An Example of a "Learning Process" Journal (using the 2 colored box format)

Book: Nexus by Mark Buchanan.

Source: New York: W W Norton, 2002.

It is 5:30 am (Sunday). I hope to make substantial progress with my note making this morning.

Chapter 1 Strange Connections

• "... the basic mathematical puzzle of the small world: Is it possible, and if so, how is it possilbe, to draw a graph of six billion dots in such a way that one could link any two dots by travelling along just six lines?" [p. 31]
• "For years sociologists have been experimenting with different ways of viewing social networks, and lots of new ideas have become available with modern computers." [p. 32]

The second quote would suggest that it may be worthwhile to try a few searches using keywords such as network and sociology.

Chapter 2 The Strength of Weak Ties

• "Suppose we draw a collection of dots and simply wire them up haphazardly. The result is what mathematicians refer to as a random graph. In the late 1950s and early 1960s, Erdos and fellow mathematician Alfred Renyi wrote a series of classic papers studying such graphs and trying to answer questions about them." [p. 35]
• In 1973 Mark Granovetter, a young sociologist at Johns Hopkins wrote a paper called " The Strength of Weak Ties". American Journal of Sociology, 78, 1360-1380. [p. 43]
• "For nearly 30 years, his simple but striking insights into the character of social networks and the importance of weak ties - especially the bridging links that some of them represent - remained virtually unnoticed by other scientists." [p. 47]

I have just downloaded the Granovetter article. The power of the Web continues to amaze me.

Chapter 3 Small Worlds

• In 1996 Duncan Watts and Steve Strogatz, two mathematicians at Cornell University, began investigating networks. They created a circle of nodes where each node was connected to a small group of near neighbors, and then added a few random links.
• "The random links, being so few in number, have no noticeable effect on the degree of local clustering that makes social networks what they are." [p. 55]
• "There was nothing in the recipe that referred to people per se, or to social connections in particular; rather the recipe reflected a very basic scheme in graph theory." [p. 56]
• "Duncan J. Watts and Steven H. Strogatz, "Collective Dynamics of 'Small-World' Networks,' Nature 393, 440-442 (1998). [p. 211]

Googling the title gave me a web site with a pdf download of the article. Amazing.

I now have two original articles to read!

Chapter 4 Brain Works

• "... the thin, gray outer layer known as the cerebral cortex. For centuries, the cortex had been thought of as an unimportant protective layer. In actuality, though never more than a few millimeters thick, this layer contains most of the brain's precious neurons. ... the cortex becomes highly convoluted and folded in creatures roughly bigger than mice, this being necessary to fit the bigger brains inside the skull. If you could stretch the cortex of the human brain out flat, it would cover the surface of a picnic table. This intricately folded and delicately packed cortex is where higher intelligence resides." [p. 63]
• "We may recall a name, whisper it to a friend, and gesture with our hands to some additional effect. These actions involve many of the brain's functional regions working in effortless combination, with information shuttling rapidly and efficiently between them. All this activity raises an obvious question: what wiring pattern does the brain use to provide this efficiency?" [p. 64]
• "... most of the neurons link up with others nearby, within the same functional region. ... However the brain also has a smaller number of truly long-distance axons that link brain regions, that lie far apart, sometimes even on opposite sides of the brain. Consequently, we have many local links and a few long-distance links." [p. 64]
• "The small-world pattern guarantees that the brain's diverse functional parts reside only a few steps from one another, binding the entire network into one intimate unit." [p. 65]
• "Are our emotions and sense of responsibility merely the consequences of the mechanical laws of physics when combined in a setting of sufficient complexity? ... Researchers have learned, for example, that consciousness always involves the activation of neurons from many regions of the brain - it seems to depend on their coherent engagement into one overall pattern. And the mechanism of this engagement, at least in part, is neural synchrony." [p. 66-67]
• "None of these findings really explain how the brain works or what consciousness is. However they do offer some clues concerning how the brain achieves the organized activity. ... Synchronous neural activity appears to play a central role in conscious functioning, and the small-world architecture facilitates the process." [p. 70]

Sheesh! I knew the cortex was highly convoluted but I never realized that it was this thin surface that actually contained almost all the neurons.

The idea of neural synchrony is another new idea for me.

Chapter 5 The Small-World Web

• "Mankind is a catalyzing enzyme for the transition from a carbon-based to a silicon-based intelligence." Gerard Bricogne [p. 73]
• In 1964 an American engineer named Paul Baran wrote a series of technical papers that described a decentralized "distributed" network for communication that would be highly redundant to enemy attack. Leonard Kleinrock at MIT and Donald Davis in England independently invented similar schemes. This idea became the Internet.
• "Peter Drucker ... suspects, it is not computers or the Internet that will be world-changing, but rather, one of their recent spin-offs: e-commerce ... a totally new, totally unprecedented, totally unexpected development. ... eBay is revolutionizing the way individuals buy and sell products" [p. 77]
• "The transactions between businesses and consumers represent only one-fifth of the total Inernet trade. The rest, the bulk of all e-commerce, takes place between businesses themselves." [p. 77]
• "The Internet is not the only network underlying the Information Revolution. ... The Internet is more or less pure hardware. By contrast, the World Wide Web is more ethereal. This is the vast network of Web pages connected together by hypertext links. ... Currently over one billion Web pages are connected together by hypertext links into one staggeringly large network. ... This network is another small-world network with a structure remarkably similar to the Internet itself." [p. 83]
• Two important mathematical curves are the Normal curve and the Power Law curve (which looks a bit like the letter L). The Power Law curve has a few very large values and then tails off toward zero. But this tailing off is much more gradual than for the Normal Curve. It turns out that if one plots the number of nodes on the Web against the number of links from each node, one obtains a graph that is very much like a Power Law curve.
• Albert-Laszlo Barabasi and colleagues at Notre Dame have devised an index of the Web, called its diameter, which is the average number of links between any two nodes. They estimate the diameter of the Web to be 19.
• "... there is more than one way for a network to be a small world." Watts and Strogatz showed one way to create a small world (local nodes plus a few random links). Barabasi showed another model: the importance of a few hubs or connectors. Cellular metabolism involves hubs as well, so does the English language.

One of the reasons I enjoy this book is the various "facts" that it mentions. I had never though about the amount of business that occurs among businesses!

I now have begun reading the book "Linked" which is by Barabasi.

Mathematics Index