Nice post & analysis, Trolololo. Following.

I made a similar observation earlier this month in Stephen Reed's thread, about what looked like an increasing time span between reaching the next order of magnitude in network size, but didn't follow up on it (

shameless plug to my own post :D).

A question.

I'm not sure about the exact type of growth of your function. What you seem to be mapping is logarithmic growth (slow) on a log chart (fast). Stripped of its minor constants, your formula is of the form 10^ln(t) for time = t. ln(t) grows extremely slow over time, but the result

*is* used as a positive, growing exponent.

It seems to fall somewhere between linear growth and exponential growth, and it isn't bounded either (like in Stephen's model). I was wondering if someone with more knowledge on functional growth could answer this once and for all for me, have been wondering about this for a while now. (EDIT: I'm wondering if it could be an instance of so called sub-exponential growth)

A critical remark.

While I personally, intuitively, find a price function with a declining growth rate (like yours) more plausible than the constant growth rate models that have been presented on this forum (i.e. the "loglinear models" you linked to as well), one problem still remains:

Price tends to "jerk around" all those models.

I remember that, late last year, when price exceeded even the loglinear model's predictions, some analysis was posted that suggested a

**superexponential** price function to model BTC price.

Then came the first leg of the 2013/2014 correction, and suddenly the

**loglinear** models were all the rage again.

Now, the correction continues, and you suggest (with good reasons, I agree) a model based on an below exponential growth assumption. But I'm afraid all it takes is another year of bear market (or perhaps, a sudden rally of huge proportions), and we need to re-adjust our assumption for what the "best" growth type for our model is...

Here's what I'm trying to say: I am using technical (i.e. historic price based) methods myself all the time for predicting price on the short term. However, I start to think that, on a long enough time scale, fundamentals govern the price function. So a model like yours (or Stephen's, or rpietila's), that are essentially an

extrapolation from an (admittedly well fitted) function on the historic price data might come to its limits.

I am thinking that perhaps the only semi-reliable way to go about mapping the "long term trend" is Peter R.'s way: finding a proxy for network size, and then modeling expected price/mcap as a function of network size.

See for example here:

https://bitcointalk.org/index.php?topic=68655.msg9059346#msg9059346He still makes a number of assumptions (Metcalfe's law, for example), and in a way, his method only

*shifts* the problem (because now we are trying to predict, i.e. extrapolate,

*network size*), but at least his predicted numbers will rarely be so out of tune with reality as the pure time series models can be at times.