Network Clustering: The Un-Google

Having finished our series on network centrality, we now approach its most natural complement: network clustering.

An easy way to appreciate the usefulness of network clustering is to try search engines that (unlike Google) are not centrality-driven. There are quite a few such search engines out there. They are great at providing a sense of direction within a previously unknown field --- when you're not yet sure exactly what question you're asking. In contrast, Google is better when your query is more specific, or when you just don't care about the rest of the forest, dammit, and want to find the biggest most popular tree ASAP.

Below are two examples of how non-centrality-based search engines display the WWW of "organizational network analysis". Click on either image to go to the search engine pictured.



There are dozens more search engines listed here by search engine junkie Bill Sebald.

I hope you enjoy the Un-Google world. Soon I'll say more about understanding this world with the help of network clustering.

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Network Clustering: Rob Cross and Kathleen Carley

Next Monday, Sept 8, begins the 2-day Network Roundtable Fall Conference. Rob Cross at UVA has led the Network Roundtable from its inception. He and his colleagues have quite an agenda planned for their time in DC.

My regular readers with sharp eyes may have noticed Rob Cross in a recent post of mine. That post introduced network clustering with an example --- a WWW clustering analysis of "organizational network analysis" computed by Grokker:

One of my favorite metaphors for clustering analysis is the table of contents. It is useful for seeing the big picture, all-inclusively, broken down into sub-categories. In an organizational network setting, a natural application would be identifying communities of practice (including those that don't yet recognize themselves as such).

Continuing with the book metaphor, we can see that the WWW authors of organizational network analysis have devoted "chapters" to these topics:

1. Social networks
2. Organizational systems
3. Public health
4. Information management
5. Knowledge
6. Tools
7. Rob Cross
8. Kathleen M Carley
9. Other

Most of these "chapters" are based on fields or methods of work. Two "chapters" stand out for being based on individual people.

Another way to view these "book chapters" is as "closed networks" (relatively speaking), as I described in my last post. I refer my readers again to that post, this time keeping Rob Cross and Kathleen Carley in mind. It's fun to speculate how the Cross and Carley camps employ stereotypes to describe their counterparts.

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Network Clustering: The Power of Reputation

As we leave our series on network centrality and begin an exploration of network clustering, who better to help us bridge the gap than Ron Burt. Burt is perhaps best known for his amazing network-based research on innovation and the source of good ideas, which brought "structural holes" to the world's attention. In Brokerage & Closure he expands these ideas into book form and brings additional attention to "closure," a key trait related to network clustering.

Very briefly, closure refers to the interconnectedness of one's contacts: When my contacts don't know each other, my network is "open," and when they do know each other, my network is "closed." Assuming that I am #1 (naturally), two extremes of open (left) and closed (right) are pictured below:"Open" and "closed" are pretty much the same as bridging and bonding, as I have discussed before:


For more discussion of network closure, I recommend Burt's online notes for his executive MBA course, "Strategic Leadership," specifically the chapter on Closure, which I would sum up with these two points:

1. The peer pressure created by closed networks builds commitment and productivity
   
2. The peer pressure created by closed networks reinforces groupthink and promotes mindless stereotypes
   

Click on the image below and you can read what Burt himself says:

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Network Clustering: Guide to Stereotyping Rob Cross and Kathleen Carley

Recently I mentioned how network clustering on the WWW indicates that Rob Cross and Kathleen Carley each have their own close-knit camps that co-dominate the world of "organizational network analysis." Before that, I shared Ron Burt's point that such close-knit camps are known not only for amazing productivity but also for stereotyping outsiders.

I am outside both the Cross and Carley camps, but I enjoy stereotyping as much as anyone, so today I provide convenient superficial labels with which my readers can simplify the contributions of these two notable network leaders.

Guide to stereotyping Rob Cross and Kathleen Carley:

1. Rob Cross provides stories for business
2. Kathleen Carley provides computer models for the military

Wasn't that easy? Now let's look at one example of each stereotype.

(1) The recent research of the Network Roundtable features Cross's "Braintrust Keynote Presentation." Here is his third slide:
Note the simple and compelling story in the top row of the table: Network density within and across departments of less than 20% indicates little collaboration. If you read the actual presentation, you'll see that the "target density" is only 9.4% because the current density is less than half that, so the target is a healthy step up towards 20%. I will skip the other rows of the table for now.

