The present essay addresses those elusive patterns in the medical data that distinguish between the healthy and pathological and how such patterns might be pinned down in order to describe the properties of complex medical phenomena. From a brief historical review it will become apparent that the working assumption that the statistics of physiologic time series are Normal, an assumption made in the nineteenth century, has become the implicit and often the explicit foundation for much of the formal theory of medicine. Here we motivate abandoning this assumption and initiate the exploration of a theory entailed by formulating a new hypothesis based on the insights into complexity made in the past few decades.

Complexity was first dealt with in the rapidly developing physics of the eighteenth century by recognizing that no experiment ever gives the same result twice. No matter how skilled the experimenter, how carefully the experiment is prepared, how precise the instruments used, an experiment repeated N times and prepared in “exactly” the same way each time will yield N different numbers. Scientists in the seventeenth and eighteenth centuries struggled with the best way to characterize the scatter in the data associated with this ensemble of measurements from ostensibly the same experiment, see for example West and Grigolini (2011) for a somewhat more extended discussion.

A breakthrough in empiricism was made at the end of the eighteenth century by introducing the arithmetic average of N measurements X₁, X₂, …, X_N:

as “the” way to characterize an ensemble of measurements. This was accompanied by introducing an arbitrary measure of the quality of the average as a way to characterize the scattered data, that being the deviation of the second moment from the square of the average:

the variance, or its square-root the standard deviation σ. This treatment of data was published independently in 1809 by the German polymath Johann Carl Friedrich (Gauss, 1809) and the American mathematician (Adrian, 1809) both of whom provided the bell-shaped curve as the representation of the Normal distribution of statistical variability. This approach evolved into the Law of Frequency of Errors as a consequence of the proof of the Central Limit Theorem provided a year later by Laplace (1810).

The Law of Errors is an interesting name for the variability in experimental data since it implies that the average is the “proper” description of the data and deviations from that value constitute errors. Consequently the Law of Errors and the Normal distribution are consistent with the mechanical laws of motion formulated by Newton. These laws of motion predict certain outcomes for experiments; the measurements of those outcomes reveal a variation about the predicted value and the Normal distribution gives the degree of variability in the measurements about the predicted or average value. The narrower the bell-shaped curve the closer the measured value is to the predicted one. Put simply, the Normal distribution implies that there is a right answer to the question being experimentally asked. Thus, even through no theory of medicine existed at the time the average value was assumed to provide the predication of that theory if and when it would be formulated.

Many phenomena are described by the Normal distribution or Normal statistics, but it is often forgotten that the measurement error in such phenomena must satisfy the four criteria of the Central Limit Theorem in order to be Normally distributed. Expressed in the language of the Law of Errors these criteria are: (1) the errors are independent; the error in a given experiment does not depend on the error in any other experiment; (2) the errors are additive; the total error made is the sum of the separate errors; (3) the statistics of each error is the same; the statistical process producing the error does not change from experiment to experiment; and (4) the width of the distribution is finite; the standard deviation converges to a finite value as the number of measured errors increases. These criteria are recorded here because complex medical phenomena often violate one or more of these conditions necessary to satisfy the mathematical proof of the Central Limit Theorem and it is here that the real world deviates from the expected variability imposed by Normal statistics (West, 2006).

There is a wide array of medical phenomena that manifest Normal statistics including height (Quetelet, 1835), birth weight (O'Cathain et al., 2002), body temperature (Mackowiak et al., 1992), and the logarithm of blink rate (Bentivoglio et al., 1997). On the other hand, there is an even greater list of non-Normal statistical medical phenomena including heart rate variability (Peng et al., 1993), neuronal avalanches (Plenz, 2012), interbreath and interstride interval variability (West, 2006), to name a few. It is the latter physiologic phenomena that we are interested in understanding and this requires the study of how the statistical properties modify our understanding. Of particular importance is our understanding of extrema and how these extrema change with increasing variability of the underlying statistics.

Recently the identification of emergent phenomena across multiple disciplines, from the swarming of insects (Yates et al., 2009), the schooling of fish (Katz et al., 2011) and the flocking of birds (Cavagna et al., 2010) observed in animal groups by naturalists; to the spatiotemporal activity of the brain (Beggs and Plenz, 2003, Fraiman et al., 2009, Chialvo, 2010) observed by neurophysiologists; to the collective and cooperative behavior observed in social groups studied by psychologists and sociologists; all demonstrate collective behavior reminiscent of particle dynamics near the critical phase transitions studied by physicists (Stanley, 1971). Each of these disciplines has demonstrated the need to investigate the dynamics of complex networks across scales in order to develop a deeper understanding of how large-scale behavior emerges from microscale dynamics and the sensitivity of the observed behavior to those dynamics.

