Saturday, September 15, 2007

Facebook statistics and close friends

I just returned from GITEX, the largest IT and consumer electronics trade show in the Middle East, held in Dubai, United Arab Emirates. Perusing Wednesday's Gulf Today, I came across an article regarding Facebook and networks of friendships in the real world. According to Will Reader, an evolutionary psychologist from Great Britain, the number of close friends that one has is essentially invariant, whether one has thousands of Facebook friends or not. Reader indicates that many studies have shown that people have about 150 people in their networks of friends. A small fraction of these are "close" friends, regardless of how big their networks grew.

The reason that this is significant is that it shows several things consonant with my analysis of the statistics of many networks. Specifically, it shows that the connectivity value -- in this case the value of a social connection -- is not distributed equally across all nodes (i.e., friends). This yet again argues against an n squared value for networks, where n is the size of the network, which would require that each connection is of equal value, or at least that the distribution of values has a mean that is a linear function of n, neither of which appears to be the case.

The site now reportedly has over 30 million users, so it also suggests that the number of actual friends and close friends does not change even though the number of potential Facebook friends connections is now substantial.

Monday, September 3, 2007

Increasing Network Value III: Front-Loading

If connectivity value is defined by the aggregation across all connections of the expected net present value of transaction streams, that leads us to a third way of increasing that value. As discussed earlier, one way is to increase the value of each transaction, and a second way is to increase the rate of transactions.

A third way is to accelerate transactions into the present. Even with the same nominal value for each transaction, and even with the same average rate of transactions, front-loading increases the net present value by decreasing the time-based discount for each transaction.

When TiVo offers "lifetime subscriptions," they are not only solving a cash flow issue, but translating a stream of future cash payments from monthly subscriber fees into a single front-loaded payment. Just like with a lottery, where one can accept winnings as a lump-sum or monthly payments stretching out over 30 years, or a life annuity, there is a breakeven point which depends on the projected expectation of payments and interest rates. Depending on these assumptions, a front-end non-recurring charge can be equal to, more than, or less than a particular finite or infinite stream of payments.

Sunday, September 2, 2007

Increasing Network Value II: Transaction Rate

Even if the value of each transaction doesn't change, another way to increase total value of some or all connections, and therefore total network value, is to increase the transaction rate. Rather than buying a more expensive item, there may be an increase in value (in the sense of net present value of the revenue stream) if a customer buys a less expensive item more frequently.

And, certainly if they buy the same item more frequently, that will also enhance the total value of the transaction stream.

Planned obsolescence, whether a result of engineering (or the lack thereof), or fashion trends, is a way to incent us to increase our consumption rate. But also, eliminating physical and societal barriers to consumption is another way to increase the transaction rate. This may mean having a coffee shop or hamburger joint on every street corner, to reduce the time and energy barrier or walking an extra 100 feet. Of course, social forces are a tricky thing: whether it is 2 button vs. 3 button suits, or SUVs vs. hybrids, trends may shift causing changes in the transaction rate.

Increasing Network Value I: Transaction Value and Pricing

Connectivity value in a network is at least one metric for valuing networks. Others, such as Reed, have proposed other metrics such as group-forming value, but let's stick with connectivity value for a moment. Even in Web 2.0 and Enterprise 2.0, groups exist due to relationships which are based on transactions across connections.

If, as I've proposed, there are conditions where connectivity value is linear in the size of the network -- be it a communications network, a producer-consumer network, or anything else -- does that mean that networks have limited value?

Of course not.

If we define the connectivity value of a link as the expected value (i.e., likelihood-adjusted) of the net present value (i.e., adjusted for time value of money) of the transaction stream of that link, then one easy way to increase the value of the connection is to increase the size (i.e., value) of the transactions.

For example, if the connectivity value between me and my car dealer is defined by buying a car every four years, that connectivity value will increase if I buy a Lamborghini every four years instead of a used Yugo. (For those that don't know, the Yugo was of note when it went on sale in the '80s as the cheapest car sold in the U. S. Presumably used ones are still for sale).

