Kirk Klasson

Innovation: Power Laws, the Adjacent Possible and the Accumulation of Continuity

plus ça change, plus c’est la même chose

 You can do some really nifty things with math. For instance, were you aware that winning the lottery is not as hard as everyone makes it out to be? If you start with a universe of say 36 or 42 or 56 possible numbers you don’t have to choose which exact numbers will be picked. You only have to pick a beginning number and then calculate the probability of the relative distance or number of integers between that number and each of the remaining ones that need to be selected. Start by writing the entire pool of possible numbers on a strip of paper. Now create a Mobius strip by twisting the paper and attaching the first integer to the last producing a kind of infinite loop. Now select your first number and calculate the probable distance between that selection and the next bearing mind that any number or interval between numbers once selected cannot be selected a second time. Go ahead, give it a try!


Recently, Vittorio Loreto, Vito D. P. Servedio, Steven H. Strogatz and Francesca Tria, serious math wizards, discovered that innovation, or perhaps more precisely, the emergence of something materially novel, adheres to established power laws. Now that’s not to say that they or for that matter anyone can predict when, where or how innovation occurs but rather to prove that when it does, its emergence will conform to variations of known power laws. Given that nearly every day billions of dollars are spent in pursuit of meaningful, monetizable innovation, it might be worthwhile to find out exactly how it might occur.

This work was based in part on the predictive capabilities of various Polya Urn models and the notion of the adjacent possible, the former the creation of George Polya and the latter the creation of Stuart Kauffman. If you’d like to pursue the entire mathematical hypothesis you can find it here: Dynamics on expanding spaces: modeling the emergence of novelties.

Polya Urn models consist of a thought experiment where a container, in this case a metaphorical urn, is filled with colored balls or unique tokens. A ball is removed from the urn and its observation recorded. Depending upon a pre-established formula this ball is returned to the urn along with more balls of the same or of a different color(s). The process is then repeated. In the instance where a ball is removed from the urn and is returned with additional balls of the same color, the probability that balls of that specific color will become repeatedly selected increases. This results in a scenario often described as “the rich get richer” and its behavior, statistically speaking, is a recognized proxy for familiar power laws, specifically Heaps’ law and Zipf’s law (see Social Subsidization and Diminishing Returns – March 2015). If you’d like to become more familiar with how Polya Urns behave you can fiddle around with them here:

Polya Urn Model

Screen Shot 2017-03-15 at 4.24.38 PM

After a few iterations strategy wonks will recognize a number of well-known concepts emerge in the model’s behavior including “first movers”, “fast followers”, “emerging disruptors”, “share lock up”, etc. The more you play with this, the more these patterns begin to repeat themselves.

The adjacent possible is a notion first developed in the evolutionary biology community to elucidate phenomena such as genetic mutation and the role it plays in evolution. The theory goes that the “new” or the “novel” already potentially exists but yet lies just beyond our ability to grasp it. Our familiarity with the known and ignorance of the new is in essence a boundary; a kind of frontier, but the frontier between what is known and unknown can be altered by incremental discoveries. The graph that accompanied the above mentioned paper lays this out in the following manner.

Screen Shot 2017-03-20 at 2.26.32 PM

The nodes on the right represent that which is known. Let’s assume you are a visitor exploring these nodes and that your travels take you to a node that lies just outside the existing frontier of known nodes. That event would instantly alter the frontier of the known and enlarge the potential of the adjacent possible that is represented by the graph on the left. Now, for simplicity’s sake, let’s assume that the nodes are actually colored balls in a Polya Urn model. What probability of colored ball discovery and rate of colored ball replenishment would have to exist in order to accurately model the rate of “new” discovery evidenced in existing phenomena? Turns out the authors believe they have discovered that formula and have dubbed it Polya Urn modeling with innovation triggering. In this model the rate and sequence of ball/node observation dictates the method and frequency of ball/node replenishment.

So while this formula does not predict the future it does suggest that innovation lies more or less at the confluence of pending coincidence. If one wants to accelerate innovation one needs to understand where those circumstances already exist in that which is known or may emerge from where frontiers of the known are expanding. The implication is that insights that produce important innovations might be more probable based on the transfer of discoveries at the cusp of existing knowledge domains and not necessarily a direct frontal assault on a specific problem known to a specific domain.(see Digital Strategies – February 2016)

Curiously, we see evidence of this notion in everyday discoveries. Last year, researchers in chemical engineering employed an innovation in materials science, a copper/carbon catalyst arranged on nanospikes, to convert CO2 into ethanol, essentially turning exhaust into fuel or the alcoholic beverage of your choice. Whether this scales to achieve economic viability has yet to be determined. But assuming it does, your buzz and your mileage may vary.