Of particular interest to us here are the medical fields in which we observe a need for a system wide approach (Richardson and Goldstein, 2010). The recent discoveries in medicine were propelled by the successes of molecular biology and genetics that have made available genomic blueprints of numerous organisms, which are complemented by extensive experimental data describing cell functions. At the same time however the realization came that biological function emerges out of the interaction of numerous molecular components, making the detailed knowledge of specific components at any level of organization insufficient to capture macroscopic functionality. One example of this limitation that we return to subsequently is the study of the cardiovascular systems in order to understand, predict and ultimately modify (in order to heal) cardiac function.

Despite experimental developments, the ability of science to make theoretical predictions of the behavior of complex networks is still in its infancy. The adoption of methods from non-equilibrium statistical physics have demonstrated limitations, resulting from the fact that living networks, in contrast to inert physical materials, are extremely heterogeneous, non-generic, highly specialized and operate far from an equilibrium state (Elsasser, 1981). We hypothesize that “The Network Effect” (TNE) is to impose a level of complexity that eludes analytic dynamic descriptions based on systems of ordinary and/or stochastic differential equations, as well as the equivalent partial differential equations describing the phase space evolution of probability density functions (PDF's).

Herein we demonstrate that what was for a very long time a niche branch of mathematics, the fractional calculus, might very well be able to span the gap between the inert materials of physics and the living networks of medicine. Consequently TNE may well be summarized as the need for a system of fractional differential equations to describe the dynamics of complex networks. The support for this hypothesis is primarily empirical as we show, with the exception of the connection established by West et al. (2014) for a model non-linear dynamic network.

Although developed along side the classical calculus, fractional differential equations have only recently been shown to be a convenient way to describe the dynamics of complex phenomena characterized by long-term memory and spatial heterogeneity (Podlubny, 1999, West et al., 2003). Fractional differential equations have been demonstrated to capture the time evolution of fractal processes, such as in anomalous diffusion, viscoelasticity and turbulent fluid flow, as reviewed by West and Grigolini (2011). In spite of the success of the mathematical descriptions of such processes there has been a lack of identification and interpretation of mechanisms that entail fractional dynamic equations in the context of complex physiological networks. Herein we suggest how this barrier might be either overcome or circumvented.

An extreme event may be thought of as the occurrence of an incident which in some phenomenon exhibits itself outside the typical region of fluctuation as measured by some appropriately chosen variable—wind gust loads on airplanes in flight, the highest temperatures or lowest pressures in meteorology, floods, and droughts in hydrology, and human life spans, all fall in this category. A perhaps even longer list of phenomena of medical importance, including heart attacks, the falling of the elderly, epileptic seizures, traumatic brain injury, could be drawn up. To be adequately prepared for such occurrences knowledge of the underlying statistical behavior of such events must be in hand.

In 1935 Emil Gumbel derived an expression for a PDF of the maxima in data sets (Gumbel, 1954, Reiss and Thomas, 1997)

where μ is related to the mean and χ to the standard deviation of the extrema variable x. In the original derivation of this cumulative distribution the variable was the oldest ages in human life spans. Subsequently, the Gumbel PDF was applied to other extreme events, such as those cited above. The derivation of the PDF was based on the two assumptions: (1) given a random variable X, such as the height of a river or the magnitude of an earthquake, the successive measurements are statistically independent of one another; (2) the statistics are stationary in time, that is, the PDF is independent of an overall shift in time. The domain of attraction of this PDF encompasses most of the commonly used distributions, that is, PDF's for which there exists a mean and standard deviation such that as the number of data points becomes arbitrarily large, the limit of the extrema PDF approaches the Gumbel form.

Figure 1 compares the Normal and the Laplace (1810) PDF's to that of Gumbel as well as to a second type of extreme value PDF due to Fréchet (1927). Note that a PDF is obtained from the negative derivative of the probability so that for the Gumbel PDF we obtain from Equation (3)

It is evident from the figure where Equation (4) is plotted that the large excursion of the variate from the central values dominate the extrema PDF's.