Nothing has changed in the order of the value of the network: it is still order (n), in other words, proportional to the number of nodes, which in this case, are many car buyers and a relative few car dealers). However, if everyone started buying Lamborghinis instead of Yugos, the connectivity value of the "global automotive sales network" would increase by several orders of magnitude.

Of course, merely raising prices or selling more expensive products doesn't do the trick. Wal-Mart's revenues are higher than Henri Bendel's. As first steps, understanding price elasticity of demand (what would happen if we charged 10% more for this product) and using dynamic pricing for yield management (this is why airline seat prices appear to fluctuate randomly) can maximize total value of the system.

Also, price targeting, discussed in extremely readable fashion in "The Undercover Economist," by Tim Harford, subtly extracts more money from price insensitive or otherwise ignorant customers. He addresses three main mechanisms: individual targeting, group targeting, and "self-incrimination." It is this last technique that enables gourmet coffee shops to sell a cheap regular coffee right next to a $5.00 super half-caf iced mocha caramel choco-frappuccino. Lest you think that this is because of special hand-picked beans which cost isn't. Tim assures us that the production and operations cost differential between cheap and expensive cups may be disregarded.

In summary, one way to increase the value of a network? Raise prices. Or lower them. Or change them dynamically. Whatever it takes to maximize the expected net present value of the connection. And, as Tim points out, in a free market economy such pricing represents the "truth" about what maximizes value to all parties in the transaction: consumers as well as producers.

Mark Cuban and the Emotional Value of Networks

At Blog Maverick, in a post titled "Metcalfe's Law and Video," Mark Cuban discusses a different perspective on network value, specifically with a view towards the intensity over time of connectivity. He comments that "the more people that see content when it is originally "broadcast," regardless of the distribution medium, the more valuable the content." Although that can be demonstrated by simple net present value calculations, he is talking about emergent effects, such as emotional attachment and the social value from real or virtual simultaneous participation.

He also hypothesizes that not only is there greater value from simultaneous delivery, but also that there is greater cost. His argument is that networks that are designed for large scale simultaneous delivery of content cost more than those that are less ambitious.

To me, this is arguable. For example, there are inherent economies in using a broadcast, content distribution network, or IP multicast to distribute content simultaneously, than to keep redelivering it on demand and sequentially. If the capital expenditure for a scalable and feature-rich network has been made, broadcast and multicast technologies and architectures actually reduce cost per bit delivered per person.

If you combine his viewpoint on the value add of "live" and simultaneous events, with my observation that such events can actually cost less, that means that there is a sweet spot, if the network is engineered properly, in delivering live simultaneous content versus delayed and on-demand content.

This conclusion is actually not surprising, since traditional broadcast TV and movie theaters were only economically viable (in their day) due to the cost reductions inherent in broadcasting program content to a large simultaneous audience rather than unicasting it asynchronously. Of course, today's technology has now reduced the marginal cost of unicasting to be an infinitesimal fraction of a customers willingness to pay for such content.

Or so it would seem. In reality though, for the foreseeable future there will be content that is too bandwidth-hungry for widespread acceptance. Maybe YouTube videos don't have that property right now, but what about HDTV to your laptop screen? How many people are willing to pay for mobile bandwidth sufficient to deliver it in real time, say for 1080p video conferencing? If not that, how about digital cinema quality images?

For the next 5 to 10 years, there will always be that dilemma. After that, perhaps not, because we will have the ability to deliver enough bandwidth to each user, whether fixed or mobile, to equal or exceed the limits of human perception. At that point, until we evolve or bio-engineer our visual cortex and other sensory modalities to become Human 2.0, any additional bandwidth will be overkill, at least for the purposes of entertainment.

Buko Obele and the Tragedy of Web 2.0

Buko Obele, in a blog post at discipline and punish called "The Tragedy of Web 2.0," observes that the lack of mergers between social network providers is yet more evidence of the lack of applicability of Metcalfe's Law in this environment. He points out that the objectives of social network service providers may not be exactly aligned with the objectives of the users, and that this misalignment prevents consolidation and, in some cases, feature enhancement.

