# Are Bitcoin bubbles predictable? Combining a generalized Metcalfes Law and the Log-Periodic Power Law Singularity model | Royal Society Open Science

### 1. introduction

In 2008, the pseudonym satoshi nakamoto introduced the decentralized digital cryptocurrency, bitcoin[1], and the innovative blockchain technology that underlies its global peer-to-peer payment network.1 From their techno-libertarian beginnings, they conceived bitcoin As an alternative to the core banking system, bitcoin has seen super-exponential growth. Fueled by the rise of bitcoin, a host of other cryptocurrencies have gone mainstream with a variety of highly disruptive use cases anticipated. cryptocurrencies have become an emerging asset class [3]. At the end of 2017, the bitcoin price peaked at almost $20,000 and the combined market capitalization of cryptocurrencies reached around $800 billion.

The explosive growth of bitcoin intensified debates about the intrinsic or fundamental value of the cryptocurrency. While many experts have claimed that bitcoin is a scam and that its value will eventually drop to zero, others believe that further growth and adoption awaits it, often compared to the market capitalization of monetary assets or stores of value. Comparing bitcoin to gold, an analogy based on the digital scarcity that is built into the bitcoin protocol, some market analysts have predicted bitcoin prices of up to 10 million per bitcoin [4]. Nobel laureate and bubble expert Robert Shiller summed up this ambiguity of Bitcoin price predictions when he stated, at the 2018 Davos World Economic Forum, that “Bitcoin could be here for 100 years, but it is more likely to crash.” completely” and “just put an upper limit on [bitcoin] with the value of the world money supply. but that upper limit is terribly large. so it can be anywhere between zero and there'[5].

Reading: Metcalfe’s law bitcoin

There is an emerging academic literature on cryptocurrency valuations [6-16] and their growth mechanisms [17-19]. Naturally, there are relationships between the value of bitcoin, adoption, and online activity (searches, tweets, etc.). macroeconomic variables should also determine the attractiveness of cryptocurrency (for example, as a hedge against failed sovereign monetary systems). however, the existence of large bubbles and shocks, which are radically non-stationary with non-linear tipping point dynamics, makes modeling these mechanisms difficult and risky with linear stationary models and conventional econometric techniques.

Many of these studies attribute some technical feature to the bitcoin protocol, such as the “proof-of-work” system on which the bitcoin cryptocurrency is based, as a source of value.2 in [21] even argued that bitcoin has value because people extract it, and not the other way around. however, as former wall street analyst tom lee [4] has proposed, one of the earliest academic proposals [22], now widely discussed within cryptocurrency communities, is that an alternative valuation of bitcoin can be based on its network of users. In the 1980s, Metcalfe proposed that the value of a network is proportional to the square of the number of nodes [23]. this may also be called the network effect, and has been found to hold for many networked systems. If metcalfe’s law holds here, fundamental bitcoin valuation may actually be much easier than stock valuation3, which is based on various multiples, such as price-to-earnings, price-to-book, or price. -cash flow- and therefore will support an indication of bubbles.

although it seems relatively obvious that bubbles exist within the cryptocurrency market, in finance and economics, the possibility of financial bubbles is often ruled out based on rationalization of market efficiency,4 which assumes unpredictable market price, for example, following a kind of random walk geometry (eg, [27]). on the contrary, didier sornette and her collaborators affirm that bubbles exist and are ubiquitous. moreover, they can be accurately described by a nonlinear trend called the periodic logarithmic power law (lppls) singularity model, potentially with highly persistent errors, but ultimately mean-reversion. the lppls model combines two well-documented empirical and phenomenological features of bubbles (see [28] for a recent review):

price shows transient growth faster than exponential (i.e. where the growth rate itself is growing), as a result of positive feedbacks such as grazing [29], which is modeled by a hyperbolic power law with a singularity in finite time, that is, endogenously approaching an infinite value and therefore needing a drop or correction before reaching the singularity;

It is also decorated with accelerating periodic logarithmic volatility fluctuations, embodying spirals of conflicting expectations of higher returns (bullish) and an impending crash (bearish) [30,31]. such periodic logarithmic fluctuations are ubiquitous in complex systems with a hierarchical structure and also appear spontaneously as a result of the interaction between (i) inertia, (ii) nonlinear positive and (iii) nonlinear negative feedback loops [32].

