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Dr. Baruch Barzel



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Spherical cows are rare

Network science, an always colorful realm of research, has just become more vibrant with the recent arXive paper by Broido and Clauset, touching on one of the sacred cows of the field – the scale free phenomenon. 

This current contribution is, perhaps, the culmination of Clauset’s consistent efforts to put the data analytics of networks, and the many power law functions that characterize them, on firm statistical grounds – efforts that have made us all transition from eye-balling the linearity of a log-log plot to running more rigorous tests on our data.

The work is truly impressive, analyzing close to 1,000 empirical networks and assessing their scale-freeness, observing that strikingly few actually pass the test.

So where should this development take us? Are we now expected to discard two decades of research driven by the concept of scale-free?

It is best, I believe, to put this contribution in context.

We often adopt a binary notion of exact sciences, in which things are either right or wrong. This line of thought is what leads, for instance, to the common idea that Einstein proved Newton’s laws to be false. Scientific reality, however, is significantly more subtle, as indeed Newtonian mechanics continue to be meaningful, even if we now know that Netown’s model of the world disregarded the intricacies later exposed by relativity theory.

Essentially, scientific models are designed to offer a smoothed out, idealized description of a reality that is blurred by wrinkles, spikes and sharp corners. In certain contexts the smoothed out version is most meaningful, in others you may actually be chasing the spikes.

Scale free networks are no exception. They provide us a smooth and simplified representation to help us conceptualize and extract meaning out of the seemingly disorganized mess of real world networks. They are convenient to handle analytically, they have a clear graphical fingerprint, and draw insightful connections to other fields of science, such as critical phenomena. Indeed, they are the well-behaved member of the mischievous family of fat-tailed distributions, and therefore they provide meaningful insight, even if they only offer a pale approximation of the exact, endlessly complicated, network statistics.

And truly, in recent years, few scientific ideas were as insightful as the concept of scale-free networks. The structural robustness of networks, their ultra-small diameter, their crucial role in epidemic spreading and their contribution to the network’s resilience, are just a few major advances, inspired by the revelation of scale-freeness. Most importantly, none of these results is sensitive to the specifics of scale-free, but rather rely on the network’s extreme heterogeneity, i.e. its fat-tailed degree distribution.

At the same time, some of these crucial advances may not have been discovered, if not for the analytical elegance and tractability of the idealized scale-free distribution. Therefore the meaningfulness of scale-free supersedes its detailed empirical accurateness, capturing the essence of many real networks, while smoothing out much of their spikes and sharp corners.

Does this dismiss Briodo and Clauset’s contribution? Of course not. Like in the case of Einstein vs. Newton, both types of analyses can (and should) coexist. Broido and Clauset took on a commendably ambitious task - to looks deeply into the statistical properties of hundreds of real networks. Such an endeavor could not have been conceived twenty years ago, being built on the two decades of research that ensued since the discovery of scale-free. 

As always in science, what you discover is closely related to the questions you ask:

In 1999 the question asked, for the first time, focused on the class of P(k) in real networks. The distinction was between the anticipated bounded family of distributions, e.g., Poisson, and the highly unexpected discovery of scale-freeness. Such a broad gap – Poisson vs. scale-free – represents a profound distinction that is indifferent to minor discrepancies.

Twenty years later Broido and Clauset ask a question, inconceivable in 1999, seeking to pin point the precise distribution within the fat-tailed family. Their microscopes are now focused precisely on the wrinkled, sharp edges and spikes of real networks, set to much higher resolution than needed to just separate Poisson from scale-free. Under such detailed scrutiny they have uncovered that the idealized, simplistic, and yet truly insightful, scale-free model is truly a rough approximation of reality.


Poetically, this, in fact, demonstrates the scale-free nature of scientific research. Indeed, new details and insights continue to emerge at every scale and level of resolution, whether through the broad strokes drawn in 1999, or by the fine brush applied in this recent contribution.

Why am I writing this response? Because I think Broido and Clauset made a crucial advance. But at the same time I fear that misinterpretation may take this advance out of context, as a call to discard the past two decades of discoveries inspired by the scale-free phenomenon. The fat-tailed distribution characterizing real networks is one of the most profound universalities of network science, underlying many of its most interesting and important phenomena. Therefore, while we have now been offered a closer look into the wrinkles the lie within this universality, we should also continue to benefit from the power of the smoothed version captured by the scale-free model.

Listening to the discussion that followed, I found that it was roughly aligned along a disciplinary divide. Physicists are nurtured to view the world through rounded models – we are aware of their, often exaggerated, simplicity, but at the same time appreciate their power to clear out the redundant details and capture only the bare essentials of a system. We feel perfectly at home milking a herd of spherical cows, even though deep down we know that some cows my deviate from a perfect sphere. We are so accustomed to this way of thought that we are often not cautious enough with our language – using simplistic terminology and approximate solutions – forgetting to explicate the roughness of our derivations – and as a consequence failing to convince researchers from other disciplines. I personally felt at times that my work was criticized, by referees or by peers (or by my wife), due to its reliance on idealized versions of complex systems.


As network scientists we live in an extremely interdisciplinary environment. We must then learn to bridge between the foreign languages we all speak. 

Spherical cow
Spherical cow
Spherical cow
Spherical cow
Spherical cow
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