As an analogy, Barabási noted, a rock and a feather fall at very different speeds even though the law of gravitation says they should fall at the same speed. If you didn’t know about the effect of air resistance, he says, “you would conclude that gravitation is wrong.”
Clauset doesn’t find this analogy convincing. “I think it’s pretty common for physicists who are trained in statistical mechanics ... to use these kinds of analogies for why their model shouldn’t be held to a very high standard.”
If you were to observe 1,000 falling objects instead of just a rock and a feather, Clauset says, a clear picture would emerge of how both gravity and air resistance work. But his and Broido’s analysis of nearly 1,000 networks has yielded no similar clarity. “It is reasonable to believe a fundamental phenomenon would require less customized detective work” than Barabási is calling for, Clauset wrote on Twitter.
“The tacit and common assumption that all networks are scale-free and it’s up to us to figure out how to see them that way—that sounds like a non-falsifiable hypothesis,” he says.
If some of the networks rejected by the tests do involve a scale-free mechanism overlaid by other forces, then those forces must be quite strong, Clauset and Strogatz say. “Contrary to what we see in the case of gravity ... where the dominant effects really are dominant and the smaller effects really are small perturbations, it looks like what’s going on with networks is that there isn’t a single dominant effect,” Strogatz says.
For Vespignani, the debate illustrates a gulf between the mind-sets of physicists and statisticians, both of whom have valuable perspectives. Physicists are trying to be “the artists of approximation,” he says. “What we want to find is some organizing principle.”
The scale-free paradigm, Vespignani says, provides valuable intuition for how the broader class of heavy-tailed networks should behave. Many traits of scale-free networks, including their combination of robustness and vulnerability, are shared by heavy-tailed networks, he says, and so the important question is not whether a network is precisely scale-free but whether it has a heavy tail. “I thought the community was agreeing on that,” he says.
But Duncan Watts, a network scientist at Microsoft Research in New York, objected on Twitter that this point of view “is really shifting the goal posts.” As with “scale-freeness,” he says, the term “heavy-tailed” is used in several different ways in the literature, and the two terms are sometimes conflated, making it hard to assess the various claims and evidence. The version of “heavy-tailed” that is close enough to “scale-free” for many properties to transfer over is not an especially broad class of networks, he says.
Scale-freeness “actually did mean something very clear once, and almost certainly that definition does not apply to very many things,” Watts says. But instead of network scientists going back and retracting the early claims, he says, “the claim just sort of slowly morphs to conform to all the evidence, while still maintaining its brand-label surprise factor. That’s bad for science.”