However, I have some criticisms of this model:

(1) He proposes this be called “Beckstrom’s Law” analogously to Metcalfe’s Law Law and Reed’s Law Laws (discussed here previously here and here).

What is a “law” good for?  In the case of other soft laws–the most famous being Moore’s Law–the value of the law comes from a scaling relation that gives some rough predictive capabilities independent of scale.  Moore’s law is a soft law because there isn’t anything about the physics of the problem that allows deriving Moore’s scaling law.  But the social/business network has behaved this way for decades.  Moore’s law is widely believed to break down as we reach the power and frequency limits of conventional integrated circuits manufacturing methods.   The physics of current circuit manufacturing techniques says that either Moore’s law will stop being an accurate predictor of the scaling of computation power, or we will make a significant shift in the physics we are using.

Thus a law provides predictive and analytical power:  How does this change as that changes?  Even a law being violated (i.e. a model breaking down) gives us signals indicating when important shifts in the relevance of a model to a social or physical system are occurring.

“Beckstrom’s Law” isn’t a law in this sense.  It derives no scaling relations.  Further, you get an idea of how difficult it is to make estimates of value in this accounting system when Beckstrom starts his discussion of phone network value by discussing the “upper limit” as GDP.

(2) Beckstrom’s Rules of Accounting. Useful laws provide an “abstraction jump.”  With powerful models, can accumulate the particulars of a system and use this leverage to understand and manipulate at a different scale, i.e., we can derive laws. For example, we sum up the location and amount of all the little masses in the earth and calculate the acceleration of falling objects at the surface of the earth–the theory allows us to think about acceleration of falling objects at the surface of the earth without worrying about the location and mass of all the little grains of stuff that make up the earth.  Keeping track of all the pieces of the earth is “accounting” (vital to the theory) but the power of the law comes from the leverage of the abstractions derived.  Beckstrom’s law doesn’t seem to have this quality. It doesn’t derive high-leverage abstractions from underlying structure.  It is an accounting method.

(3) Beckstrom’s valuations are essentially relative.  In particular, benefits and costs are calculated compared to the prevailing contextual way of performing the transactions.  They depend on the path through history rather than depending on form. In his example of Amazon.com, for example, the benefits of for a user of Amazon.com are realized in savings over driving to the local bookstore. He points out that Amazon.com also gets a “network benefit” from the transaction because they have wholesale and operational costs less than the purchase price of the book.  These are “benefits” of Amazon.com over the local bookseller based on cost savings for a commodity assumed to be available through both channels.

Before you discount this problem with a “what else is there?” consider that the system of agents and interactions may reach a novel state through losses–a new state not used in the relative benefits calculation.  The network may enable new states of the system that were not contemplated before the existence of the network.  This has important implications for the network security arguments in Beckstrom’s paper.  Losses may exceed benefits.  More importantly, losses may come from reaching system states that were not visited on the way to realizing benefits.

Backing this thinking up a bit, we realize that relative gains pose a problem too: the system of agents and interactions may enable transactions that were not possible in the prevailing contextual way of performing transactions.  Then the benefits can no more be calculated than the losses using an accounting method.

(4) Transactions don’t seem to neatly belong to a single network. In the Amazon.com example, presumably, Beckstrom was calculating the value of the Internet, or maybe just the World-Wide Web. Or maybe the value of the network of the network of roads allowing me to drive to the store vs. the value of the Internet?  Or maybe the network of all booksellers (since they all buy from the same publishers)?  Which network does the transaction belong to?  Or do we allocate a fraction of the transaction to each of the overlapping networks that can be argued to enable the benefits of the transaction?

(5) Very carefully (!) add historical benefits to predicted losses. (I think Taleb would argue never!) These are apples and oranges. See multiple explanations of the current state of the world’s banking system.  Beckstrom argues that “optimal security investment occurs on the loss function line where it is tangent to the 45 degree line, or where one dollar of security investment equals one dollar of decrease in expected losses.”  This is true in two places on Beckstrom’s curve–once for very low security investment and again for moderate security investment.  In fact, my losses are bounded by “everything I value” for $0 security spending (I can lose everything, and no more). So the existence of two inflexion points seems accurate and how we choose between them based on Beckstrom’s arguments isn’t clear.

