This is a comment on the recent arxiv by Voitalov, van der Hoorn, van der Hofstad and Krioukov titled Scale-free networks well done, and the ongoing debate of scale-free networks. As usual, I take a laid-back spectator position—no papers, no research of my own, just another blog post of my own reading of this contribution and the state of scale-free networks in network science.
Earlier this year Broido & Clauset put out a preprint called Scale-free networks are rare. This brought back network scientists’ all-time favorite discussion topic of to life after a decade-long sleep. Simply speaking, there are two camps: those seeing scale-freeness as an emergent property, and those seeing it as a statistical property. My very cartoony summary is as follows:
- View scale-free networks essentially as outlined in Barabási & Albert’s Emergence of scaling in random networks and subsequent works of Barabási et al.
- Think of empirical networks as snapshots of growth processes. If these growth processes asymptotically lead to power-law degree distributions, then they’re scale-free networks.
- Didn’t lose faith in the buzzwords of the nineties’ complexity science: universality, fractals, self-similarity, criticality, emergence.
- . . and since these concepts are only well-defined in the limit of large systems, their minds tend to wander off to the spectacular blue skies of infinite networks, even though they are standing on the ground, amid finite networks.
- Reasons that scale-freeness is not scientifically important if it is not testable.
- . . and to test if a finite, worldly network is scale-free, one cannot use anything else than the network itself. Thus, one should deal with the finite reality on the ground.
- Are on top of the latest data science trends.
The discussion between these two camps was by no means less fierce in the early days of network science—check this (I-don’t-even-know-what-to-call-it-kind-of) paper by statisticians Handcock, Holland Jones and Morris.
The benefit of the emergence perspective—in addition to the fantastic phenomena in the infinity limit—is that math is easier when N → ∞. So there are many theoretical results about scale-free networks valid only there. Statisticalists can’t logically incorporate those results, because they have already given up the infinity limit with the assumption that only the network itself should decide its status as scale-free or not. If they deem an empirical network scale-free, it could still be that it, grown to infinite size, wouldn’t follow a power-law (and vice versa—it could seem to be scale-free, but scaling it up, it would not be).
The arxiv by Voitalov et al. essentially takes the statisticalist perspective, but doesn’t fight against the emergentists as hard as Broido and Clauset. Starting from different premises than Broido and Clauset, they develop an alternative, less restrictive, method and consequently discover more scale-free networks. After a quick read-through, I find it sound and well-researched. The disappointing realization is that whether scale-free networks are rare, medium or well-done is really a choice that needs to be argued by words (via the assumptions that go into the inference machinery). It’s a bit like how the emergentists’ dream of a universal mechanism for the emergence of power-law degree distributions was quenched by the discovery of a multitude of growth models all giving power-law degree distributions. Will we see yet more statistical ways to infer scale-freeness?
All in all, I think this debate and the different perspectives are great. The fact that there is so much to say about something so simple just goes to show how complex the topic we study is. If I may wish one thing for the future, it would be more work on statistical inference of network-evolution processes in general (not only preferential attachment).