快猫短视频

IT WAS the late 1990s, and a group of physicists had it all figured out. A universal rule seemed to explain a vast range of behaviours in social, biological and computer networks. Everything from how ecosystems evolve to how the internet works seemed to follow the same statistical pattern, called a 鈥減ower law鈥�. What came as a surprise was that this simple law implied a deep underlying principle for all these networks.

Researchers began working with power-law models of the internet and other systems 鈥� and that鈥檚 when they made a startling observation. If their models were correct, eliminating the most highly connected computers would cripple data flow. Their work appeared to show that the internet and systems like it had an 鈥淎chilles heel鈥�, and that contrary to popular belief a few carefully targeted attacks could bring down the entire network. The finding spawned a whole branch of research known as 鈥渟cale-free鈥� network theory (快猫短视频, 13 April 2002, p 24). The name, which derives from the idea that members of a power-law network have no 鈥渢ypical鈥� number of connections, has since graced the covers of the most prestigious scientific journals.

Now a growing number of biologists, mathematicians and computer scientists are complaining that the idea has been overhyped, and that the power-law pattern does not reveal anything fundamental about what makes these networks tick. John Doyle, an expert in control and dynamical systems at the California Institute of Technology in Pasadena, is among those who dismiss the idea that scale-free theory can make useful predictions. 鈥淭he problem isn鈥檛 hype, the problem is it鈥檚 wrong,鈥� he says. Researchers like Doyle are developing more sophisticated tests of power laws and models of what they mean. They have proposed theories that take into account the evolution, design and structure of specific networks, and their ideas have led to statistical methods for modelling forest fires, protein-protein interactions and other biological phenomena.

At heart, power laws are simple. If you plot the proportion of 鈥渘odes鈥�, or members of the network, having a certain number of connections versus that number of connections, a power law appears as a curve shaped like a suicide ski slope: declining steeply at first, and then ever more gently. This reflects the fact that most nodes have only one or two connections to others, while just a handful of nodes have hundreds, even thousands of links.

So where does the controversy come from? In physics, power laws give powerful insights into simple systems like phase transitions from liquids to gases. The systems that researchers began referring to as scale-free, however, were more complex. 鈥淚t was a fascination for many of us,鈥� says physicist Albert-L谩szl贸 Barab谩si of the University of Notre Dame in Indiana, a leading author of the original scale-free papers. 鈥淪o many networks have absolutely nothing to do with each other, but they all end up being scale-free.鈥�

Perhaps they should not have been so impressed, says Michael Mitzenmacher, a computer scientist at Harvard University. 鈥淚 think that鈥檚 a sort of lack of historical knowledge.鈥� He says the notion of a power law is ill-defined, and what, if anything, it signifies outside of simple systems has been debated for the past 80 years.

The most popular model to explain why power-law distributions occur in networks is known as preferential attachment 鈥� the idea that in general well-connected things tend to garner ever more connections. The first widely cited appearance of this model was in a paper in 1925 that described the power-law distribution of species among genera. It also proposed an explanation: genera with many species were more likely to have a random mutation in one of the species that then spawned a new one, so genera with many species added more species faster than those that were species-poor. In 1959, in the journal Information and Control, Benoit Mandelbrot, famous for his work on fractals but also prolific in statistical analysis, confronted Nobel prizewinning economist Herbert Simon in a heated debate over whether the idea of preferential attachment has any validity. The argument is still going strong today, as Mitzenmacher pointed out in a 2004 paper in the journal Internet Mathematics.

鈥淧ower laws don鈥檛 mean anything. The devil lurks in the details of the network鈥�

Preferential attachment has earned itself the most play on the World Wide Web, where search-engine companies Google and AltaVista have used ideas from scale-free theory to justify their system of ranking the most connected web pages at the top of their search results. Scale-free thought, however, doesn鈥檛 go much further than noting the existence of these highly connected pages and predicting that they should become ever more highly connected.

The problem lies in making the leap from scale-free statistics to the underlying process that determines how a particular network behaves. 鈥淧ower laws don鈥檛 necessarily mean anything,鈥� says Mark Newman, a physicist at the University of Michigan in Ann Arbor. 鈥淭here鈥檚 this tendency for people to see a power law somewhere and assume that this process is going on. This is a logical fallacy, like saying, 鈥楤ears like honey, my wife likes honey 鈥� therefore my wife is a bear.'鈥�

Consider the router network that directs data through the internet, and which was the subject of a 2000 Nature paper, co-authored by Barab谩si, that brought scale-free thought into the mainstream. The router network is not unlike a traditional telephone network, only with lines stretched over the globe connected to routers that receive information packets from one link and send them out on another. If you are sitting in London and send an email to a friend in Washington DC, your email gets routed from your computer to your internet service provider (ISP), which may send your email to a data router in New York, which passes your email to a router in Washington, which tosses it to your friend鈥檚 ISP and on to your friend鈥檚 computer (see Diagram).