How do we know the curve doesn’t look like this all the way down?

"Pathelogical" risk-cost curve. Risk Investment on the x-axis; Cost of losses on the y-axis.

More troubling is that security investment is historical accounting while the loss curve represent one of the infinite possible world lines.  What if security spending creates value rather than merely stemming losses–then we have more to lose than when we started security spending.  Maybe the curve looks like this?

Risk-cost region. Risk Investment on the x-axis; Cost of losses on the y-axis.

What will a law describing network value look like?

• Scaling laws from underlying size, structure and dynamics
• It is highly likely that the values of constants essential to practice will be missing from the law.  Actual risk and benefit calculations will depend on the particulars of the system we are analyzing.
• Provide some level of causal explanation for the generative, emergent qualities we observe from real networks every day.  It seems clear that networks enable new transactions. So using relative transaction value seems to miss the most exciting source of value from networks. A useful network value law will say something about the way that the future is not always a simple extrapolation of the past.
• (There may be no such law–then maybe Beckstrom’s “Rules of Accounting” are the best we can do.)

Both my Facebook and LinkedIn social networks have gone through periods of rapid grown in the past year. Although I have been a member of LinkedIn for many years, my network expanded rapidly last year, approximately doubling in a couple of months.  I have had a similar experience in the last few months with my Facebook network.

This is likely a convergence of factors.  My contemporaries are reconnecting with high-school and college acquaintances on Facebook.  More of the people I have worked with in the past have joined LinkedIn and started added colleagues from past employers.  This has led to a critical mass of people both connecting and inviting others to connect.

In each case, it seems that when enough people in my circle of acquaintances are connected, new connections form rapidly because it isn’t far (in terms of network hops) to other people I know.

So am I already "socially networked" with everyone I know? I think the answer is yes.  Because I am not very active at inviting new friends, it might be more accurate to say that I could be completely socially networked with an evening’s work.  In terms of network dynamics, this is because the network of my local group of friends has undergone the phase transition network mathematicians refer to as "percolation."

A simple model of network percolation

Where are without a model? Not very far on this blog. Instead of modeling a complete social network, let’s start with a simplified model and see if the behavior I want to describe pops out.

Here is a model that is easy to visualize. Imagine I am about to tile my kitchen floor. I draw a grid on the floor showing where each tile will be placed.  Looking at the kitchen floor as I prepare to lay the tile, you see a simple grid of lines, like graph paper. Now, start adding tiles. But instead of working from one side of the kitchen to the other in rows (like a normal person), I add tiles randomly.

I lay one tile by the sink, then two by the fridge, then another by the stairwell, a couple in the center.  The floor is getting cover red with tiles, but in a hodgepodge sprinkling.  Not too many of them are sitting right next to each other at first.  As I put down more and more tiles, however, there start to be little clusters of adjacent tiles. After a little while, the floor might look like this:

But then, along comes my nephew Hayden, wanting to go to the sink for a drink of water. I don’t want him to walk where there are no tiles, because he will track glue everywhere. But he is small, and can’t take steps bigger than one tile. So I need to hurry to put down enough tiles so he can walk to the sink.  But I don’t want to ruin my beautiful experiment with the random tiles.  How many tiles do I have to put down (still placing them at random locations) before Hayden can walk on adjacent tiles from the living room to the sink to get a drink?

When he can do this, the edges of the network of tiles are connected.  The network of tiles is said to have a percolating group of tiles.

A slightly more complex simple network percolation model

A mathematical description of the tile model goes like this.  Each tile (graph nodes or vertices) can have at most 4 edges connecting it to other tiles (graph edges or links).  The connections are created by placing adjacent tiles.

