By Fon Sundaravej
Nexus: Small Worlds and the Groundbreaking Theory of Networks
By Mark Buchanan
Abstract
This book is an introduction to the network theory. It is readable with a
simple explanation of complex network rules. Major concepts of the theory
include a small-world network, degrees of separation, strong and weak ties,
and a tipping point. At the end of the book, several examples that the theory
can apply to are discussed. The theory is applicable to diverse fields such
as ecology, biology, politics, economics, and computer science. Even so,
more studies on the network theory in different fields are still necessary to
help us better understand a structure of a complex network in diverse areas
and gain more benefits from the studies.
Author Biography
Mark Buchanan is a physicist and science writer whose
articles have appeared in numerous professional and general science
magazines including Scientific American and New Scientist. He
earned a Ph.D. in Theoretical Physics from the University of
Virginia in 1993. He has worked for several years in the areas of nonlinear dynamics,
Hamiltonian systems theory and plasma physics. Formerly an editor of the international
science journal Nature, and the science magazine New Scientist, he has extensively written
for publications ranging from Science to the Harvard Business Review.
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Introduction
A psychologist, Stanley Milgram, initiated the lost letter technique which later
became a concept of the six degrees of separation. This concept implies to a characteristic
of our social networks that everyone in the world is separated by only six other people.
From this finding, Duncan Watts and Steve Strogatz, two mathematicians from Cornell
University, discovered a theory of the small-world networks. This theory was soon applied
to different areas ranging from biology, ecology, economics, politic, computer science, etc.
Chapter 1: Strange Connections
The small-world networks by Watts and Strogatz is simply a graph that contains a
network of dots connected by lines. The basic idea behind this diagram is to explain the
structure of a social network that formulates a small world. The Oracle of Kevin Bacon
game (http://www.cs.virginia.edu/oracle) was invented by two graduate students from the
University of Virginia to imitate a social world of Hollywood actors. The game reveals that
the social network of Hollywood actors can be described as a small-world network,
indicated by a small number of links from all actors to Bacon. This result refers to our
social network as a small-world network.
Chapter 2: The Strength of Weak Ties
In a social network, there are two types of connectors: strong and weak ties. A
familiar or close person such as a family member, friend, or co-worker is considered as a
strong tie, whereas an unfamiliar person is a weak tie. The author claims that a secret of the
small-world networks or six degrees of separation is the performance of a weak tie that
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acts as a social bridge or crucial connection tying networks together. A network with
clustering connections does not have a small-world property. Even though, a strong-tie
connector is eliminated from networks, networks can still be connected as long as a weaktie connector is not omitted.
Chapter 3: Small Worlds
A technique to create a small world is to add some random links into a network.
From an experiment by Watts and Strogatz, with a very few random links added to a
clustered network, the degrees of separation can be tremendously reduced. This
phenomenon happens because an added random link acts as a shortcut between distant
points. In an ordered network, in contrast, moving from one point to another may take
several steps.
Chapter 4: Brain Works
How a human brain works represents a small-world network. The brain is an
assembly of distinct modules. Each module is responsible for a different basic function of
perception, for instance, speech, vision, emotion, language, etc. These modules are
connected in order to coordinate overall brain activities. Many Dendrites or neuron’s
receiving channels link nearby modules that perform the same function, while few Axons
are longer channels that connect to more distant modules. If some Dendrites are damaged,
a person still can talk, see, or feel. If Axons are destroyed, however, brain may not perform
some functions.
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Chapter 5: The Small-World Web
The Internet is considered as a redundancy network. A major benefit of a
redundancy network is its flexibility. That is, if a route to transmit data is destroyed, the
network computers still can reroute the messages to the destination. The Internet is also
proved to be a small-world network because tracing a connection between computers is not
more than ten links.
A Webpage can reflect a random network with a different number of hyperlinks on
each page. There are a few Web pages that operate as a hub or contain a lot of links
connected to numerous Websites, while most Web pages contain approximately three to
four links per page. This pattern refers to a power law, which means that each time the
number of links doubles, the number of Web pages with that many links becomes less by
about five times. A logarithm is used to measure a diameter of a Web page or a distance
between documents. An experiment verifies that even though new Web sites are
continually established, the number of clicks from one document to another is raised only
from about 19 to 21.
Chapter 6: Accidental Science
The Internet has a complicated structure. There is no regular framework like other
scientific experiments such as water heated in a pan or sand shapes after a disturbance.
These types of experiments have orchestrated historic events that can predict the formation
of a future pattern. Even if the Internet growth is truly random, its unorganized evolution
establishes that the Internet shares a similar globally organized pattern.