The model thus characterizes a process in which, as the speculative frenzy intensifies, the bubble matures towards its endogenous critical point and becomes increasingly unstable, so that any small disturbance can trigger a collapse. this has been further formalized in the so-called jls model, in which the rate of return accelerates towards a singularity, offset by the increasing rate of collision risk [30,33], giving a generalized risk-return relationship . We emphasize that one should not focus on the unpredictable instant trigger itself, but rather monitor the increasingly unstable state of the bubbling market and prepare for a correction.

Other studies have considered bubbling dynamics within bitcoin: in [18] (followed by Garcia & Schweitzer [34]), a four-variable autoregression was introduced. Finding evidence of positive feedback between price and online activity, the possibility of bubble formation was suggested. however, the model focuses on moderate short-term effects and integrates to produce a linear price trend, without producing large bubbles or justified fundamental value. the lppls model has been applied to bitcoin bubbles (eg, [35-37]). in particular, in [36] it was stated that the fundamental value of bitcoin is zero. furthermore, explosive unit roots have been detected in the value of bitcoin (for example, [37-39]). these tests can identify bubbles, insofar as the bubbles can be explained by a constant slightly exploding unit root, while perhaps also allowing for a linear logarithmic trend, but they are not specific [40] and have limited descriptive and predictive power.

Here, we combine, as a fundamental measure, a generalized metcalfe’s law and, as a technical measure, the lppls model, to diagnose bubbles in bitcoin. when both measures coincide, this provides a convincing indication of a bubble and an impending correction. if, in hindsight, such signals are followed by a correction similar to the one suggested, they provide compelling evidence that a bubble and crash did indeed occur.

This document is organized as follows. In the first part, we document a generalized metcalfe’s law that describes the growth of the population of active bitcoin users. We show that generalized metcalfe’s law provides a level of support, and that the relationship between market capitalization and “metcalfe value” provides a relative valuation relationship. On this basis, we identify a substantial but not unprecedented current overvaluation in the price of bitcoin. In the second part of the article, we discover a universal superexponential bubble signature in four bitcoin bubbles, which corresponds to the lppls model with a reasonable range of parameters. the lppls model is shown to provide early warning, in particular with confidence intervals for the critical time of explosion based on profile probability. An pppls fitting algorithm is presented, which allows the bubble start time to be selected and provides an interval for the shock time, in a probabilistically sound manner. We conclude the article with a brief discussion.

### 2. fundamental value of bitcoin: active users and metcalfe’s law generalized

metcalfe’s law establishes that the value, in this case the market capitalization (cap), of a network is where *u* is denominated the number of active users, imperfectly quantified by a proxy, being the number of active users addresses.5 is a single factor model for a fundamental valuation of bitcoin, and plausibly for other cryptocurrencies. From Figure 1, we actually see a surprisingly clear logarithmic linear relationship. Instead of taking Metcalfe’s law for granted, we estimate the relevant parameters using a logarithmic linear regression model, which we refer to as (generalized) Metcalfe’s law,

figure 1. (*a*) scatter plot of bitcoin market capitalization versus the number of active users, with logarithmic scales. the dots get darker as time progresses, and the last three crashes are indicated by colored dots and arrows indicating the size of the fix. the generalized metcalfe regression is given in solid red, and with the slope constrained to be 2 given by the red dashed line. (*b*) active users (rough black line), again in logarithmic scale, as a function of time, with linear scale insertion. a climbing bitcoin market capitalization overlaps the gray line. the red and yellow dashed lines are the active users nonlinear regression fits, fitting to different time windows.