(2) Kathleen Carley's camp responds to the above story with the following article:
As far as stories go, this article sucks. But look, it is classified under "statistical simulation," because the researchers use computer programs not only to analyze networks, but also to create the very networks that they study (no pesky data collection necessary).

For those whose eyes are glazing over, let me summarize the computer model punchline with a picture. The following three networks all have exactly the same density, 20%; and so according to Cross each of the three networks below has exactly the minimum recommended allowance of connectivity to indicate collaboration:
As you can see, density of 20% means different things depending on how many nodes are in the network.

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Do small world networks promote innovation?

After reading the HBS Working Knowledge interview of HBS Professor Lee Fleming, I was happy to connect with Lee in person early this week. We discussed his working paper, "Small Worlds and Regional Innovative Advantage," co-authored with Charles King and Adam Juda.

This paper seeks to answer the question: Do small world networks promote innovation?

[By "small world" they mean that people tend to connect to their neighbors (hence we feel like local actors) but there are enough other links so that there is a short path of relationships between any two people, even across great distance.]

Fleming et al note that many have argued "yes" to this question but with scant evidence. So they have collected an impressive body of data to put the question to a more rigorous test.

Their results? My reading of the paper is that small world networks (at least the ones they studied) aren't so closely related to innovation after all. When I put that to Lee, he declined to sound so definitive and said, "The jury is still out."

Roughly speaking, here is what they did: Lee's team collected records of 2.5 million US patents filed by 2 million inventors over the last 25 years. Then they created a multi-component network by linking inventors with jointly filed patents. Finally, they looked at the largest connected component in each of 337 metropolitan areas, calculated 19 different characteristics of each (from both economic and network perspectives) and determined the correlations of these characteristics across all 337 regions.

Which characteristics correlate to more patenting? The strongest predictors of a productive patenting network of inventors are (1) one type of technology dominates the network of inventors, rather than many different kinds of technology, (2) young technology, (3) one firm dominates the network, rather than many different firms.

As for small world characteristics, Fleming et al found that clustering (connectedness of neighbors) has a negative relationship with innovation. Short path length has a statisticaly robust positive relationship, but its magnitude is several hundred times weaker than the positive effect of a regional monopoly (#3 above), for example.

Lee has left the "30,000-foot" view of this research, and his next project is to investigate a more closeup view that studies individual inventors in much greater detail. So far he is finding similar results (clustering is bad, short path lenghts marginally good) but again, the jury is still out.

After reading how BzzAgent succeeds at viral marketing without even considering SNA, and now reading Lee's research, my sense of social network analysis has just become significantly more abstract. These results are more than a little discouraging to anyone hoping to sell SNA as a engine of productivity. But the SNA practitioner in me is still encouraged, on balance, by results such as Burt's study of Raytheon managers. And the SNA researcher in me is excited to see how these studies continue to unfold, either way.

Web science, Webwhompers

I have just unveiled Webwhompers, which bears the fruit of four years of my teaching Web science at Boston University. The site features a few interests of mine:

• A solid layman's introduction to Web science, focusing on the intersection of mathematics, sociology, and the Web as it is used and built by regular people. It is all presented as an online textbook you can read here.
• A case study in educational methodology. Unlike the online textbook, which is meant to be read, the rest of Webwhompers is meant to be experienced. It provides the online portion of my answer to the question, "What can 70 non-technical college students do together in 12 weeks that will result in their learning as much as possible about the Web?"

The course mission statement puts it this way:

Technology is often created by "experts" and then used by "regular people." Webwhompers celebrates the "Web builder": a regular person who creates his own Web technology.

Sometimes it helps to distinguish between "regular people" who use technology and "experts" who create technology. For example, a regular person might want a home stereo; he pays experts to create hi-fi technology for him. In other cases, regular people create technology without even considering asking for expert help—for example, making a snowball.

Much of the Web technology that regular people want is within their power to create, just like a snowball. Webwhompers seeks to unleash the technical creativity of the regular person: By highlighting Web building resources, by bringing together aspiring Web builders, by providing expert guidance when necessary, and by encouraging regular people to try on the idea that they can create their own Web technology.