This corresponds to Odlyzko and Tilly's analysis "A refutation of Metcalfe's Law and a better estimate for the value of networks and network interconnections." Although, as I've observed, there are many cases when network connectivity value may only be linear, even if it is n log (n), as discussed by Odlyzko and Tilly, there still may be relatively weak incentives for consolidation.

Nick Carr and Customer Lifetime Value

Nicholas Carr, author of "Does IT Matter," refers to an interesting analysis regarding Customer Lifetime Value (CLV) and the Network Effect, on his blog, Rough Type. Specifically, he addresses a paper by three professors, one at Harvard Business School, regarding "free" customers. These are customers, such as buyers of real-estate or at on-line auctions, who don't pay a service provider, such as a realtor or auction house, directly. The professors are able to, for a specific unnamed real-world auction house, identify the economic value of each seller, (roughly $500), and an economic value for each buyer (roughly $550). The main -- and somewhat counterintuitive -- point of the paper is that "customers" who don't spend money can be worth more than those who do. However, a hidden impact of the analysis, which took into account numerous factors including word-of-mouth recommendation value and discount rates and Lagrangian multipliers and Jacobi-Bellman conditions and many other elements that most of us have never heard of, is that the value of each customer is essentially a constant. The implication is that, in a network of buyers and sellers, the total value generated by the network, and therefore revenue to the service provider intermediary, is linearly proportional to the size of the network. Again, this provides support for the hypothesis that network value can be linearly proportional to the size of the network.

Linked by Albert-Laszlo Barabasi

Dr. Barabasi's book "Linked" is very readable overview of the math underlying a variety of network architectures, with many real-world examples in a variety of contexts. Briefly, three main types of networks are addressed. One, a random network, where links between nodes are created at random. Secondly, a "small-world" network, where most nodes are linked to nearby "neighbors," and a few links span across clusters. As studies by Watts and Strogatz, these types of networks lead to six degrees of separation types of architectures, where any node can reach any other node in a small number of hops.

However, he points out that neither type of network represents the type of structure one might find in, say, a telecommunications network or the World-Wide Web. Therefore a third type of network, a "scale-free" network, comprising a few larger hubs and many smaller hubs and endpoints is introduced. His research indicates that the node degree distribution matches the power-law distribution of many real-world structures, including neural networks and the World-Wide Web.

These real-world networks arise when two phenomena are present: one, growth, and two, preferential attachment for these growing networks. Also, implicit in the model is that each new node links to a fixed number k of existing nodes. Based on these assumptions, where a new node will tend to prefer to connect to existing nodes with more connections, a scale-free architecture emerges.

Interestingly, if we define the value of a link as a constant when it exists and as zero when it doesn't exist, the overall connectivity value of any of these networks is provably linear, based on the assumptions.

Is Metcalfe's Law Way Too Optimistic?

I recently wrote an article addressing Metcalfe's Law and related analyses from Reed and Briscoe, Odlyzko, and Tilly of network value. The summary of my analysis is that a number of factors can cause real world networks to have value substantially less than n squared. One factor is convergent value distributions, where each connection does not have equal value. Instead, if the distribution of connection values from each node converges to a limit, that drives the total network value to be only of order (n), in other words, linearly proportional to the size of the network.

Another factor is limits of consumption that are intrinsic to the type of network. If each user can hit an upper bound in money or time spent extracting value from the network, then the value of the network is also just linear. The actual article was published in Business Communications Review, but is available here as a pdf.

The analysis also applies indirectly to Reed's 2^n valuation of Web 2.0 networks based on their group-forming capabilities. Briefly, while it is true that there are 2^n (2 to the nth power) subgroups of a network, it is unlikely that they are all equally valuable. This makes the total value substantially less than 2^n.

The Network Effect

Welcome to this new blog. Its purpose is to explore and discuss networks: whether abstract, such as random graphs; technical, such as business continuity architectures; application, such as remote desktop; or social, such as Web 2.0, Facebook, Blogger, and MySpace.