A physicist probes an elegant law that governs cells, cities, and almost everything in between

July 11, 2017
Albert-László Barabási

A dog owner weighs twice as much as her German shepherd. Does she eat twice as much? Does a big city need twice as many gas stations as one that is half its size? Our first instinct is to say yes. But, alas, we are wrong. On a per-gram basis, a human requires about 25% less food than her dog, and the larger city needs only 85% more gas stations.

If we double the population of a city, we also need roughly 15% fewer water pipes and electrical wires than linear thinking would predict. In other words, if you wish to live in a green city, you should forget bucolic rural settlements and consider Manhattan instead.

Moving to a city that’s twice as big will not only offer you 15% more income on 15% less infrastructure, you’ll also be 15% more likely to patent an invention. It is perhaps no surprise, then, that Boston, San Francisco, and New York City have emerged as the unbeatable incubators of creativity.

As Geoffrey West explains in Scale, the reason behind these intriguing phenomena is a universal law known as allometry—the finding that as organisms, cities, and companies grow, many of their characteristics scale nonlinearly.

West is a theoretical physicist who traded in particle physics for complex systems in 1997. He eventually landed at the Santa Fe Institute, a hotbed of complexity research, where he became an eloquent spokesman on behalf of his newfound subject.

Allometric scaling has deep roots in ecology, dating back to the century-old work of J. B. S. Haldane, D’Arcy Thompson, and Julian Huxley. Yet, for about a century, it remained a puzzling empirical observation. This changed two decades ago when West and collaborators offered the first quantitative explanation of allometric scaling. With that came a newfound enthusiasm toward the subject as researchers began discovering its relevance to everything from cities to companies.

Scale offers a fascinating journey into the genesis, applications, and implications of allometric scaling. Did you know, for example, that regardless of size, all animals have about 1.5 billion heartbeats in their lifetime? And that thanks to this, the bigger an animal, the longer it lives? Only humans defy this law, living twice as long as allometric scaling predicts for our weight.

Readers also learn that, on average, people commute about an hour each day, regardless of city size or mode of transportation. Our walking speed, however, depends on the size of the city we live in: Big-city residents walk twice as fast as the locals of small towns, sometimes creating logjams when the two populations attempt to traverse the same streets. This observation prompted the British city of Liverpool to create fast walking lanes, offering an unobstructed path for city dwellers through the masses of leisurely visitors.

West’s enduring contribution to our understanding of complex systems is his explanation of the roots of allometric scaling. He observes that complex systems—from cells to cities—require networks to ensure that every component has access to the resources needed. These networks have evolved to optimally transport resources, minimizing, for example, the energy our hearts exert to circulate blood or the time we spend traveling from work to home. In Scale, West patiently describes these foundational observations, eventually arriving at the scaling laws that have resisted explanation for more than a century.

Given the central role that networks play in West’s theoretical framework, it is puzzling the degree to which the narrative is divorced from network science, the field that focuses on the scaling properties of real networks. For example, a universal feature of all networks discussed in the book—from the metabolic network that supplies energy to a cell, to the social and professional networks that contribute to the amazing vitality of a big city—is the presence of major hubs that hold the smaller nodes together. Yet the scale-free property of these networks, which explains how the size of these hubs scales with the number of nodes in the system, is never considered in allometry.

Scale offers a deeply personal narrative about the origins and evolution of allometric scaling that is enriched by West’s distinctive voice. It’s a journey with many fascinating digressions that do make the nearly 500-page book a true time investment. Yet, for those willing to commit, West’s insightful analysis and astute observations patiently build an intellectual framework that is ultimately highly rewarding, offering a new perspective on the many scales with which nature and society challenge us.

Originally Published by Science (2017)

Photo Credit: SCALE cover Geoffrey West, Penguin Press

Figure 1. How hard is to distinguish random from scale-free networks? To show how different are the predictions of the two modeling paradigms, the scale-free and that or the random network models, I show the degree distribution of four systems: Internet at the router level; Protein-protein interaction network of yeast; Email network; Citation network, together with the expected best Poisson distribution fit. It takes no sophisticated statistical tools to notice that the Poisson does not fit.
Box 3: All we need is love

If you have difficulty understanding the need for the super-weak, weakest, weak, strong and strongest classification, you are not alone. It took me several days to get it. So let me explain it in simple terms.

Assume that we want to find the word Love in the following string: "Love". You could of course simply match the string and call it mission accomplished. That, however, would not offer statistical significance for your match.

BC insist that we must use a rigorous algorithm to decide if there is Love in Love. And they propose one, that works like this: Take the original string of letters, and break it into all possible sub-strings: 


They call the match super-strong if at least 90% of these sub-strings matches Love. In this case we do have Love in the list, but it is only one of the 14 possible sub-strings, so Love is not super strong.  

They call the match super-weak if at least 50% of the strings matches the search string. Love is obviously not super-weak either.

At the end Clauset's algorithm arrives to the inevitable conclusion: There is no Love in Love.

The rest of us: Love is all you need

‍Figure 3. Differentiating model systems Curious about the reason the method adopted by BC cannot distinguish the Erdős-Rényi and the scale-free model, we generated the degree distribution of both models for N=5,000 nodes, the same size BC use for their test. We have implemented the scale-free model described in Appendix E of Ref [1], a version of the original scale-free model (their choice is problematic, btw, but let us not dwell on that now). In the plot we  show three different realizations for each network, allowing us to see the fluctuations between different realisations, which are small at this size. The differences between the two models are impossible to miss: The largest nodes in any of the Erdős-Rényi networks have degree less then 20, while the scale-free model generate hubs with hundreds of links. Even a poorly constructed statistical test could tell the difference. Yet,  38% of the time the method used by BC does not identify the scale-free model to be even ‘weak scale-free,’  while 51% of the time it classifies the ER model to be ‘weak scale-free.’


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