The course overview puts it this way:

Our course introduces Web science. It has no prerequisites and has been used by non-technical undergraduates at Boston University since 2006. Our curriculum is guided by the following passage adapted from "Web Science: An Interdisciplinary Approach to understanding the Web," by James Hendler, Nigel Shadbolt, Wendy Hall, and Tim Berners-Lee:

Web science, an emerging interdisciplinary field, takes the Web as its primary object of study. This study incorporates both the social interactions enabled by the Web's design and the applications that support them.

The Web is often studied at the micro scale, as an infrastructure of protocols, programming languages, and applications. However, it is the interaction of human beings creating, linking, and consuming information that generates the Web's behavior as emergent properties at the macro scale. These properties often generate surprising properties that require new analytic methods to be understood.

For example, when Mosaic, the first popular Web browser, was released publicly in 1992, the number of users quickly grew by several orders of magnitude, with more than a million downloads in the first year. The wide deployment of Mosaic led to a need for a way to find relevant material on the growing Web, and thus search became an important application, and later an industry, in its own right. The enormous success of search engines has inevitably yielded techniques to game the algorithms (an unexpected result) to improve search rank, leading, in turn, to the development of better search technologies to defeat the gaming. More recent macro-scale examples include photo-sharing on Flickr, video-uploading on YouTube, and social-networking sites like mySpace and Facebook.

The essence of Web science is to understand how to design systems to produce the effects we want. The best we can do today is design and build in the micro, hoping for the best; but how do we know if we've built in the right functionality to ensure the desired macro-scale effects? How do we predict other side effects and the emergent properties of the macro? Further, as the success or failure of a particular Web technology may involve aspects of social interaction among users, understanding the Web requires more than a simple analysis of technological issues but also of the social dynamic of perhaps millions of users.

Given the breadth of the Web and its inherently multi-user (social) nature, its science is necessarily interdisciplinary, involving at least mathematics, computer science, sociology, psychology, and economics.

Four important themes of Web Science are

• Micro: an individual acts
• Macro: the world responds (or not) to an individual's action
• Synthetic: something is created to produce a desired result
• Analytic: laws are stated to explain observed phenomena

We focus on these themes as they apply to Web builders -- people who contribute links and other content to the Web:

	Synthetic	Analytic
Micro	An individual builds a Web site to produce a desired result.	(We do not speak to this quadrant.)
Macro	"The world" builds a Web site to produce a desired result.	Laws are stated to explain large-scale Web phenomena.

Some Web builders consider themselves Web developers; others consider themselves bloggers; others merely post an occasional comment on someone else's blog or discussion forum. We say "Web builder" to encompass the full spectrum of people who contribute links and other content to the Web.

Our lab curriculum provides an informal hands-on approach to the task of building a Web site. Our Search and Share pages help Web builders leverage collectively engineered resources (such as WordPress). The formal chapters of the Study page (which you are now reading) explain large scale Web phenomena; they also explain the Amazon recommendation algorithm and the Google PageRank algorithm.

The sociology, psychology, and economics of this course follow Duncan Watts' Six Degrees, which we recommend as a narrative companion to our own material. Our complete suggested reading list is below.

Online safety

Protecting yourself from evildoers “The Difference Between a Virus, Worm, and Trojan Horse” by Webopedia "Secret geek A-Team hacks back, defends Web" by Joshua Davis, Wired, Nov 2008 "Thieves winning online war in your computer" by John Markoff, NY Times, Dec 2008 Privacy, trust, and ownership "Google Knows All" by Sarah Elton of Macleans.ca. "Wikipedia: What do they know; when do they know it, and when can we trust it?" by Susan Youngwood, Vermont Today, April 2007 Choosing a license by Creative Commons.

Networks

Basic mathematical foundations of networks: Set Theory Sets Explicit Notation for Sets Cardinality Subsets Venn Diagrams Union and Intersection Ordered Lists Implicit Notation for Sets Logical Expressions Compound expressions with "or" Compound expressions with "and" Union and intersection defined formally Similarity of Sets Graph Theory Graphs Undirected and Directed Neighborhood and Degree Density and Average Degree Paths Paths in undirected graphs defined formally Paths in directed graphs Length Distance See also Facebook and Touchgraph

Network Structure

Hubs, clusters, and other basic structural features of the Web: Network Structure Connected: a word of many meanings Induced Subgraphs "Connected" defined formally Connected graphs and connected components Hubs Clusters Defining clusters, part one: connected components Defining clusters, part two: cliques Defining clusters, part three See also: "Bow Tie" Structure of the WWW, by Chris Sherman, Information Today, May 22, 2000. Six Degrees Chapter 2 (skim pp 48-55 & 62-68)

