Power Laws in the Real World
https://www.quantamagazine.org/scant-evidence-of-power-laws-found-in-real-world-networks-20180215/
In this article, correspondent Erica Klarreich presents conflicting claims about the prevalence of power laws, or “scale-freeness,” in real-world networks, such as those found in metabolics and on the web. When former physicist Albert-László Barabási first started research on the behavior of networks in the 1990s, he was adamant that the scale-free paradigm was consistent across complex networks, and that power laws could, in fact, be used to describe the majority of said networks. Before this research, network connections were thought to follow a scale, or bell curve, in which each node would have a constant probability of connection with no strong outliers. Despite the idea of a scale being somewhat applicable, however, Barabási’s studies of mega web pages like Google found that there is vast inequality in the number of links a network has (hence the term “scale-free” to refer to a power law-shaped distribution, which can be modeled by the basic 1/kc equation, where c is usually 2 or 3). This research is the basis of the rich-get-richer phenomenon in networks, which is perpetuated by the mechanism of preferential attachment: in this system, newly-joined nodes in a network are highly likely to connect to already-popular hubs, either directly or through copying other nodes, and so the rich nodes gain power and the poor nodes get poorer. In a side point, Barabási states that the scale-freeness of power law-abiding networks is their triumph and their downfall: as long as the hubs remain intact, the network can sustain much damage elsewhere, but it will quickly unravel if the hubs are attacked. This is somewhat related to the concept of network effects, in the sense that if people become disillusioned to a hub or highly popular commodity, they might suddenly find it not worth their money, and the company can fail instantaneously.
Skeptics of the aforementioned ubiquity of power laws, statistician Clauset and his grad student Broido, deigned to embark on a statistical analysis of Barabási’s work, and garnered highly controversial results. In fact, Clauset concluded that, out of around 1,000 networks, two-thirds exhibited distributions that were unexplainable by power laws (as for the rest, no conclusion could be made about the existence or nonexistence of scale-freeness). From this, Clauset strengthened his proposition that few real-world networks are actually explainable through power laws: only around 4% met all scale-free criteria and passed all high-threshold tests. Overall, these disagreements highlight the debate about the real nature of networks in the world today. Are they defined by preferential attachment, or does preferential attachment occur only after a scale-free network has established itself? This, as well as many other questions, are as of yet unexplainable. The only thing scientists are sure of is the multiplicity of factors at play when deciding what goes into a network, making it almost impossible to resolutely conclude that a single model can explain any single network.
Clearly, this article has many relations to what we have learned about power laws in class. It details the early assumption of Gaussian normality when modeling networks, which we briefly touched upon, and then goes into the more plausible version of network modeling: the power-law equation. The article makes the same distinction as we did between exponential/logarithmic functions and the power-law distribution, which has a fat or “heavy” tail that makes it slightly more robust on the unpopular end of the power spectrum. In class, we touched upon real-world examples of fat tails; for example, in retail, there are many items or brands that fail to become popular, thus fattening the tail of the distribution to a great extent. The article hypothesizes that, in fact, the fat tail model could be a slightly better explanation for real-world networks, as being fat-tailed does not necessarily mean following a power law to the T. Finally, we discussed the idea of preferential attachment in the context of the “rich get richer” copy model, where new nodes either link directly to a ‘random’ node or copy the link of an existing node. In either case, it is highly likely that a new node will end up linking to a hub, especially in a well-established network, as it would be almost impossible to find a way to bypass all connections to said hub. Overall, this article highly corroborates what we learned in class, only introducing the new detail that the models we have been taught might not explain the real world as perfectly as we may believe.