The result of this adjustment, on 2782 daily values, from July 17, 2010 to February 26, 2018, is a slope *β* = 1.69 (s.e. 0.0076), intersection *α* = 1.51 (0.087), and the coefficient of determination *r*2 = 0.95.6 to force the exponent *β* to be equal a 2 would result in an intercept of −2.01 (0.018), but this regression is significantly worse than the previous one.7 Furthermore, a slope of 2 (or greater) is strongly rejected in moving windows.8 On this basis, it seems that the value 2 proposed by metcalfe is too large, at least for the bitcoin ecosystem.9

It should be noted that this regression seriously violates the assumption that the errors are independent and identically distributed, as there are persistent deviations from the regression line. this statement deserves to be made in more prominent terms: the residues are in fact the bubbles and the shocks! this is the focus of the second part of this document. Ignoring this blatant violation of the so-called Gauss-Markov conditions is well known for giving the false impression of accurate parameter estimates. furthermore, endogeneity is an issue, as the number of active users can determine long-term market capitalization, but large fluctuations in market capitalization can also trigger fluctuations in active users on shorter time scales (figure 1). we address this by smoothing out active users,10 assuming this will average out the short-term feedback effects of market capitalization on active users. a multiplier effect is also a plausible consequence of this endogeneity: a jump in user activity causes an increase in market cap, which triggers a (smaller) jump in user activity, feeding back market cap, etc. therefore, we do not intend to isolate the effect of a single increase in active users on market capitalization and not need it. Finally, we omit formal tests of causality, given the plausibility of the general mechanism behind Metcalfe’s law, as well as the very turbulent and long-term adherence to it.11

In view of these limitations, Metcalfe’s law generalized here is still quite impressive and will prove to be very useful, despite its radical simplicity and uncertain parameter values. Of course, other variables can be added to the regression, which further characterize the network, such as the degree of centralization, transaction costs, volume, etc. however, actual volume (value of authentic transactions), for example, is not only difficult to know, but, in general financial markets, is known to be highly correlated with volatility, of which bubbles and busts they are the most formidable contributors and thus may be too endogenous to clearly indicate a core value. therefore, the ‘active users’ variable is kept as the focal quantity.

Regarding Figure 1, a clear and important feature is the declining growth rate of active users, which we modeled using a relatively flexible ecological nonlinear regression that saturates at a ‘carrying capacity’ , *u* → *e**a* as *t* → ∞, and where the logarithmic transformation stabilizes the noise level. As in the case of the generalized Metcalfe regression, there is a clear structure in the residuals here, as feedback loops develop between the number of active users and price during speculative bubbles. We choose to fit the curve (2.3) by OLS (ordinary least squares) and treat it as a rough estimate: adjusting from January 1, 2012 to February 26, 2018,12 the annual growth rate is expected to decline over the next 5 years from 35 to 21%, bringing the expected level of active users from 0.79 million today to 2.60 million in 2023 with standard errors of 5 and 8%, respectively. compared to an adjustment that started earlier on October 24, 2010,13 a similar declining growth rate is again confirmed, but with predictions for 2018 and 2023, respectively, being 7% and 28% higher than predictions for the first fit. more generally, within the sample, the fitted curves are similar, but, beyond the sample, the differences explode in such a way that there is a difference of 4 orders of magnitude between the predicted carrying capacities. here, the uncertainty of the model dominates the uncertainty of the estimated parameters. There is also likely to be some non-stationarity and regime shifts as the bitcoin network evolves and matures, contributing another level of uncertainty in the long-term extrapolation of our models. therefore, accurate inference based on a single model, particularly the omission of any limitations imposed by the physical bitcoin network, is misleading and long-term predictions are meaningless. however, smoothing past values is not problematic and short-term projections may be reasonable.

given the number of active users and calibrations of generalized metcalfe’s law, which maps to market capitalization, we can now compare the predicted market capitalization with the true one, as in figure 2. additionally, using users smoothed assets, the local endogeneities, where price drives active users, are assumed to average out. the regression estimated by ols, by definition, fits the conditional mean, as shown in figure 2. thus, if bitcoin has evolved based on fundamental user growth with transient overvaluations at the top, then the estimate of ols will give an intermediate estimate and therefore above the fundamental value. for this reason, support lines are also given and, although their parameters are chosen visually, they can give a stronger indication of the fundamental value. in any case, the predicted values for market capitalization indicate a current overvaluation of at least four times. in particular, the ols fit with parameters (1.51,1.69), the support line with (0,1.75) and the metcalfe support line (−3,2) suggest current values around 44, 22 and 33 billion of dollars, respectively, in contrast to the current current market capitalization of 170 billion dollars. Furthermore, assuming continued user growth in line with active user regression from 2012, Metcalfe’s end-2018 predictions for market capitalization are $77, $39, and $64 billion, respectively,14 which still it is less than half of the current market. Cap. these results are found to be robust with respect to the chosen fitting window.15