Network Dynamics

How randomness, homophily, and cumulative advantage shape the Web: Network Dynamics Limitations of traditional graph theory Introduction to network dynamics Three models of dynamic graphs Random graphs Demonstration of random graph dynamics Random graph algorithm Clusters and homophily Triadic closure Triadic closure algorithm Hubs and cumulative advantage Preferential attachment algorithm See also: "Is Justin Timberlake a product of cumulative advantage?" by Duncan Watts, The New York Times, April 2007 "Science Online: Too Much of a Good Thing?", NSF Press Release featuring sociologist James Evans, July 17, 2008. All the above are summarized in the following table: Random graphs Clustering Centrality Real-world phenomenon explained by model Giant component forms quickly when |E| ≅ |V|. Clusters emerge, providing "table of contents" overview. Hubs emerge, indicating popularity and/or influence. Web sites N/A Clusty, iBoogie, Grokker Google et al Sociological force Chance Homophily Cumulative advantage Mathematical model Random graph algorithm Triadic closure algorithm Preferential attachment algorithm

Variables, Probability, and Scale-Free Networks

Understanding that the Web is a scale-free network requires some probability theory: Variables and Probability Variables in mathematics Variables in algorithms Random variables Discrete vs. continuous variables Probability distributions Degree distributions General discussion of scale-free networks: Six Degrees Chapter 4, pp 101-114 From previous chapter on Network Dynamics Hubs and cumulative advantage Preferential attachment algorithm

Information and Computation

Applying fundamental concepts of computer science to the Web Information and computation Information, computation, and algorithms Summation: an example of what computation is HTML: an example of what computation is not Computing distance, part one: Information diffusion Computing distance, part two: Example Computing distance, part three: Algorithm Examples of information diffusion on the Web: Social bookmarking in plain English by Common Craft RSS: RSS in plain English by Common Craft What is RSS? by Software Garden XML: Introduction to XML by Jan Egil Refsnes How can XML be used? by Jan Egil Refsnes See also: The Machine is Us/ing Us Skim Six Degrees pp 135-139: "Is six a big or a small number?"

Collaborative Filtering

How to compute personalized recommendations: Collaborative Filtering "Expert opinions" without the experts Delicious: example of CF Bookmarks: content of Delicious Tuples: content of CF Bipartite graphs: structure of CF Structural equivalence: computation of CF Delicious: algorithmic summary The four steps of collaborative filtering

The Long Tail

Niches and blockbusters in the world of Web commerce: The Long Tail Macro-analytic view of collaborative filtering Power law revisited Niches, megahits, and the neglected middle Macro-analytic view of the long tail Macro view of Web programming See also: The Long Tail, by Chris Anderson. Wired, October 2004. Going Long, by John Cassidy. The New Yorker, July 2006. Six Degrees Chapter 7, pp 207-215: Information Externalities & Market Externalities

Influence in Networks

How to compute the influence of a Web page: Influence in Networks Popularity, influence, and centrality Introduction to PageRank NetRank: a simplified version of PageRank Normalization and convergence The NetRank algorithm Dividing by outdegree: the NR* formula The PageRank formula The damping factor: PageRank as probability See also PageRank Explained by Phil Craven

Competition and Cooperation

What happens when Web builders seek to increase their influence?

Games: Competition and Cooperation

• Dynamics of popularity and influence
• PageRank competition
• Doing the right thing
• Mutually assured construction
• Authority, reciprocity, reputation
• Game theory
• Winners' dilemma

See also

• Six Degrees Chapter 7 pp 202-204: Diners' Dilemma and Tragedy of the Commons
• Prisoners' Dilemma by Serendip; also "What's so important about this game"
• Six Degrees Chapter 7 pp 215-219: Coordination Externalities
• Tragedy of the Anti-Commons (aka "The Permission Problem") by James Surowiecki, The New Yorker, August 2008

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Sunbelt Goes to the Oscars

I am dried off from the record-setting rains of Sunbelt XXV and ready to start reflecting... about this weekend's big Oscar hoo-ha (sort of).

Some of the things I like about social network analysis include (1) using something fundamentally simple like a set of nodes and edges to illuminate very complex behavior, and (2) producing powerful images that convey social dynamics very intuitively.