Let’s add a small generalization to the model to make it slightly more realistic. The tiles are connected by the fact that they are placed next to each other.  But in a social network, you can be "near" someone, but choose not to connect.  So let’s add the condition that each connection between adjacent people can be a friendship or not. And let’s allow this independently for each near neighbor. In the social network analogy, this means everyone person (node) can have at most 4 friendships (relationships, links to other nodes) but they don’t have to have be connected to someone they are "near."  Now people and friendships might look like this:

The analogous percolation question for a social network: how many friendships have to be created before you can get anywhere in the network by traversing friendships?  In other words, at what point is it possible to find everyone you know on a social network?

Further, regarding a period of rapid network growth: Is there some point in the evolution of the network when connections are created more rapidly?

To model this simple network, I created a simulation. You can download the Python script and run it yourself.  I created a grid 6 x 6, giving 36 nodes.  Central nodes can have at most 4 friends.  Edge nodes can have 3 and corner nodes only 2. For the entire network, the number of possible edges is 2 x 6 x 5 = 60.

One reason to use a computer model, is that since I am placing the friendships randomly, each time we run the model on a 6 x 6 network, the details will come out slightly different. So we want to understand what happens on average.  Using a computer model lets us run many (50 in this case) copies of the network experiment (without tiling a kitchen floor, and ripping it up over and over).

Instead of asking just about the path from the living room to the sink,  or one side of our generalized network to another, let’s look at the overall connectivity of the nodes. How close are we to being able to traverse the graph from any point to any other point? For this, we measure the size of the largest group of connected nodes (people) as we add more edges (friendships).

The size of the largest group of connected people looks like this as we add friendships between the people:

From the plot above, you can see that the size of the largest group gets large very fast after about 30 friendships form.  The network is fully connected after 40 friendships form–even though there are 60 friendships possible.

Okay, let’s close the loop on the analogy. If I had 36 high school friends on Facebook 3 months ago, they would have been difficult to find through each other.  But over time, a few of them talked on the phone, met on the street etc. and invited each other to friend them on Facebook.  This process was slow at first, but after about 30 friendships were formed, the process took off. One I was connected to a couple of sub-groups, I could find anyone I knew of my 36 high-school friends. Once the network percolates, I am socially networked with everyone I know.

To get some idea how the tall head evolves, compare the tall head on day 0 with 19 successive days. The plot below shows the Top 10, Top 50, Top 100, Top 500 and Top 1000 URLs for day zero and the fraction of the top URLs on day zero appearing in the tall head on the nth day.

While the Top 10 and Top 50 URLs show stability day after day, the Top 500 and Top 1000 roll over at a fairly constant rate after day one.  The plot can be used to estimate the size of the persistent tall head of URLs for this population and the rate at which the tall head evolves.

First, look for a change in behavior from maintaining a constant fraction of the day-0 URLs to a steady decline from day to day. By this heuristic, estimate the persistent tall head to be between 50 and 100 URLs.

Secondly, to estimate the turnover of the tall head, choose the approximate desired tall head, e.g., the Top 500 URLs (cyan), and look at the slope of the line for days 1-19. (Alternately, choose a timescale for which the tall head should turn over to a given fraction remaining, say, 75%, giving a timescale of approximately 15 days.)

The plot above shows the Top 500 URLs rollover about 0.5% per day from days 1 to 20.

The tall head is the short list of sites that get the most visits.  We could call it the Top 10, Top 100 or whatever we think is relevant.  Later I propose a couple of heuristics for determining tall head membership in real time.

The tall head is made up of URLs that much of the population visits.  These are the "winner take all" URLs of user attention–URLs like cnn.com, google.com, or facebook.com.  For this reason, we might expect that while the long tail is growing with more unique URLs and the number of URLs with 2-3 hits is growing rapidly, the tall head is relatively stable.

This is the case.

One way to think about how stable the tall head might be is to ask how well a subset of the population predicts membership in the tall head for the entire sample.  The data from the last two posts is well-suited to look at this question.

Below is a plot of the accuracy of various subsets of the population (the same subsets we used previously) in predicting the Top 10, Top 20, Top 50 and Top 100 URLs of the entire population.  Just over 40% percent of the population predicts the Top 100 URLs with 90% accuracy.  The Top 10 are predicted to 90% accuracy by 10% of the population.

The composition of the tall head depends relatively weakly on the subset of the population doing the predicting.

