Hubs and the rich-get-richer

A world without celebrities

Lesson 1 ended with a question about random networks. Suppose links among people were laid down purely by chance: for each pair, roll a die, and draw a link if it comes up six. The result is predictable. If the average person has five links, about nine people in ten will have between two and eight. A person with fifty would be so unlikely that in a random world of eight billion people you would not expect to meet one. The mathematical name for the shape is the Poisson distribution, and it is much like the familiar bell curve: a clear typical value, with the exceptions close to it.

The Hungarian mathematicians Paul Erdős and Alfréd Rényi worked out the properties of such networks in papers published in 1959 and 1960. (The American mathematician Edgar Gilbert described the version with a fixed chance for each pair, the one just described, in the same year.) It was a beautiful theory, and for forty years it was the default picture of a network when nobody knew better. But in a random society there would be no celebrities, and no one would be left out. Look at the person on your left and the person on your right, and they would have about as many friends as you.

Real life is not like that. Some people are known to millions. Some web pages are linked to by millions of others, while most are linked to by a handful. So the question, around 1998, was a plain one: what does the map of a real network look like?

Mapping the Web

The Web was a good place to look. Its nodes are pages and its links are hyperlinks, and, most important, anyone can read the links on a page and follow them. A program that does so automatically is called a crawler, and it is the basic tool of every search engine. In the late 1990s a research group at the University of Notre Dame crawled their own university’s part of the Web, the nd.edu domain: 325,729 pages joined by 1,469,680 links. A page links to about four or five others on average. The random picture predicts that nearly every page would have about that many links into it, too.

It was not so. Most pages had one or two incoming links, and a handful had a great many. When the team plotted the share of pages against the number of links they had, using logarithmic axes on both scales, the points fell along a straight line. A straight line on log-log paper means a power law: the share of pages with k links falls as k raised to a negative power, written k^−γ. For incoming links the exponent γ was about 2.1, and for outgoing links about 2.45. (A larger study of crawls from the search engine AltaVista, in 1999, with more than 200 million pages, found an incoming-link exponent of about 2.1 again.)

What does it mean? A bell curve has a typical scale. Adult men are about 180 centimeters tall, and the tallest and shortest men on record, 272 and 57 centimeters, differ by a factor of less than five. No one is 20 meters tall. A power law has no typical scale: pages with 10 links, 100, 1,000 and 10,000 all occur, each category rarer than the one before, but not impossibly rare. Networks like that came to be called scale-free. Their defining feature is not the mathematical formula but the picture it stands for: very many small nodes, and a few large hubs that hold the network together.

There is an analogy in the road map and the airline map. A country’s highways are much like a random network: most cities have a handful of roads and no city has a thousand. The airline map looks different. Hundreds of small airports each have a few direct flights, and a few big hubs have connections to nearly everywhere. Delta began routing its flights through Atlanta in 1955, and the other US airlines followed after the Airline Deregulation Act of 1978; Atlanta’s Hartsfield-Jackson has been the world’s busiest airport by passengers every year since 1998, apart from the pandemic year 2020. A map of Chinese airline routes looks the same, with Beijing, Shanghai, and Guangzhou as the obvious hubs. One careful study of the world’s air traffic (Guimerà and colleagues, 2005) added a twist: the airports with the most connections are not always the most central. Anchorage and Port Moresby have few flights but sit at the crossing points between regions.

The idea that wealth, too, has this shape is more than a century old. The Italian economist Vilfredo Pareto, in his lectures of 1896 to 1897, showed that if you count how many people earn more than a given amount, the numbers fall along a straight line on log-log paper. The popular “80/20 rule,” that about 80 percent of the land or wealth belongs to about 20 percent of the people, is a later simplification, and the story of the garden peas (20 percent of the pods yield 80 percent of the peas) is an anecdote that we could not trace to Pareto’s own writing. But the shape is real: a few people with a very great deal, and many with little.

The same shape inside a cell

If hubs appeared only in networks that people built, such as the Web, the airlines, and the phone system, the explanation might be that people like to build hubs. So scientists looked at a network that no one designed. In 2000 a team compared the metabolic networks (the webs of chemical reactions that break food into the molecules a cell uses) of 43 species from all three domains of life, and found a power law, with an exponent of about 2.2, in all of them. The same few small, widely used molecules were the hubs in every species. In 2001 the same group mapped the 1,870 proteins of baker’s yeast and their 2,240 binding interactions. About 93 percent of the proteins had five or fewer links, and about one in five of those proved essential for the cell’s survival. The 0.7 percent with more than fifteen links were about three times as likely to be essential: knocking out a hub was far more often deadly than knocking out a minor node.