James Moody demonstrated both these points spectacularly in his Freeman Award Presentation. Moody was recognized at Sunbelt XXV as the most outstanding young researcher in SNA.

Moody described a simple principle involving three individuals, A, B, and C: If A likes B, and B likes C, then A will probably also like C. Otherwise (if A does not like C) things can get tense between friends A and B, producing social instability.

Using little more than this principle of transitivity (also known as clustering), Moody has built a social network model that behaves with an amazingly lifelike quality. If you've got broadband, then I recommend you get some popcorn and take a look at this movie (18MB). The nodes and edges do an amazing job of conveying the dynamics of a cocktail party. And even more amazing is the simplicity of the model behind this behavior. Bravo!

To see the other movies nominated for "Best Network Visualization," take a look at "Dynamic Network Visualization" by Moody, Daniel McFarland, and Skye Bender-deMoll.

Influence and social capital of 21st century leaders

My previous post summarized "four fundamentals of networks" with special emphasis on the context of leadership. Today I'll take a closer look at the foundation of the four fundamentals: personal influence. This foundation is highlighted in the bottom two quadrants below, which share a network focus on influential positions and roles:
These two quadrants provide a good foundation for at least a couple reasons:

First, most of us naturally equate leadership with positions of personal influence. In their excellent article "Social Capital of Twenty-First Century Leaders," Dan Brass and David Krackhardt begin by saying, "Accomplishing work through others has always been the essence of leadership"; later in the chapter they simplify this to "Influence is the essence of leadership." As I summarized in this post, Brass and Krackhardt then describe how aspiring leaders can use social networks to gain as much influence as quickly as possible. (Their article really is outstanding, FYI.)

Second, centrality and structural holes--the network concepts underlying the highlighted two quadrants--are the two most intuitive notions of network structure. If you find "structural holes" less intuitive than "centrality," then just substitute "clustering" in place of "structural holes." Clustering refers to groups, structural holes to the gaps between groups: Just like foreground and background, they define each other in complementary partnership.

The topic of personal influence in social networks gets lots of attention. For example, this announcement crossed my desk last week: "'Influence is the future of media'. Influence is the hottest topic in marketing, advertising, media and social media today. Find out how to tap the power of influence." It's not too late to sign up for http://www.futureofinfluencesummit.com/.

Another view of influence and social networks crossed my desk a month ago: Duncan Watts, Columbia sociologist and principal research scientist for Yahoo, told Fast Company magazine his opinion of the idea that a subgroup of "influentials" is largely responsible for trend-setting: "It sort of sounds cool, but it's wonderfully persuasive only for as long as you don't think about it." Later in the article, Watts concludes: "If society is ready to embrace a trend, almost anyone can start one--and if it isn't, then almost no one can."

Are these views of influence hopelessly at odds? Perhaps not. As I explore that, I'll move to the top half of the four fundamentals of networks.

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NSF and Google-induced stupidity

The NSF has just published Fostering Learning in the Networked World: The Cyberlearning Opportunity and Challenge. Reading it reminds me of why I bailed out of academia. The introduction starts: "To address the global problems of war and peace, economics, poverty, health, and the environment, we need a world citizenry with ready access to knowledge about science, technology, engineering, and mathematics."

Wow. Another thing the world citizenry needs is a ban on vapid topic sentences whose only purpose is to inflate the perceived importance of the author's pet project.

In the NSF-funded land of cyberlearning, there is a five-tiered hierarchy of human interaction, represented by the cool picture below: The report explains the picture thus: "[The figure above] depicts historical advances in the communication and information resources available for human interaction. Basic face-to-face interaction at the bottom level requires no resources to mediate communication. The second wave of resources offered symbol systems such as written language, graphics, and mathematics but introduced a mediating layer between people. The communication revolution of radio, telephony, television, and satellites was the third wave. The outcomes of the fourth wave—networked personal computers, web publishing, and global search—set the stage for the fifth wave of cyberinfrastructure and participatory technologies that are reviewed in our report."

So, we are going to solve the "global problems of war and peace" with a framework that explicitly omits mediation from the realm of face-to-face communication. I wonder how much cyberinfrastructure South Ossetia would need to put this framework to use.

Next time I will get back on my network clustering thread again...