How can I predict the tall head for the day by 9 am in the morning? This is the real-time problem of long tail distributions.  The dynamics of the system are that real time Web exploration data appears as a time-ordered list of URLs from whatever users happen to be surfing.  This means that a real time heuristic for determining top URLs for the day has to rely on the properties of the time series including a small surfer sample size and recent counts of visits.

Fortunately, by the results illustrated above, a small sample size is a pretty good bet for determining tall head URLs. What we are still missing is metrics or intuition for how the long tail distribution evolves over time.

We do know that for a URL to end up in the tall head, it must be visited by many Web explorers.  This means that we can rule out all URLs that are visited by only one or two users.  This assumption also leads to a heuristic based on the time between visits–URLs visited by many people should have the same visit/time distribution as the users/time distribution of the entire sample.  More specifically, we might guess that if the time between visits has an average near 1 day/number of visits and relatively low variance, it is likely to be in the tall head.  A project for a little later…

Next post: How does the composition of the tall head change from day to day?

• How much overlap is there between the URLs 10 people visit and those in the 11th person’s click stream?
• How about the 100th or 100,000th person? Does the millionth user explore any unique URLs at all?
• Can we build a model to answer How many people are required to crawl 10% of the Web?

The second part of the answer is to look at how the model of URLs and visits evolves as we add users.  To get samples with of different sizes using the same click stream data set, randomly select a subset of the users and run the analysis from the previous post.  Through everyone back into the pot and randomly select a slightly larger set.  Repeat.

I reran the model for 3%, 4%, 5%, 7%, 9%, 11%, 15%, 20%, 30%, 40%, 50%, 60%, 70%, 80% and 90% of the overall users in the 1-day data set. The sample sized ranged from 3,000 to 91,200 users. For the entire data set, the average user made 184 URL visits during the day. In the randomly chosen subsets, users made an average of between 181 and 187 URL visits with most of the variation in the smaller sample size as expected.

Do I expect the number of unique URLs be linearly proportional to the number of users? Or if users are visiting many of the same URLs and URLs tend to have the "winner take all" properties we looked at before, we might expect the number of unique URLs added by the 90,000th user to be fewer than the number of unique URLs added by the 1,000th user.

I first plotted the number of unique URLs against the number of users in each sample.  The curve looks straight but may be slightly concave downward.  It is very subtly.  I needed to look at the data in a way the amplified the change over the various subsamples.

Below is a plot of the number of unique URLs/user vs. the number of users. This line is flat if the number of URLs is growing linearly with the number of users.

The blue curve is the best fit to another power function ( f(x)=ax^k ). The first few thousand users are contribute more original URLs (>90 URLs per user) to the sample than the 100,000th (83 URLs).  If you are the first explorer of the a new world, all of your discoveries are original; when you are a late comer, your contributions are around the margins. It may be surprising how much original content being explored by the 100,000th explorer.

Does the long tail get relatively longer or shorter?  For simplicity, I use the URLs with only one visit to represent the long tail.  Then ratio of 1-visit URLs to unique URLs decreases subtly. For the smallest samples size 70.0% of the unique URLs are hit only once; for the overall data set, the ratio is 69.2%. To amplify this change like above, the plot of 1-visit URLs per user is shown below.

At 100,000 users, the long tail is growing at 57 URLs per additional user.  The decrease with each additional user is slowing.  The blue curve is the best fit to another power function.

If the power function is the best explanation of the underlying dynamics, the number of unique URLs and the long tail both continue to grow no matter how many people are exploring.  Since an increasing number of people need to explore to keep the exploration rate constant, the cost of exploration per URL goes up as explorers are added.

Does anything interesting happen in the tall head where the big winners are? That will have to wait for another post.

In order to understand the pattern or people discovering content on the Web, I looked at the click streams of real Web users.  (For privacy hounds, I have real data, but the user names and URLs are hashed so I can’t personally identify anyone or look up the actual URLs they visited.)

How much overlap is there between the URLs 10 people visit and those in the 11th person’s click stream?  How about the 100th or 100,000th person? Does the millionth user explore any unique URLs at all? Can we build a model to answer How many people are required to crawl 10% of the Web?