This is a surprising match. The metabolic networks are the product of some four billion years of evolution; the Web dates from 1991. If both have hubs, perhaps the reason has nothing to do with the details of either, and has something to do with how any network grows.

Where hubs come from

The random model leaves out two things that real networks have. The first is growth. Erdős and Rényi started with a fixed set of dots; real networks start small and add nodes one at a time. The Web was opened to the public in 1991 from the CERN laboratory near Geneva, with a single site. Every web page, every person on social media, every protein, joined a network that already existed.

The second is preferential attachment. When a new node arrives it does not choose its partners at random. A new web page links to pages it has heard of, and the pages most likely to be known are the ones that already have many links. A new researcher cites the papers that are already well cited. A new traveler goes through the big airport. So the chance that a newcomer links to an existing node is taken to be proportional to the number of links that node already has: twice as many links, twice as likely. The result is the rich get richer, or cumulative advantage.

A model that has only these two rules, growth and linear preferential attachment, was published in 1999 (the paper is in the Sources list). It has a clear mathematical solution. Every node’s number of links grows in proportion to the square root of time, so the oldest nodes are the biggest, and the share of nodes with k links falls as k⁻³, a power law with exponent 3, whatever the number of links each newcomer brings. Try it. The activity grows two networks side by side, one with random choices and one with preferential ones.

The idea itself is older than network science. The statistician Udny Yule used it in 1925 to explain why a few biological genera have very many species; Herbert Simon applied it to words and cities in 1955; and in 1965 and 1976 the historian of science Derek de Solla Price used what he called cumulative advantage to explain why a few scientific papers collect most of the citations. In 1968 the sociologist Robert Merton described the same pattern in science and named it the Matthew effect, after a line in the Gospel of Matthew that says that those who have will be given more. In Chinese it is called 马太效应 Mǎtài xiàoyìng. Price’s model of citations is, in effect, an earlier version of the same model. The 1999 paper used the idea to explain maps of the Web and other networks, with a version simple enough to be solved exactly and tested against data.

Can a latecomer win?

The model has a flaw that is also a prediction. If each node’s number of links grows in step with the square root of its age, the first arrivals always lead. In business, that is called the first-mover advantage. And it often fails. AltaVista launched in December 1995 and was one of the leading search engines of the late 1990s; Google was incorporated in September 1998. By March 2002, one panel count (Nielsen//NetRatings) had Google reaching about 25 percent of US web users and AltaVista about 5 percent. Facebook, launched in February 2004, passed MySpace, launched in August 2003, in worldwide visitors in April 2008 (116.4 million against 115.7 million in the comScore count) and in the United States in May 2009. Anyone starting out hopes that a latecomer can become a hub, and the fitness model says it can.

For this the model needs a third ingredient: a node’s fitness, the quality that makes it attractive once it has been found. Links tell you how visible a node is. Fitness tells you whether, having found it, you want to link to it: whether a web page is worth returning to, whether a person you have met is worth calling again. In the 2001 fitness model, the chance that a newcomer links to a node is proportional to the product of its fitness and its number of links. Now every node still grows in a way that can be described by a power of time, but the power is larger for fitter nodes. A node that arrives late but is fit enough will, given time, overtake the early arrivals. The next activity shows it in the simplest setting.

The theory has a dramatic limit. If one node’s fitness is far above everyone else’s, it can take a fixed share of all the links in the network, however large the network becomes, instead of a share that dwindles with size. In the 2001 paper on this, the authors saw a mathematical resemblance to a quantum phenomenon, Bose-Einstein condensation, in which a large fraction of particles falls into the lowest energy level: the nodes play the role of energy levels and the links of particles. It is a formal analogy, not a claim that networks are quantum. Its business reading is a winner-takes-all market, and the authors themselves cited Microsoft’s operating system as an example, not as a measurement. Windows did hold more than 90 percent of the operating systems on PCs from the 1990s until about 2012, and about 78 percent of desktops in 2025 by one web-traffic count. The point of the analogy is the question it asks: is a market’s leader the best, the earliest, or just the one with the most customers already?

A similar question was asked about size. In 2010 Facebook.com passed Google.com in the share of US visits, according to the analysts at Hitwise: in the week ending 13 March (7.07 percent against 7.03 percent), and over the whole year (8.93 against 7.19 percent). That was not true worldwide, and not when all of Google’s sites are counted together. The lesson is how carefully “biggest hub” has to be defined.