Figure 2. Comparing Bitcoin market cap (black line) with predicted market cap based on various generalized Metcalfe regressions of active users. The rough red line is given by plugging the true number of active users into the generalized Metcalfe regression shown in figure 1, having OLS estimated coefficients (*α*, *β*) = (1.51, 1.69). The remaining lines plug smoothed active users (non-parametric up to 2018 and the nonlinear regression starting in 2012 to project beyond) into the generalized Metcalfe formula with different parameters: the smooth green line for the estimated coefficients (1.51, 1.69); the orange dashed line is proposed as a ‘support line’, having coefficients (0, 1.75) specified by eye; the blue dashed-dotted line being a Metcalfe support line with coefficients (−3,2). The lower inset plot with grey line is the price per bitcoin in USD.

On this basis alone, the current market looks similar to that of early 2014, which was followed by a year of sideways and downward movements. there would have to be some separate fundamental development to justify such a high rating, which we are not aware of.

### 3. bitcoin bubbles: universality of unsustainable growth?

#### 3.1. identification and main properties of the four main bubbles

Using the generalized metcalfe regression on smoothed active users, as well as their support lines, four main bubbles can be identified in Figure 2 that correspond to the largest upward deviations of the market capitalization from this estimated fundamental value. These four bubbles in market capitalization are highlighted in Figure 3 and detailed in Table 1; in some cases they show a 20-fold increase in less than six months! in all cases, the bursting of the bubble is attributed to fundamental events, listed below, in particular for the first three bubbles, which quickly corrected at the time of the clearly relevant news. the fourth and very recent bubble was much longer, and it is plausible that the headline news was actually the $20,000 value of bitcoin, i.e. it eventually collapsed under its own weight.16 market participants often lament that crashes they are unpredictable due to the unpredictability of bad news.

Figure 3. Upper triangle: market cap of Bitcoin with four major bubbles indicated by bold coloured lines, numbered, and with bursting date given. Lower triangle: the four bubbles scaled to have the same log-height and length, with the same colour coding as the upper, and with pure hyperbolic power law and LPPLS models fitted to the average of the four scaled bubbles, given in dashed and solid black, respectively.

However, focusing on the news that may have triggered the accident is like waiting for “the last straw”, instead of monitoring the unsustainable load that is developing on the poor camel’s back. Of particular interest here is that, although the height and length of the bubbles vary considerably, when scaled to have the same logarithm of height and length, nearly universal superexponential growth is evident, as diagnosed by the general upward curvature in this linear-logarithmic plot (lower figure 3). and in this sense, like a mound of sand, once the scaly bubble gets steep enough (angle of repose), it will precipitate, while the precise trigger push is essentially irrelevant.

listed below17 are the events believed to cause bugs/fixes, corresponding to bubbles 1 to 4 in table 1:

June 19, 201118: mt. gox was hacked, causing the bitcoin price to drop 88% over the next three months.

August 28, 2012: ponzi fraud of perhaps hundreds of thousands of bitcoins under the name of bitcoin savings and trust; charges filed by the securities and exchange commission.

April 10, 2013: major bitcoin exchange, mt. gox, breaks under high trading volume; the price falls more than 50% in the next 2 days.

December 5, 2013 – Following Strong Adoption Growth In China, People’s Bank Of China Bans Financial Institutions From Using Bitcoin; bitcoin market capitalization falls 50% in the next two weeks. Feb 7, 2014 – Operational issues on major exchanges due to distributed denial of service attacks, and two weeks later mt. gox closes.

December 28, 2017: Reports that South Korean regulators were threatening to shut down cryptocurrency exchanges.

#### 3.2. log-periodic finite-time singularity model

following sornette et al [30,33,42], as mentioned in the introduction, we consider bubbles to be the result of unsustainable growth (faster than exponential), achieving infinite return in finite time ( a finite time). temporal singularity), forcing a correction/regime change in the real world. we adopt the lppls model, parameterized in [43], for the log market capitalization, *p**i* at time *t**i *, where 0 < *m* < 1, ln(*p**c*) = *a* and *t*1 ≤ *t*me < *t**c*. *t*1 is the start time, and *t**c* the end or so-called critical time at which the bubble should burst. this model combines two well-documented empirical and phenomenological features of bubbles: (1) a transient ‘faster than exponential’ growth with singularity at *t**c*, modeled by a law of pure hyperbolic power (the above equation with *c* = *d* = 0), resulting from positive feedbacks, which is (2) decorated with accelerated periodic volatility fluctuations, incorporating spirals of fear and clashing expectations.