Of particular interest to us here are the extreme properties of fluctuations in medical observables. If the maximum fluctuation in body temperature doubles an individual may be in some distress but if the maximum interval between heart beats double the pathology may be life threatening. Part of the reason for this difference can be traced to the changes in the behavior of extrema properties for a Normal statistical process and non-Normal processes, say one described by Pareto, i.e., IPL statistics. Extrema in stock market fluctuations, Taleb's black swans Taleb (2007), generated by an underlying Pareto distribution are unpredictable and have a distribution that does not approach the Gumbel form, but does converge on the Fréchet form. On the other hand, these extrema could also be Sornette's dragon kings (Sornette, 2009) when they are transiently organized into extreme events that are statistically or perhaps mechanically different from those in the underlying distribution that produces the smaller values of the fluctuations. An example of the latter mechanism was worked out in detail by Montroll and Shlesinger (1984) who leverage the moments of a distribution with finite central moments using a renormalization group argument to generate the Pareto tail in the income PDF's for Western society independently of the initial distribution. We present this argument subsequently.

The lack of interchangeability of black swans and dragon kings in the interpretation of medical pathologies has not yet been addressed explicitly, as we subsequently discuss. This distinction becomes important in a medical context given the recent progress by de Souza Cavalcante et al. (2013) in real time forecasting of an impending extreme event (“dragon king”), but more importantly that it is demonstrably possible to perturb the system to suppress the onset of the extrema for certain chaotic mechanisms.

In Section 2 the idea is developed that the PDF's generated by complex phenomena, particularly those in medicine, require a new method of quantification; one based on fractal statistics, self-similarity, and the fractional calculus. Some renormalization group scaling ideas are taken from Zaslavsky (2002) but other sources for the mathematical infrastructure developed are also discussed. The renormalization group properties of the statistical distribution are discussed, including the existence of complex fractal dimensions that explain the empirical harmonic modulation of the scaling observed in some data. Thus, information in fractal phenomena is coupled across multiple scales, as for example, observed in the architecture of the mammalian lung (West et al., 1986, Nelson et al., 1990, Weibel, 2000); manifest in the long-range correlations in human gait (Hausdorff et al., 1996, West, 1999); measured in the human cardiovascular network (Peng et al., 1993) and observed in a number of other physiologic contexts (West et al., 2008). These all appear to be consequences of TNE.

Fractal processes are dynamically rich in interconnected scales with no one scale or set of scales dominating. The solutions to the fractional equations of motion for a category of such processes are given in Section 3 using scaling arguments. The alpha-stable Lévy distribution is one such solution that has been suggested to describe the statistics of heart rate variability (HRV) of both healthy and diseased individuals (Peng et al., 1993). A discussion of these statistics is given in Section 4 where the fractional calculus is shown to provide a truncated Lévy distribution to describe healthy individuals. It is argued that the truncation is determined by a physiological control process that is suppressed in individuals that suffer a cardiac induced death. Extreme value theory is used to distinguish between black swans and dragon kings using the length of time one would wait before the reoccurrence of an event of a given magnitude. This might translate into how long one can survive with a critical illness given the time one has already survived with that illness.

In Section 5 we present some conclusions based on the application of the fractional calculus and scaling to physiological time series and discuss the implications of the TNE hypothesis.

In this section we introduce the kinetic equation for the evolution of the probability P(x, t)dx of a dynamic process X(t) along a fractal trajectory having a value in the interval (x, x + dx) at time t. Zaslavsky (2002) provides an excellent description of the relation between chaotic trajectories generated by non-linear equations of motion and the fractional kinetics of the PDF. He considers the infinitesimal changes of P(x, t) in time along chaotic trajectories whose local averages yield a FKE, which is to say an equation of motion for the PDF in phase space in terms of fractional derivatives. There are a number of alternative derivations of the FKE including the continuous time random walk (CTRW) of Montroll and Weiss (1965) as reviewed by Metzler and Klafter (2000); the extension of the CTRW using subordination (Gorenflo et al., 2007); as well as the fractional generalized Langevin equation, see for example, Lutz (2012), West and Grigolini (2011). One of the simplest form of the FKE is

and the non-integer parameters α and β are scaling exponents that characterize the fractal structure of the trajectories in Zaslavsky's approach and consequently yield fractional derivatives in time and space, respectively. Of course it is also possible that the diffusion coefficient is dependent on t and/or x, but we do not consider these cases here and instead refer the reader back to the literature (Zaslavsky, 2002, Klafter et al., 2012). It remains for us to interpret the symbols indicating the fractional derivatives in time and in space, which we do in Section 3.