The first part of the answer is to look at the distribution of URLs created by a group of users.  The sample has 102,000 user’s click streams for 1 day.  I use only 1 day, because using multiple days complicates the estimates due to nearly all users having a set of URLs they visit daily. In the sample, users make 18.6M visits to just under 8.4M unique URLs.

When the URLs are ranked by number of visits, the distribution of visits over the 8.4M URLs shows that a few URLs get many hits while the tail of the distribution (way out to the right) is a long flat curve with many URLs getting only a handful of hits.

The distribution of visits can be fitted fairly accurately with a power law, p(x)=ax^k.  I don’t plot the curve, because the head is so tall compared to the tail and the distribution falls off so quickly that the plot is a very sharp "L" shape and we don’t get much from looking at it.  It is more useful to look at the cumulative distribution function (CDF) of the distribution.  This is the sum of the probabilities over the rank from highest to lowest.  Summing the probabilities to the lowest ranked URL, gives 100% of the visits recorded. Using the CDF perspective gives insight we can apply to practical situations.

Below is a plot of the CDF.  The red dots are the data points calculated from the sample while the blue line is the best fit to the CDF of the proposed power law distribution.  (The fit parameters are a=0.00219 and k=-0.690.)

One conclusion that comes out of this view of the data is that URL visits follow the so-called "80/20 Rule." This predicts 80% of the visits for the day went to roughly 20% of the URLs. Actually, for this data, the proportion is about 80/50–80% of the traffic when to the top 4.5M URLs or the top 57% of URLs.

This view shows that the tall head of big visit winners takes a significant fraction of the overall attention of the group.  These are likely URLs like www.google.com or www.cnn.com.

What does the long tail look like? The tail is just as surprising. For this data set, 5.8M URLs or 69% of the URLs visited during the day were visited only once. The number of URLs visited twice is 1.4M.

How do these numbers scale with the size of the group? That’s coming in the next post. 

Many words we use imply a ratio.  Here are the top 5 words whose meaning fizzles when the implied denominator becomes large:

1. "Most"
2. "Average"
3. "Typical"
4. "All"
5. "None/No"

It is not only large denominators that undermine these words (more accurately, the success we have using our intuition to make predictions with these words). They also fall apart when probability distributions are anything but Normal (peaked, symmetric and short-tailed).  We don’t seem to do a great job of helping people understand multi-modal or skewed distributions with our basic statistics education.  It is increasingly important in planning and executing many business ideas that we become more sophisticated with long-tailed, asymmetric and multi-modal probability distributions.

Chris is the author of The Long Tail and editor of WIRED magazine and curator of TED.  He is working on a new book about "free" products and services.  All recommended–check them out.

Tournament 1 was one in which everyone played once each round.  This seems more like a competitive environment in which the other players don’t get complete information about the how play proceeded in that round–maybe more like competing for jobs or making friends in high school. 

Tournament 2 seems to give more complete information at each round as a public auction or public disclosure pricing might provide.  I am sure the merits of using one style of tournament over another have been debated plenty.

The difference in outcome illustrates an important characteristic of complex interactions among agents: The initial conditions and the rules of play make all the difference in the world.  For those who are committed to free markets, there seems to be a parallel assumption that the ultimate achievement in free markets is one with now rules at all. Citizens of a connected and crowded planet may choose to design the rules of interacting based on our initial conditions. When I play with these simulations for awhile, I start to understand Case’s argument against depending on the simplistic dogma of free-markets to answer all questions of our common wellbeing.

Data tables … 

Match Win-Loss-Tie

Match Scores

The simulation runs 200-game matches for 40 rounds of play.  I have coded both everybody-plays-once-per-round and everybody-plays-everybody-per-round tournaments, although the results here are only from  everybody-plays-once-per-round tournaments. Tournaments start with 25 agents of each strategy type.

