better possibly to a scientist than an executive. BernieHogan

Random connection synchronize massively parallel algorithms like the CollectiveIntelligence.

- This is an empowering concept of WikiWorld, and lies at the heart of its belief that a few individuals can have an powerful or exaggerated influence on the WE. This is why it is a featured stop on the TourOfWikiWorld.

## ToDo

Please clean up/Refactor **NetworkTheory** as appropriate (Owner StarPilot)