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Economic externalities and network science

Thanks to Six Degrees: The Science of a Connected Age, which I use as the textbook for my course on web science, I have learned about economic externalities. I don't always include the topic of economic externalities in the class, but last fall I did--with intriguing results. It was all part of the major course redesign that preoccupied me more or less 24/7 from Columbus Day until Kwanza.

Here are the four types of economic externalities according to Duncan Watts, with explanations and examples by yours truly:

1. Information externalities: Knowing how others have acted under similar circumstances saves me the effort of evaluating all the options "objectively." Example: I am hungry. McDonalds has sold 30 billion Big Macs. They must be OK.
2. Coercive externalities: Anticipating the impact of my decision on others influences my choice. Example: Everyone is drinking at this party. What will they think of me if I don't drink?
3. Market externalities: As a particular option is chosen by more and more people, that option becomes more and more valuable to all those who have chosen it. Example: In 1980 very few people had email and so email was of very limited use. In 2007 many people have email and that popularity makes email exponentially more useful.
4. Coordination externalities: I will sacrifice my short-term selfish interests for long-term gains that depend on favors from others, to the extent that (1) I care about the future, and (2) I believe my actions affect the decisions of others. Example: When my friend lends me $10, I will pay him back the next time I see him. I lose $10 when I pay him back but gain more than that in the long run.

Watts discusses the above concepts almost exclusively in terms of their relationship to "tipping point" network dynamics. He also emphasizes the top of the list at the expense of the bottom. Having swallowed Watts' taxonomy of economic externalities into the same pipeline that is slowly digesting Tim Berners-Lee's framework for web science, I now have (1) a chronic case of indigestion, and (2) the following mapping of economic externalities onto basic concepts of network science:

1. Information externalities --> cardinality and centrality
2. Coercive externalities --> connectivity and clustering
   
3. Market externalities --> structural equivalence
   
4. Coordination externalities --> symmetry and asymmetry
   

As I prepare for spring '08, the above appears to be the backbone of my web science curriculum. If it doesn't make sense, dear reader, please be patient with me and stay tuned!

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Holy Trinity of Network Power

Last Thursday the US Supreme Court ruled that prisoners at Guantanamo Bay have a right to hear and to challenge the reasons for their detention.

Eric M. Freedman, a habeas corpus expert at Hofstra University Law School, called the decision "a structural reaffirmation of what the rule of law means," and said it was as important a ruling on the separation of powers as the Supreme Court has ever issued, according to the NY Times.

Dating back at least to ancient Greeks, the separation of powers traditionally splits state power into three parts: executive, legislative, and judicial.

Over the next few posts, Connectedness will celebrate the separation of powers by comparing each of its three components to three notable pillars of the network perspective: centrality, clustering, and structural equivalence.

Stay tuned for something like this:

Politics	Networks	Easy-to-Remember Stereotype
Executive	Centrality	Tyrannical Dictator
Legislative	Clustering	Mob of Special Interests
Judicial	Structural Equivalence	Politically Unaccountable Intelligentsia

Hopefully by July 4th, we'll have celebrated all three.

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KM 0.0 by Dave Pollard

Recently I was invited by HP's knowledge management (KM) connector Stan Garfield to join a conference call that featured Dave Pollard. It was the first I heard the expression "KM 0.0", which was perhaps coined by Dave here. Dave describes KM 1.0 as "content and collection," and KM 0.0 as "context and connection." This not only makes for a poetic KM checklist, but it also reminds us that the better we get at KM, the more our KM draws from pre-historic roots of humanity.

My attempt in the conference call to agree with Dave did not get very far. Too many ideas in my head and not enough sense out of my mouth, I think. Nevertheless, those who want to support Dave's "KM 0.0" notion will do well to notice how 1920's anthropological study of archaic societies anticipates this 2006 MIT Sloan Management Review cover on "Enterprise 2.0."

Dave's poem also deserves more consideration:

Content, collection;
Context, connection.

I interpret this poem as a tribute to Amazon.com and other exemplars of the Long Tail phenomenon--digital hosts who provide not only content but also ways for users to interact through their experience of that content. It's an amazingly successful network recipe cooked with equal measures of centrality, clustering, and structural equivalence.

Too many ideas in my head now, so I must sign off.

This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License and is copyrighted (c) 2007 by Connective Associates LLC except where otherwise noted.