Scant Evidence of Power Laws Found in Real-World Networks

In grad school scale-free networks were the soup du jour and my advisor was hammering on me to show that the human metabolic networks I studied had that property. "Guaranteed paper in Nature, get on it!", he exhorted. I sensed folks were not using the right statistical tests to show that distributions were scale-free (hint, doing a log-log-plot and doing linear regression to get a slope does not give you the base of the power law if there is no power law in the first place) and found a few notes on homepages that this was the case, but no publications (this was 2006 or so).

There seems to be an attitude in the physical sciences that math is there as sort of a glaze or condiment you can throw on top of bad data to make it palatable. I didn't make many friends in the biology department by telling them their pet model they'd based their careers off of wasn't only bolstered by $trendy_math_analysis but actually weakened by it. People seemed less interested in truth and more interested in appearance and publication prestige.

Always a good time to re-read Cosma Shalizi's classic "So You Think You Have a Power Law — Well Isn't That Special?" (http://bactra.org/weblog/491.html)

I love when facts contradict socially propagated truth, and we actually change our minds as a consequence. That update mechanism is one of the biggest sources of faith in humanity I can think of.

This would be tremendously comforting. After reading the book 'Linked', I have been worried. It presents a possibility that I hope no one is immoral enough to pursue. If it's claims of how social networks cluster, and how both important and fragile the most critical edges in those networks are in terms of enabling the society-wide spread of information/viewpoints/etc are all true.... it opens a very dangerous door.

In the book, they claim that something like 'Six Degrees of Kevin Bacon' works not because of those heavily-connected nodes, but primarily because of nodes which bridge mostly-disconnected clusters. So like a member of a biker gang who plays bridge with his elderly aunts knitting circle on Sundays would be an example of that. Ideas can flow from the group of bikers to the group of elderly knitters pretty much only through him. And almost by necessity, those links are weak. Once broken, for information to travel from one of those groups to the other immediately becomes extremely difficult.

So conjecture that there were an organization that had the ability to observe the social network of peoples communications. Conjecture that they also felt that they had a mandate to protect the status quo, at least on the largest scale, and to do what was within their power to prevent things like widespread social unrest, formation of disruptive political movements, etc. If they had the ability to interfere with those communication networks even in a very mild way, they could affect the most successful and quiet oppression in history. By bouncing a few emails, dropping a few packets here and there, communication between these weak links would break down pretty easily. And once mostly-connected clusters only talk amongst themselves primarily, it becomes fundamentally impossible for widespread social change to occur.

Now this is a very 'blind' approach, of course. You don't get to pick which ideas get isolated and which are permitted to spread. But, you do gain a guarantee that even if an idea is very powerful, it can never spread far enough fast enough to gain widespread acceptance. Sure there might be "large groups" that get very loud about it... but they wouldn't have an 'inside man' to introduce the idea in ways acceptable to the mostly-disconnected group, so it simply wouldn't spread. I've been wondering for several years now if this kind of manipulation would leave telltale fingerprints.

This is not new, at least for biological networks. For example, see this paper by Lima-Mendez and van Helden: The powerful law of the power law and other myths in network biology: http://pubs.rsc.org/en/Content/ArticleLanding/2009/MB/b90868... and several blog posts by Lior Pachter: https://liorpachter.wordpress.com/2014/02/10/the-network-non...

For someone unfamiliar with power laws and scale in networks, that introduction has me totally bamboozled with multiple repeating negatives.

A scale-free network is one that follows power laws, which manifests as having specific hubs that are much more interconnected than others.

A random network does not follow power laws. Hubs and edges are distributed relatively evenly.

Is that right?

I think Metcalfe's law is a bit overstated. Logic seems to suggest it starts out close to exponential but eventually becomes more like log n. Because each participant only has X amount of friends, that they derive benefit from.

For network of one-to-many broadcasts it may be more like n * k where k is the number of active broadcasters.

A classic from Lior Pachter (not mentioned in the article) from 2014: https://liorpachter.wordpress.com/2014/02/10/the-network-non...

What are the implications here for internet service providers and the big social networking companies?

Now, why do so many systems seem to follow log-normal??

A central claim in modern network science is that real-world networks are typically "scale free," meaning that the fraction of nodes with degree k follows a power law, decaying like k−α, often with 2<α<3. However, empirical evidence for this belief derives from a relatively small number of real-world networks. We test the universality of scale-free structure by applying state-of-the-art statistical tools to a large corpus of nearly 1000 network data sets drawn from social, biological, technological, and informational sources. We fit the power-law model to each degree distribution, test its statistical plausibility, and compare it via a likelihood ratio test to alternative, non-scale-free models, e.g., the log-normal. Across domains, we find that scale-free networks are rare, with only 4% exhibiting the strongest-possible evidence of scale-free structure and 52% exhibiting the weakest-possible evidence. Furthermore, evidence of scale-free structure is not uniformly distributed across sources: social networks are at best weakly scale free, while a handful of technological and biological networks can be called strongly scale free. These results undermine the universality of scale-free networks and reveal that real-world networks exhibit a rich structural diversity that will likely require new ideas and mechanisms to explain.

Paper referred: https://arxiv.org/abs/1801.03400

“Power laws” are the golden ratio of network science.