Equation (5) is sometimes called the fractional Fokker-Planck equation (FFPE) with zero potential because it can be generalized by introducing a potential function in complete analogy with the historical Fokker-Planck equation. When α = 1 Equation (5) reduces to the anomalous diffusion equation (Seshadri and West, 1982, Metzler and Klafter, 2000, West et al., 2003)

Here the anomaly arises from the heterogeneity in the phase space variable captured through the non-local character of the spatial fractional derivative. It is not necessary to review the fractional calculus in order to understand the solutions to Equation (5) or (6); the understanding required for our purposes can be achieved from their scaling properties.

Zaslavsky (2002) applied a renormalization group (RG) transformation to the system dynamics such that the scaling properties of the incremental changes in space and time are

which apply after some averaging in a restricted space-time domain and (λ_x, λ_T) are scaling parameters. He goes on to say that a basic feature of renormalization group kinetics is that the FKE is invariant under the operation of a renormalization group transformation R{·}:

implying that the FKE satisfies the scaling behavior

The scaling results from the fractional powers of the differentials in the basic definition of the fractional derivative.

This renormalization procedure may be applied an arbitrary number of times to the FKE and consequently the resulting fractional differential equation remains valid only if the ratio of the scaling parameters satisfies

The FKE is linear so that the sum of the individual fixed-point solutions is also a solution. In this way the fixed point equation Equation (10) has an infinite number of solutions

where the fixed point solutions have integer n. Consequently, indexing the time parameter with the fixed point index the ratio of the time to space parameters becomes

Indexing the time parameter by n is arbitrary and is only intended to distinguish among the various fixed point solutions to the FKE.

In Section 3 we show that the solution to Equation (5) has the general scaling form

using the zero-order solution to Equation (12). Consequently the first moment of the dynamic variable takes the form

where the coefficient can be calculated to be Zaslavsky (2002)

However, if we include the higher-order fixed point solutions from Equation (12) into the average we obtain

resulting in periodic variations in the average value, with variations in ln t, the logarithm of time for time series data, having period ln λ_T.

The expansion coefficients A_n in Equation (16) can be explicitly calculated when the kinetic equation of motion has a discrete renormalization invariance, see, for example, Hanson et al. (1985), Montroll and Shlesinger (1984), West et al. (1986). The lowest-order coefficient is given by A₀ = D⁽⁰⁾_αβ. On the other hand, these coefficients can also be used heuristically as expansion coefficients much like what is done with empirical Fourier series fitting the expansion coefficients to time series data.

It is important to note that the concept of fractal dimension implies that the underlying process is self-similar, but the fact that the dimension is complex implies even richer properties of the process. The real part of the complex dimension yields the inverse power-law index for the Pareto distribution. The time series data is seen to have bursts of activity followed by long quiescent intervals during which no event occurs. However, when one of these bursts is examined with higher resolution it is seen to be composed of bursts separated by relatively long quiescent intervals. The relative amount of bursting and resting is apportioned by the scaling index. The imaginary part of the dimension indicates that there is a memory superposed on the Pareto statistics producing log-periodic modulations of the moments. It is useful to understand that this apparently bizarre phenomenon is not new.

Over 40 years ago Novikov (1966) in his investigation of the properties of turbulent fluid flow discovered the periodically modulated scaling properties of intermittent stochastic processes. He considered a general Poisson process supplemented by “nested” pulses of activity. The power spectral density function was consequently determined to satisfy a scaling relation of the renormalization group form and the predicted spectrum was a modulated IPL. Novikov's prediction regarding the log-periodicity in turbulence has subsequently been observed by Zhou and Sornette (2002). Thus, whether the functional form of Equation (16) arises in the study of the moments of a processes, or in its spectrum, it indicates a process that is void of a characteristic scale and has long-range correlations induced by nested bursts of activity as also pointed out by West and Fan (1993).

