LINKED: The New Science of Networks (Paperback available!). "LINKED: The New Science of Networks". by Albert-László Barabási. Review by 'Jim'

Last year I read a book called Linked which introduced me to this power law concept.
LINKED: The New Science of Networks (Paperback available!). "LINKED: The New Science of Networks". How Everything is Connected to Everything Else
http://www.nd.edu/~networks/linked/
In his latest book Linked; The new science of networks Barabasi continues to explore the Web's connectedness and network theory. Barabasi has found that the websites that form the network (of the WWW) have certain mathematical properties. The conditions for these
properties to occur are threefold. The first is that the network has to be expanding, growing. This precondition of growth is very important as the idea of emergence comes with it. It is constantly evolving and adapting. That condition exists markedly with the world wide web. The second is the condition of preferential attachment, that is, nodes (websites) will wish to link themselves to hubs websites) with the most connections. The third condition is what is
termed competitive fitness which in network terms means its rate of attraction. Babarasi, as quoted by Boyle said: "We have this very inhomogeneous Web structure — many pages and
nobody points to them — and there are a few sites that have millions of links pointing to them…… "Yes, you could have a Web site out there, but will anybody point to it" Boyle (2003)
Barabasi's team says that there is another common kind of network that has hither to been neglected: the 'scale-free' network, in which there is no meaningful average number of links--no 'scale' to it, in other words. In a scale-free network the number of nodes with a given number of connections simply declines as that number of nodes increases. Many nodes are linked to the network via just one connection; fewer have two, even fewer have three, and so forth. Unlike an exponential network, there remain small but significant numbers of nodes with many connections. Ball (2000)
In this book Barabasi makes these main propositions. One, the Internet is a scale free network. A scale free network means that many nodes (websites, on the web) will have only a few links to them. Other sites will have more, and that a very few will have lots of links, and in network terms be transformed into "hubs". Two, that a Power law operates in these self-organising networks and is a property of scale free networks. The power law is driven by preferential attachment, that is nodes in a network will associate themselves with the larger hubs of the network, and that this phenomena breaks the Internet into 4 continents. Those he calls the
central core as well as islands and tendrils. The size of which roughly approximate each other in size, being 1.2 billion webpages (as of 2002) divided by 4.The distribution of links to all sites on
the web approximates a "power law", that is, a small number of sites receive the majority of links and most sites receive very few links.
As Barabasi (2002) has explained in his article in physicsweb a `scale free network "is similar to an airline route map where there are few hubs (fatly connected) which link almost all other
airports(nodes). He makes the point that the majority of objects on the web have only a few links. Growth and linkage preference are key characteristics of this and other scale free networks. "According to a recent study by Steve Lawrence of the NEC Research Institute in New Jersey and Lee Giles of Pennsylvania State University, the Web contains nearly a billion documents. The documents represent the nodes of this complex network and they are
connected by locators, known as URLs, that allow us to navigate from one Web page to another."
"The finding that the Web is a scale-free network raised an important question: would such inhomogenous topology also emerge in other complex systems?Recently an answer to this question came from an unexpected direction - the Internet itself. The Internet forms a
physical network, the nodes of which are "routers" that navigate packets of data from one computer to another, and groups of routers and computers that are called "domains". The links that join the nodes together are the various physical connectors, such as phone wires and optical cables (figure 2). Due to the physical nature of the connections, this network was expected to be different from the Web, where adding a link to an arbitrary remote page is as easy as linking to a computer in the next room. To the surprise of many, the network behind the Internet also appears to follow a power-law distribution…. This indicates that the wiring of the Internet is also dominated by several highly connected hubs. "
Barabasi (2001) The Internet is represented as a network, as can strategies for solving a problem, topics in a conversation, and even words in a language. Many networks, turn out to be small worlds.Network works as a descriptor because we can invoke a mental picture
of connectedness to describe something that is not physical like an object, but is real and occurring over time. You can record "its" traces and thus measure "its" existence. What we know about the Internet is that it is scale free. There is a power law in operation that allows that powerful property. It is also self-directed and operates by preferential attachment. Just as in nature one species preys on another but never the other way around.
"Today we know that, though real networks are not as random as Erdos and Renyi envisioned, chance and randomness do play a n important role in their construction. Real networks are not static, as all graph theoretical models were until recently. Instead, growth plays a key role in shaping their topology. They are not as centralized as a star network is. Rather, there is a hierarchy of hubs that keep these networks together, a heavily connected node closely followed
by several less connected ones, trailed by dozens of even smaller nodes. No central node sits in the middle of the spider web, controlling and monitoring every link and node. There is no single
node whose removal could break the web. A scale-free network is a web without a spider."
Barabarasi (2002, page221) The Power law is a property of scale free networks. Preferential
attachment is the driver of this property. As networks grow, new nodes don't attach themselves randomly. They prefer to connect to the nodes that are already the best-connected. This is a "Power law distribution".
"Power laws rarely emerge in systems completely dominated by a roll of the dice. Physicists have learned that most often they signal a transition from disorder to order. thus the power laws we spotted on the web indicated, for the first time in precise mathematical terms,that real networks are far from random. Complex networks finally started to speak to us in language that scientists trained in self organization and complexity could finally understand."
Barabarasi (2002 page72 ) "Nature normally hates power laws.In ordinary systems all quatities
follow bell curves,and correlations decay rapidly, obeying exponential laws. But all that changes if the system is forced to undergo a phase transition.Then power laws emerge-natures unmistakable sign that chaos is departing in favour of order" Barabarasi (2002, page 77)
Using a power law we know that as the number of nodes (websites) on the network (WWW) rise, the curve becomes more extreme. This would seem to be contrary to what one would expect, but none the less increasing the size of the system increases the gap between the top
spots (hubs) and the median spots of the nodes. Secondly the majority of the nodes (websites) have below average links or connectivity because the curve is weighted towards the hubs on the
network.
"Scale-free networks" " refers to the connectivity "embodied by the average node and fixed by the peak of the degree distribution. A "scale-free" network can in graph form many weakly-connected nodes at one end, and a few highly-connected nodes at the other and has no
intrinsic scale. What Linked shows here is the notion that these are universal properties of networks of all kinds and are both their source of power and their source of vulnerability. A contagion spreads quickly through a human network -- aided by well-connected nodes. The distribution of links in the network is key to how rapidly a contagion, be it an idea or a software virus, spreads.
Some contagions are good, such as new ideas, others are bad.

References
Barbarasi A (2001) "The Physics of the Web", Physics World, July
2001.
Accessed 10/10/2002,available as of 17/2/2003:
http://www.physicsweb.org/article/world/14/7/09
Babarasi A.L. (2002) "Linked " , Perseus Publishing, Cambridge Mass.
Ball P. (2000) "Missing Links", Nature, March 2001.
Accessed 7/8/2002,available as of 17/2/2003:
http://www.nature.com/nsu/000727/000727-9.html
Boyle A. (2003) " Internet Navigator Think Small"
Accessed 9/1/2003, available as of 17/2/2003:
http://www.msnbc.com/news/750507.asp?cp1=1#BODY