See also: Reports show scammers cashing in on crypto craze | Federal Trade Commission

the model must fit the data ((*y*1, *t*1), …, (*y**n*, *t**n*)), in a window (*t*1, *t*2), where *t*1 ≤ *t*1 < · · · < *t**n* ≤ *t*2 < *t**c*. the window (*t*1, *t*2) must be specified, and the selection of the start of the bubble *t*1 is usually less obvious. as is typical in time series regression [44], the *ε**i* errors are correlated and can have a changing variance (heteroskedasticity), which if ignored leads to suboptimal errors. estimates and confidence intervals too small (too optimistic). in this case, generalized least squares (gls) provides a conventional solution, which has been used with pppls [45-47] and, if well specified, has optimal properties. here, we opt for a simple specification of the error model, being autoregressive of order 1.19 to model fairly persistent deviations from the general trend. then we estimate the lppls model by profiling nonlinear parameters (*m*, *w*, *t**c*, * ϕ*), which allows conditionally linear parameters (*a*, *b*, *c*, *d*) to be estimated analytically, by gls, or by ols if *ϕ* = 0. assuming white noise normal errors *η**i*, this is the maximum likelihood, and allows profile probability confidence intervals for all parameters.

Here, we focus on *t**c*, the critical time when the bubble is most likely to burst. Before accounting for metcalfe’s fundamental value, and to provide a curve for comparison with the data in Figure 3, we fit the pure hyperbolic power law (obtained by putting *c* = *d * = 0 in (3.1)) and the pppls model to the average of the four scaled bubbles,20 with the results summarized in Table 2. The hyperbolic power law and pppls fits provide a similar trend, and the critical / breakout time hugs the lower bound of 1.01 (true peak is by construction at 1).

perhaps curiously, despite fitting to an average of disparate non-synchronized bubbles with similar overall trajectories, the pppls fit is significantly better, based on log-likelihood (*p* < 10−5) as it captures some of the persistent fluctuations and allows for significantly less *ϕ*, i.e. memory time reduction ∼1/(1− *ϕ*) of the residuals by a factor of 13.21

#### 3.3. bubbles in the labor market-quality relationship

Given our proposed fundamental value of bitcoin based on the generalized metcalfe regressions presented above, we define the ratio of market value to metcalfe value (mmv) as the actual market capitalization (*p**i* at time *t**i*) divided by the market capitalization predicted by the metcalfe support level, with parameters (* α*0 = −3, *β*0 = 2) in (2.1), with smoothed active users (*u**i*) connected .22 we sampled the value every 3 h during the time periods corresponding to bubbles 1-3 and 5 in table 1.

as shown in figure 4, the bubbles are persistent deviations of the mmv above support level 1, which are well modeled by the lppls model. In particular, the hyperbolic power law parameters and the fitted lppls models on the mmv relationship data, for the full bubble lengths, are given in Table 4. For the different bubbles, the key nonlinear parameters lie within of similar ranges, and the calibration of *t**c* is exact. again, the pppls fits dominate the pure hyperbolic power laws, according to likelihood ratios.

Figure 4. (*a*) MMV ratio over time. The apparent bubbles, which radically depart from the fundamental level 1, are coloured and given in table 1 as bubbles 1-3 and 5. (*b*) For the same four bubbles, the MMV ratios are shown in log-scale as a function of linear rescaled time, with 0.25 vertical offset for visibility. The hyperbolic power-law and LPPLS fits on the four full bubbles are shown. Values of the MMV ratio after the bubble peak are shown on the grey background, where the coloured vertical lines indicate the upper limit for *t**c* of the 95% profile likelihood confidence interval for each of the four bubbles. The three thin vertical black lines gives the rightmost edge of the 95, 97.5 and 100% data windows on which fits were done, with parameters summarized in table 4 and appendix table 5.

