Seeing Sigmoids

One of the most basic but misleading heuristics that the human mind uses is that of linearity. If we see a progression (say 0, 1, 2), our first instinct is that the next step follows linearly (namely 3). But there is no a priori reason to prefer the linear interpretation to any other, say quadratic (which would suggest the next step in the sequence is 4). For whatever reason – probably due to the relative simplicity of linearity – our brains seem wired to prefer linear explanations to non-linear.

The problem with this state of affairs is that many phenomena in the world, especially in the biological, social and technical realms are non-linear. One basic reason that non-linearity crops up is that events and internal states of objects are often dependent on one another. Despite our desire to simplify and assume independence, the weather in Kansas is affected by the butterfly flapping its wings in China (and a multitude of other factors, some more significant, some less). Often times these dependencies are weak enough that the independence assumption is a good approximation, and thus our linear predictions work out very well. It took thousands of years before we discovered that light does not travel in a straight line, but rather is affected by gravity. We would do well, therefore, to never assume independence, even if the available evidence suggests it. The pursuit of understanding complex systems could reasonably be defined as exploring what happens when we look at things anew after removing the independence assumption from our thinking.

The first thing we see when looking for these heretofore hidden dependencies is that what was linear now becomes exponential. A canonical representation of this effect occurs when you compare the growth of two different types of networks, both with the same exact nodes, but different ways of attaching (i.e. linking) one node to another. In network A, links are added randomly between pairs of nodes. But in network B, nodes that already have links are more likely to get new ones (the more existing links, the more likely a node will get a new one). This is known in the literature as “preferential attachment” and entire books have been devoted to exploring its consequences.

But just as linear growth is an approximation, so is exponential. Why? Because even though our mathematical models of the world work with infinities, the world we live in is finite. There is a finite amount of mass, energy, space, time, etc. in the universe, if not actually, then at least for all practical human experience. In the network example, we began with a fixed set of nodes and so while early on the number of possible new links is large, eventually the network becomes fully connected and there are zero new links to be created. In the real world, even if we allow for growth in the number of nodes, eventually this growth too will run out of steam.

While exponential curves are better than linear for modeling and extrapolating, better still are sigmoids, or S-shaped curves. Sigmoids look like exponentials initially, but instead of unrealistically appearing as unchecked growth that gobbles up all available resources, sigmoids mercifully abate due to the self-referential source of their growth. I say mercifully because we’ve all been subjected to terrifying predictions that are inherently wrong because they ignore the simple truth of the sigmoid: that there’s no such thing as unchecked growth. Growth checks itself, one way or another. Malthusian prophecies will never come true, for the very ability for humans to replicate is dependent on resources that are consumed increasingly by that self-same replication. Pandemics are self-limiting as well, their rate of spread being inversely proportional to their effectiveness at killing their hosts. And while we are seemingly on an exponential train ride to destroying our environment, that too is a misconception; we may very well be on track to wipe out humanity (and many other species with us), but we will never truly destroy our “environment” because as soon as the last of “we” bites the dust, the destructive force is gone too. Actually long before then it will have petered out on the sigmoid.

Although the mercy of the sigmoid as presented seems to offer little solace, we cannot forget that inherent in the self-referential nature of such tragedies is the means for salvation. It may appear at first that I am letting us off the hook by saying that it will all work out by virtue of the sigmoid, that we needn’t lift a finger. However, the truth is just the opposite. We don’t get a sigmoidal soft landing without recognizing the downstream effects of our actions and feeding them back into our decision making process for future actions. If we want the softest landing possible, we must open our eyes to the faintest of connections, realizing they exist whether we can prove so (yet) beyond reasonable doubt.

By understanding the dynamics of complex, non-linear systems — the sigmoidal reality — we can appreciate even more how important our actions and inactions are to our own wellbeing. Whenever we see non-linearity, we should be thinking not of a Greek tragedy beyond our control but rather how this non-linear footprint must be the result of a feedback process, putting us not only in the audience but also on stage. If a linear reality is the mark of the simpleton, and an exponential reality the mark of the doomsayer, then a sigmoidal reality is the mark of optimism… but also of activism.

Like exponentiation before it, sigmoidality brings new understandings that were masked by the less nuanced model. One curious fiction that goes by the wayside in a sigmoidal world is the concept of a tipping point, made popular by Malcom Gladwell’s eponymous book. While it makes for vivid imagery, the critical “point of no return” is rarely found in real systems. This is not to say that we can’t identify the straw that broke the camel’s back, but rather that the back’s breaking is not the end of the story.

An avalanche does not continue to gain momentum indefinitely, it eventually comes to a stop. A meme (such as tipping point itself) eventually slows down, once it has been passed to a significant portion of the population. In fact, the switchover point from speeding up to slowing down makes for a more meaningful use of a tipping point concept, albeit quite different from the original. Ultimately, though what one gains from looking at the world with sigmoidal glasses is more accurate models, better prediction, and hence deeper truth.

  • norcalmetsfan

    this reminds me of something i read on radar about the growth of social graph sites.

    it also reminded me of something william saffire wrote in the times when the left was disillusioned after bush beat kerry in ’04. he wrote, “in democracies, the pendulum always swings.”

    -ncmf

    p.s. first time poster, regular reader

  • Thanks for the pointer. Interestingly, the Radar post mentions this as an argument for why Ray Kurzweil’s argument about the singularity is probably wrong. I was going to make the exact same point in an upcoming post. Although I think it’s more complicated than that. I think the “singularity” is not near, it’s here. It’s not some cataclysmic, tipping point, but rather a continuum.

  • Brady Firth

    good posts. i’m currently studying team/organizational dynamics as a PhD hopeful, and you’ve got some good stuff going on over here. cheers.

  • dylan

    In Chapter 9 of “The Singularity is Near”, Ray addresses this argument.

    I can’t tell by your statements whether you agree or disagree with Ray or if you just agree somewhat..

  • I have not read “Singularity is Near” yet, but if you point me to the pages that address this I will take a look. I elaborate on the teaser above on this forum (search in page for my name):

    http://www.singinst.org/blog/2007/12/27/sotala-mccabes-singularity-objections/

    Let me know whether that means I agree or disagree with Kurzweil :-)

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