I programmed 7 agent strategies:

• Defect (always snitch)
• Random (fair coin toss)
• Tit-for-tat (Do what your opponent did last play–T4T is best overall according to the literature)
• T4T with occasional snitches
• T4T with occasional forgiveness
• Prey on cooperators
• Average T4T (T4T based on the average of all plays to date)

Below are the results of 40 rounds of play. After each round, the total points earned by each strategy are tallied, then the population is redistributed according to the point breakdown.  Predatory strategies tend to become extinct after 15 to 20 rounds.

Average T4T seems to be the population winner in my simulations, settling in at about 40% of the population.  I haven’t seen any documentation on this effect.  It may be an artifact of my particular mix of agents when the tournament starts.  Or I may have discovered a new, effective agent strategy.  This doesn’t seem likely as it is such an obvious strategy. Has anyone seen this effect before?  Please drop me a line.

The game win-loss-tie record between agent’s strategies was a little surprising to me.  Defection wins most of the matches. But it is clear that winning games doesn’t correspond closely to achieving the highest scores per round.  Coming in a close second against a wide range of strategies is what explains a particular population’s survival and growth.

Match Win-Loss-Tie

AvgT4T-Random:	0.492, 0.508, 0.000
T4TDefect-Random:	1.000, 0.000, 0.000
T4TDefect-T4TForgive:	1.000, 0.000, 0.000
AvgT4T-T4TForgive:	0.000, 0.000, 1.000
Defect-T4TDefect:	1.000, 0.000, 0.000
Defect-Pred1:	1.000, 0.000, 0.000
T4T-AvgT4T:	0.000, 0.000, 1.000
Pred1-T4TForgive:	1.000, 0.000, 0.000
T4TDefect-T4T:	0.974, 0.000, 0.026
Pred1-AvgT4T:	1.000, 0.000, 0.000
T4T-T4TForgive:	0.000, 0.000, 1.000
Defect-AvgT4T:	1.000, 0.000, 0.000
T4TDefect-AvgT4T:	1.000, 0.000, 0.000
Random-T4TForgive:	1.000, 0.000, 0.000
Defect-Random:	1.000, 0.000, 0.000
Pred1-Random:	0.250, 0.750, 0.000
Defect-T4T:	1.000, 0.000, 0.000
Defect-T4TForgive:	1.000, 0.000, 0.000
Pred1-T4T:	1.000, 0.000, 0.000
Random-T4T:	0.750, 0.000, 0.250
T4TDefect-Pred1:	1.000, 0.000, 0.000

If you compare the percentage of points earned in the (7!)/(2!)(5!) = 21 possible pairings, it is clear that the point leaders fairly evening split the points with all opponents, while the agent strategies the result in rapid extinction lose to some strategies by wider margins.

Match Scores (Fraction of Points)

T4T-T4TForgive:	0.500, 0.500
T4TDefect-T4TForgive:	0.510, 0.490
Random-T4TDefect:	0.492, 0.508
Defect-AvgT4T:	0.506, 0.494
T4TForgive-Random:	0.491, 0.509
Pred1-AvgT4T:	0.522, 0.478
Defect-T4T:	0.506, 0.494
T4TForgive-AvgT4T:	0.500, 0.500
T4TDefect-AvgT4T:	0.518, 0.482
Pred1-Random:	0.397, 0.603
Defect-Random:	0.858, 0.142
T4T-T4TDefect:	0.496, 0.504
AvgT4T-T4T:	0.500, 0.500
T4TDefect-Pred1:	0.512, 0.488
Pred1-T4T:	0.503, 0.497
Pred1-T4TForgive:	0.510, 0.490
Random-T4T:	0.502, 0.498
Random-AvgT4T:	0.503, 0.497
Defect-T4TDefect:	0.506, 0.494
Defect-T4TForgive:	0.537, 0.463
Defect-Pred1:	0.506, 0.494

You can download the code (zip, gzip) here to play with ideas for strategies or see how the dynamics of the system change based on the mix of agents. There is a Readme file explaining how to create new agents with your favorite strategies. Adding new agent types is fairly straight forward and requires only a few lines of new code in most cases. Python 2.5  is required to run the simulations.  The analysis programs require MatPlotLib to create the plots.  This software is available under a non-commercial Creative Commons License.  Have fun!