The log-periodic variability of an observable average was first discussed in a physiologic context by Shlesinger and West (1991) to describe the scaling behavior of the bronchial airway network of the mammalian lung. In the application to physiology the solution Equation (16) takes the heuristic form

where 〈x; t〉 is the average diameter of the bronchial airway at generation t, A₀ and A₁ are empirical constants, and the generation t denotes the number of branchings of the bronchial tree starting from the trachea t = 0. Consequently, a result of TNE on the bronchial network is that it is characterized by having a complex fractal dimension

whose imaginary part produces the modulation of the dominant IPL behavior of the average diameter. This periodic modulation is clearly seen in Figure 2 for four distinct species, human, dog, rat, and hamster for all generations and is apparently a general property of mammalian species and is called the fractal lung model (West et al., 1986).

The log-periodic modulation of the IPL behavior of the average diameter in the bronchial airway network is only one of many such phenomenological regularities observed in physiology. Another is cerebral blood flow (CBF) velocity measured using transcranial Doppler ultrasonography, which is not strictly constant. West et al. (1999) use the dimensionless relative dispersion, the ratio of the standard deviation to the mean, to show by systematically aggregating the data that the correlations in the beat-to-beat CBF time series data is the modulated IPL depicted in Figure 3. This scaling of the CBF time series indicates the existence of long-time memory in the underlying control process. They argued that allometric control (West and Grigolini, 2010) enables the CBF to maintain a relatively constant perfusion.

The CBF time series for 2 h of data from each of six subjects is processed to obtain the relative dispersions depicted in Figure 3. The processing procedure is to first calculate the relative dispersion using all the data as indicated by the zero point on the horizontal axis in the figure. Next each of the neighboring data points is added together to obtain half the original number of data points and the relative dispersion is calculated again. This is the 1 indicated on the horizontal axis in the figure. These data points are aggregated again in the same way and the relative dispersion is calculated a third time to obtain point 2. This process is repeated six more times with the data and plotted as shown. For each subject the relative dispersion is successively aggregated in this way and plotted against the size of the aggregation and is seen to yield a modulated IPL PDF. The underlying theory on which this processing technique is based is given by Bassingthwaighte et al. (1994) for fractal time series where the relative dispersion is shown to satisfy a renormalization group relation.

These are only two examples of physiologic phenomena with complex fractal dimensions that are revealed through modulation of the processed time series data as a consequence of the TNE. Sornette (1998) also emphasized the log-periodic corrections to scaling produced by the imaginary part of the complex fractal dimension in his extensive discussion of the concept of discrete scale invariance. He Sornette and Johansen (1997) implemented the renormalization group idea to postdict stock market crashes, that is to “predict” historical stock market crashes, fitting the log-periodic modulation of the solution to the renormalization group relation to historical data. This distinctive modulation is obtained using the Dow Jones time series financial data for the United States stock market crash of 1929 depicted in Figure 4. The solid line segment is the RG solution fit to the data. Predicting the occurrence of a crash from historical stock market time series subsequently became equivalent to predicting the arrival of a dragon king (Sornette and Johansen, 1997).

This analysis suggest the possible use of extrema in the prediction of mortality as we subsequently discuss.

The Fourier transform of the symmetric Reisz-Feller operator ∂^β_|x|[·] acting on an analytic function f(x) is as shown in Appendix 6.2 (Seshadri and West, 1982, Metzler and Klafter, 2000):

Uchaikin (2000) inverse transformed Equation (22) for arbitrary α and β to obtain the analytic form for the PDF; but that level of mathematical detail is not necessary for the present discussion. The desired insight is provided by utilizing the scaling properties of the characteristic function Equation (23) and considering the PDF in the form of the inverse Fourier transform

In the previous section we noted that a process described by an alpha-stable Lévy distribution satisfies the general scaling solution to the FKE. More explicitly we note that the Fourier transform of a PDF is the characteristic function for a process. Consequently, the inverse Fourier transform of the characteristic function given by Equation (23) yields the PDF

In Section 3.3 we determined that the solution to the FKE with α = 1 is the alpha-stable Lévy distribution given by Equation (33). It was noted that such distributions have fat or IPL tails that decay more slowly than the typical exponential. Such fat tails can generate diverging variances, which are not plausible for healthy physiologic data. Consequently, it is necessary to find a PDF that behaves as an IPL for intermediate amplitudes but manifests physiologic control to mitigate the occurrence of extreme events such as dragon kings. For this reason we assume that the HRV statistics of healthy individuals are determined by a physiologic feedback mechanism in which the tails of a Lévy distribution are truncated.