In this whole sequence, the only computers linked to more than a few others are those operated by the ISPs, which will connect to hundreds or thousands of subscribers. The major data routers will typically have two to six links. If a graph is plotted showing how many of the internet鈥檚 computers have a given number of connections to other computers, the vast majority will pile up on the far left, linked to just a few others. Trailing out at the far right of the graph will be the handful belonging to the ISPs. The resulting curve, which falls steeply to start with and flattens out at its tail end, represents the power law.

Knock out the net

After finding that this power law described the statistics of internet routers, Barab谩si and colleagues used a theoretical network with the same proportion of highly connected routers to model the net, and from these models came the idea that eliminating highly connected routers could shut the net down. Doyle argues that this approach, while superficially attractive, ignores a simple fact: the highly connected routers are ISPs on the edges of the network, close to end users (Proceedings of the National Academy of Sciences, vol 102, p 14497). Take down highly connected routers around the US, and you鈥檒l knock out ISPs that serve users in certain neighbourhoods. It would, for sure, be an annoyance for localised clusters of internet users, but the majority of traffic flowing around the world would continue unscathed. The bulk of internet data 鈥� be it financial trading, web surfing or massively multi-player online games 鈥� will flow unimpeded, through routers that have only a few links each. 鈥淭here are so many routers that you鈥檇 have to destroy a ridiculous number of them before you鈥檇 really cripple the internet,鈥� Doyle says.

The router example reveals the weakness of scale-free models as a predictor of how a system will behave. They simplify systems, leaving out the details in which the devil can lurk. 鈥淭he approach of scale-free models was diametrically opposite to the types of models that are truly useful, which are grounded in specificity,鈥� says Evelyn Fox Keller, a historian of science at the Massachusetts Institute of Technology. A useful model would specify what the nodes do, where they are in the network and how their connections work.

A few researchers have proposed ways to model power-law networks to make more useful predictions. While this has raised some hackles among supporters of scale-free theory, even Barab谩si concedes that the original ideas are not the whole story. 鈥淚t is absolutely correct that there are lots of other properties of the networks that are just as important as the scale-free: some are fractal, some are not, some preferentially attach, some don鈥檛,鈥� he says. Research on networks has evolved considerably in the seven years since scale-free ideas made their entrance, he says.

One leading alternative is known as 鈥渉ighly optimised tolerance鈥�, or HOT. It originated in the late 1990s, and is the basis for a more realistic internet model. Its most vocal proponent is Doyle. 鈥淩eal engineering and real biology are really complicated. Yet we want simple models,鈥� he says. 鈥淲ith HOT we鈥檙e trying to explain, in as simple models as the scale-free models that are more faithful to the specifics of the domain, what is general about complex networks.鈥�

The key idea of HOT is that networks evolve according to what they are designed to do and their physical constraints. Real systems behave differently at different size scales and locations. Take biological cells and tissues. At one level they look like a sea of proteins. Zoom out and you see organelle structure. Zoom out further and you see bunches of cells stuck together to form a particular tissue. HOT鈥檚 recognition of complex systems鈥� 鈥渟elf-dissimilar鈥� structure sets it apart from scale-free theory, which treats systems as the same if they have the same statistics.

Yet both theories share an appreciation for power laws. HOT uses them to gauge which aspects of the system are important. The exponent in the power law controls the steepness of the curve, and is in turn determined by the specific goals and constraints of the network. In the simplest type of HOT model, called the 鈥減rofit-loss-resource鈥� model, a complex system is boiled down to a conflict between the resource and the loss, and HOT assumes there is one optimal way to set up the system.

As an example, take the management of forest fires. If the profit is timber, the loss is the amount of land that burns and the resource is land used for fire barriers, the essential trade-off is planting trees versus protecting them with fire barriers (Proceedings of the National Academy of Sciences, vol 102, p 17912). Another application of HOT is working out how to create an optimally navigable website, in which the conflict is between small file sizes for fast downloading and minimising the number of clicks before the user finds the desired information. The router configuration of the internet is more complicated 鈥� there are multiple goals, such as speed and volume of information flow 鈥� but Doyle says HOT can treat it as an optimisation problem too.

Where do networks go from here? The work that Doyle and others are doing is 鈥済enerally excellent鈥�, Mitzenmacher says. 鈥淭hey鈥檙e trying to define what it means to be scale-free.鈥� However, he cautions that although HOT is alluringly useful, optimisation arguments cannot be the whole answer any more than preferential attachment was. If the universal law of networks is wrong, it seems, researchers must continue to press on for better models, better testing and better understanding 鈥� all seasoned with a healthy dose of scepticism.