In addition to backchecking the lppls bubble, one would like to have a strong advance warning of the bubble’s existence and a reasonable confidence interval for the burst time. First, we provide a simple indication of this potential with two additional sets of fits for each bubble: fit with bubble data up to 95% and 97.5% of the bubble length. overall parameter estimates (see Appendix Table 5) are similar to the 100% window of Table 4, with key nonlinear parameters typically in ranges of 0.1 < *m* < 0.5 and 7 < *w* < 11. Focusing on the critical burst time, Table 3 gives the estimated *t**c* and 95% confidence intervals, showing a fairly stable early warning . that is, the point estimates and confidence intervals are consistent with the actual explosion time, taking into account that *t**c* is, in theory, both the most probable time as the last for the bubble burst. [30,33,42], since the market is increasingly susceptible as it approaches *t**c* and therefore can be affected by bad news.

Next, a more extensive demonstration of predictability is made for the case of the recent large bubble, summarized in Figure 5. at each *t*2 (from 1 year before the inflection point , here specified from December 17, 2017, until two weeks later) a critical weather forecast is made and the confidence interval is calculated. furthermore, for each *t*2, a range of bubble start times is considered, *t*1 (from 360 to 250 days before the inflection point), and the combined time critical times are estimated.23 As summarized in the figure, a fairly consistent parenthesis of the critical time achieved is obtained. moreover, the uncertainty is reduced and generates a strong alarm about two weeks before the possible turning point. the mean square error of the regression increases by 50% within two weeks (around the gray vertical line in the figure) after the turning point is made, providing a simple statistical indication of the end of the bubble regime.

Figure 5. In the lower panel is a plot of the Bitcoin MMV ratio, from 1 year prior to the turning point (here defined as 17 December 2017, and called *t***c*) to a few weeks afterwards. An exemplary LPPL fit is given (orange line), and Bitcoin price values are labelled at some points. In the upper panel are the aggregated 99.5% confidence intervals for the estimate of the critical time corresponding to each time, *T*2. The intervals are shifted such that the origin is the time of the realized tipping point, and the diagonal line defines the lower bound for the prediction (i.e. *t**c* ≥ *T*2).

### 4. discussion

In this paper, we have combined a generalized metcalfe’s law, which provides an approximate fundamental value based on network characteristics, with the lppls model, to develop a comprehensive diagnosis of the bubbles and their collapses that have marked history of the cryptocurrency. In doing so, we were able to diagnose four distinct bubbles, being periods of high overvaluation and LPPL-like trajectories, which were followed by declines or sharp corrections. although the height and length of the bubbles vary substantially, when scaled to the same logarithm of height and length, almost universal superexponential growth is documented. this is in stark contrast to the view that (crypto) markets follow a random walk and are essentially unpredictable.

Furthermore, in addition to being able to identify bubbles in hindsight, given the consistent characteristics of lppls bubbles and demonstrated early warning potential, lppls can be used to provide ex-ante predictions. for example, a reasonable confidence interval for the critical moment indicates a high risk of a correction in that neighborhood, as any minor event could topple the volatile market. the success of such an approach was shown for the big bubble of 2017. false positives and misses are of course possible, but somewhat ambiguous in view of the limited number of big bubbles, and also depend on the specific decision rule being used (possibly including human judgment). furthermore, massive exogenous shocks, although rare, could occur at any time, and the lppls model cannot provide any warning there.

Focusing on the outlook for bitcoin, active user data indicates a declining growth rate, which a number of parameterizations of our generalized metcalfe’s law translate to slowing market cap growth. furthermore, our metcalfe-based analysis indicates current support levels for the bitcoin market in the range of $22-44 billion, at least a factor of four less than the current level. On this basis alone, the current market resembles that of early 2014, which was followed by a year of sideways and downward movements. Given the high correlation of cryptocurrencies, short-term movements in other cryptocurrencies are likely to be affected by corrections in bitcoin (and vice versa), regardless of their own relative valuations.

### data accessibility

all data used is openly available, with relevant sources mentioned in the text. data is also provided in the dryad digital repository: https://doi.org/10.5061/dryad.22k10nd [50].

### author contributions

all authors contributed to the design, analysis and writing, with authorship according to relative contribution.

### conflicting interests

We declare that we have no competing interests.

### financing

We did not receive funding for this study.

See also: The Game Theory of Cryptocurrency