We hypothesize physiologic feedback to produce an exponential suppression of very large fluctuations. This exponential decay of large fluctuations can be formally incorporated into the anomalous diffusion equation Equation (6) in the following way

The solution to the modified FKE is again given by means of Fourier transforms. The Fourier transform of the shifted operator is determined by the binomial expansion in Appendix 6.2 resulting in the equation for the characteristic function

Integrating Equation (37) yields the characteristic function after including a term to insure proper normalization of the PDF, that is, P˜(k = 0, t) = 1, and introducing the phase ϕ for the inverse tangent

The inverse Fourier transform of Equation (38) yields the truncated symmetric Lévy PDF

The truncated Lévy PDF was first studied numerically in the context of stock market fluctuations by Mantegan and Stanley (1994) and soon thereafter Koponen (1995) provided a formal derivation of the characteristic function for a truncated Lévy flight. Matshshita et al. (2003) explain that the resulting process is infinitely divisible by scaling x and γ with (K_βt)^1/β. Consequently, in terms of the scaled variables x_s = x/(K_βt)^1/β and the scaled parameter γ_s = γ(K_βt)^1/β the probability that an RR interval occurs in the interval (x, x + dx) is

where L_β(x_s) is the stable Lévy PDF with Lévy index β.

It has been hypothesized that disease is associated with the loss of complexity (Goldberger et al., 1990). This hypothesis has been repeatedly tested using HRV data. For example, a study of combat casualties in an emergency department in Iraq involving 70 acutely injured adults determined that the complexity of HRV dynamics over a range of time scales was lower in high-risk than in low-risk combat casualties (Cancio et al., 2013) using a multiscale entropy (MSE) as a measure of complexity (Costa et al., 2002). In this last reference Costa et al. (2002) show that MSE uses the same coarse graining procedure that was implemented to obtain the relative dispersion in Figure 3. The scaling of the data established that the MSE was higher for healthy subjects than for those with congestive heart failure or with atrial fibrillation. It should be noted however that the loss of complexity only became evident after a certain level of coarse graining was carried out.

Consequently, in contradiction to the central limit theorem these statistical fluctuations do not converge to a Gaussian distribution. Struzik et al. (2008) emphasize an interpretation in which healthy-heart rate represents the upper bound on HRV, and reduced variability of heart rate fluctuations is of clinical risk. They call into question the complexity paradigm and its clinical interpretation. In particular they find that there is an increase in fluctuations and in complexity of heart rate in chronic-heart failure patients, in particular those at risk of death, just as those observed in Figure 7.

A quarter century ago we (Goldberger et al., 1990) proposed that the scaling behavior of the statistics may provide a measure of the complexity of the underlying process. Thus, it would appear from Figure 7 that complexity has increased in those patients that are the more severely diseased rather than the other way around as hypothesized. This interpretation proposes that complexity is proportional to variability and therefore is greatest for a non-ergodic statistical process with diverging central moments. A more thoughtful analysis of complexity reveals something different.

First of all the scaling of the statistics for the central or Lévy part of the PDF is the same for both sets of curves. However, there is the additional scaling of the truncation parameter in the truncated Lévy PDF that reduces the expanse of the fluctuations for those that do not expire by cardiac death. This second scaling, a scaling that would be produced by a control mechanism, certainly adds to the overall complexity of the process. It is the loss of this control that enables the dragon kings of cardiac death. So that although those that expire due to cardiac death have greater variability they do not have greater complexity.

The question of how to distinguish between the extrema generated by the Lévy and the truncated Lévy PDF's can be answered using extreme value theory. In Appendix 6.3 type-I and type-II extrema are discussed. Type-I extrema, those represented by a Gumbel PDF Equation (63), are generated by underlying processes with normal, log-normal, Poisson or Weibull statistics; all of which have a finite variance. Thus, we would expect that extrema generated by the truncated Lévy PDF to be attracted to the type-I PDF, since the exponential asymptotically dominates in the truncated Lévy. On the other hand, type-II extrema, those represented by a Fréchet PDF Equation (67), are generated by process with diverging second moments such as in the case of IPL's and Lévy PDF's. Thus, we hypothesize that black swans are more like type-I extrema and dragon kings are more like type-II extrema. So how readily can we distinguish between the two?

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.