Living on the fault line

 This work is neither the first nor the only attempt by mathematicians to model the future and determine how one might actually get in front of it. For instance, Cliodynamics is dedicated to using big data to model our social future. Currently one of their predictions is that wide scale riots and terrorism will break out beginning in 2020. The good news is you still have time to bounce over to Rural King and pick up that jerky dehydrator that you always dreamed about.

Before Cliodynamics there were other attempts to describe change mathematically. One was the idea of catastrophe theory. Back in 1976 E.C. Zeeman proposed that rapid change could be modeled topographically. Like the original Polya Urn models, where the rich get richer, catastrophe theory used a model of cusp topologies where more of the same begets more of the same until suddenly it doesn’t. Both Cliodynamics and catastrophe theory attempt to predict how large-scale episodic change comes about. (see Hacking the Future 2.0 – January 2015)

Cusp Image

In catastrophe theory the shape of a surface or topology continues to increase or maintain its continuity until it reaches a pleat or a fold which allows the system in question to collapse back to its original state or, in some instances, into a new state altogether. At the time this was introduced it was being applied to such things as stock market crashes and unexpected social behaviors. Once change occurs the system would go back to an interval of stasis, what some prognosticators might call a “new normal” and continue that way until another episode of change occurs. If you substituted change in Zeeman’s model for the notion of innovation in the Polya urn hypothesis you might discover they are actually trying to describe the same thing.

Of Power Laws and Ponies

 If all of this seems a tad familiar, if not down right axiomatic, there’s a good reason for it. Phenomena with established life cycles lend themselves to change in a predictable manner. So, for earthquakes that tend to occur periodically, the probability of them happening at the end of the current intervening period of no activity would increase. The batteries in your smoke detectors will need to be replaced eventually and the longer you live, the greater the probability that you may soon die. That kind of thing. Although one might not actually recognize where in the life cycle things currently are. Cosmologically speaking, we suspect things started with a Big Bang, how close we might be to the other side of that is anybody’s guess and most of us aren’t in a hurry to find out.

Other phenomena are far less predictable and for change to occur requires a significant share of happenstance. For instance, when will the Chicago Cubs next win the World Series? Predicting this can be a random walk to nowhere regardless of where you think the frontier of the known might lie.

Still other novelties, if not completely obvious, are indeed noticeably latent in the reality to which we have all become inured. This would be an instance of when the new and the innovative seem to be one and the same. Some things may be new or they might just be new to you.

Long before there was an Android operating system, Linux was being unpacked and stuffed into specialized computing devices. This was considered, even then, a distinct and recognized market segment called embedded computing. But most of the major players in enterprise computing couldn’t be bothered with it since it was relatively small, fragmented and operating at a price point below the pennies per share they were trying to manipulate. So unpacking a Linux variant to build a smart phone OS wasn’t necessarily new; it was merely a matter of time. (see Requiem for a Business Model – January 2011)

Another example would be the burgeoning Internet of Things, a phenomenon where power laws abound. For starters the sheer number of devices is beyond Sagan-esque, there are billions and billions of cameras, sensors, microphones, toasters, etc. just waiting to be monetized but yet when it comes to the value of these devices all we ever hear about are the same hackneyed anecdotes about wind mill power generation telemetry and how much we’ve learned from it. Say for instance the number of dead bats per rotation. Yet, hiding somewhere in all that noise is another innovation that has yet to emerge.

One can just imagine the founders of Palantir sitting in a conference room tossing tennis balls against the wall and murmuring, “…look at all this data, look at all this data…there must be a pony here somewhere.”


Graphic courtesy of Vittorio Loreto, Vito D. P. Servedio, Steven H. Strogatz and Francesca Tria, Dynamics on expanding spaces: modeling the emergence of novelties




5 Comments to "Innovation: Power Laws, the Adjacent Possible and the Accumulation of Continuity"

  1. source says:

    bookmarked!!, I like your blog!

  2. source says:

    Hi there, I read your blog like every week. Your story-telling style
    is witty, keep up the good work!

  3. It’s arduous to seek out knowledgeable folks on this topic, however you sound like you realize what you’re speaking about! Thanks

  4. top financial advisors says:

    I’d like to thank you for the efforts you have put in penning this website. I really hope to check out the same high-grade content by you later on as well. In truth, your creative writing abilities has inspired me to get my very own website now 😉

  5. education degree says:

    Fantastic items from you, man. I’ve understand your stuff prior to and you’re just too wonderful. I really like what you’ve bought here, really like what you’re saying and the best way during which you are saying it. You are making it entertaining and you continue to care for to keep it sensible. I can’t wait to read far more from you. That is actually a terrific web site.

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Insights on Technology and Strategy