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Chapter 7: The Rich Get Richer
The small-world networks or six degrees of separation can be summarized into two
types: egalitarian and aristocratic. The egalitarian type has been explained by Watts and
Strogatz. They believe that the network does not grow and each connector in the network
links to almost equal number of other connectors. In contrast, the aristocratic type is
explicated by Albert-Laszlo Barabasi, Reka Albert, and Mark Gronovetter. They consider
that a network is complex with some connectors acting as a hub to dominate the network.
A key characteristic of the aristocratic network is the rich get richer pattern. A
Website is an example. Generally, a Website tends to link to popular sites that contain a lot
links. As such, popular sites gain more connections. However, it is also possible that a
popular site loses its name recognition when time passes and its connections to other sites
become less until below a tipping point or the point that differentiates between two stages.
Chapter 8: Costs and Consequences
From the concept of a tipping point, an aristocratic network can be transformed to
the egalitarian one. An airport hub is a case. An airport hub allows a high volume of
travelers to diverse cities. However, a too congested airport hub results in flight delays.
Many airlines will gradually vanish to other less congested airports.
This occurrence can also apply to a computer network. If a data transfer encounters
congested traffic, the congested route can be avoided and the data are rerouted to a
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different path. However, for a network safety issue, redundancy of a network is not
enough. A network protection such as firewall or data encryption is needed.
Chapter 9: The Tangled Web
Chapters 9 to 12 demonstrate examples of small-world networks in various areas.
Chapter 9 is a small-world network in ecology. An ecosystem has a complex food-web
network which is unstable even with a small disturbance. Destroying only one type of
species wildly and rapidly affects the entire community. A proof illustrates only two or less
than ten degrees of separation in an ecosystem. The ecosystem can be described as an
aristocratic small-world network. The predator that eats only one other specie is perceived
as a strong connector in the network. If this predator is eradicated, the entire network will
be disturbed.
Chapter 10: Tipping Points
Chapter 10 applies the theory of networks to sociology. The notion that ideas,
rumors, crimes, and other influences spread through a society is explored. Meme is a
genetic element to the logic of spreading or imitating ideas, catch-phrases, fashions, etc. If
Meme continues its existence, its stage will be above the tipping point. On the contrary, if
Meme disappears, its position will be under the tipping point. If Meme is on the critical
stage or balance point, this means it is changing between two different conditions.
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Chapter 11: Breaking Out Small-World Style
This chapter relates the small-world networks to biology. The spread of AIDS
epidemic has strongly existed in the past fifteen years. It is believed that the origin of
AIDS is in Africa and it has lasted for more than a half century. The disruption of post
colonial Africa is considered as a major cause of the disease expansion. Soldiers, doctors,
nurses, and many other people traveled to Africa, while more Africans departed their
country by newly constructed communication channels such as trains or roads. However, if
an infected person has only one sexual partner, the disease will not spread so rapidly. Until
one day, the disease will be below a tipping point and disappear. However, the truth is
opposite. Infected people have more than one sexual partners, especially prostitutes. This
results in a continuing existence of AIDS. To control AIDS, we should concentrate on the
group of strong tie patients who have a sexual contact with many people rather than the
weak tie group.
Chapter 12: Laws for the Living
The last example of the theory of networks applies to economics. Wealth can flow
within a network when someone purchases goods and services or earns a salary. Normally,
in a society, there are few rich people but many poor ones. Wealthy people tend to get
richer, while meager people remain poor. Rich people have a capability to purchase more
lands or do a huge investment to earn more money than poor people. Taxation, as a result,
is used as a mechanism to erode the difference in wealth of people in a society. However,
there is a probability of a collapse of wealth. This represents that a tipping point exists in
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economics. In addition, a collapse of wealth can dramatically transform a society such as a
shift of political power to a small group of wealthier people.
Chapter 13: Beyond Coincidence
An aim to study a science of complexity is to discover patterns in every kind of
complex network, ultimately to better understand ourselves and our world. The
consequence of the study also proves that the small-world networks or small degrees of
separation found in a complex society such as the human brain, food webs, distribution of
wealth, or Internet is not simply a coincidence. The main characteristic of a small-world
network is a random system that still remains a high degree of clusters with few weak ties
to connect the entire network together. Both strong and weak ties play a significant role in
a small-world network. A group of strong ties has a history and frequent interaction, share
the same norms, and have a dense relationship, while weak ties keep a relationship with the
rest of the community and access diverse information. In conclusion, a small-world
network can gains benefits from both ties.
Author Biography References
http://www.randomhouse.com/crown/catalog/results.pperl?authorid=3655
http://www.orionbooks.co.uk/9827-0/Author-Mark-Buchanan.htm
http://www.informatics.indiana.edu/colloquia/default.asp?archive=true&id=523&st=&year
=2005
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