## Abstract

We document and discuss two different modes of evolution across multiple systems, *optimization* and *expansion*. The former suffices in systems whose size and interactions do not change substantially over time, while the latter is a key property of *open-ended* evolution, where new players and interaction types enter the game. We first investigate systems from physics, biology, and engineering and argue that their evolutionary optimization dynamics is the cumulative effect of multiple independent events, or *quakes*, which are uniformly distributed on a logarithmic time scale and produce a decelerating fitness improvement when using the appropriate independent variable. The appropriate independent variable can be physical time for a disordered magnetic system, the number of generations for a bacterial system, or the number of produced units for a particular technological product. We then derive and discuss a simple microscopic theory that explains the nature of the involved optimization processes, and provide simulation results as illustration. Finally, we explore the evolution of human culture and technology, using empirical economic data as a proxy for human fitness. Assuming the overall dynamics is a combined optimization and expansion process, the two processes can be separated and quantified by superimposing the mathematical form of an optimization process on the empirical data and thereby transforming the independent variable. This variable turns out to increase faster than any exponential function of time, a property likely due to strong historical changes in the web of human interactions and to the associated increase in the amount of available knowledge. A microscopic theory for this time dependence remains, however, a challenging open problem.

## 1 Introduction

Evolutionary optimization is a decelerating process that is seen across physical, biological, and sociotechnical systems as well as in engineering process control and machine learning. Its similarity with the relaxation of disordered magnetic materials and with other types of glassy dynamics has been discussed at length in the literature (see, e.g., [22] and references therein). In all cases, frequent reversible dynamical fluctuations are interspersed with increasingly rare irreversible changes of macroscopic observables, a phenomenon known as punctuated equilibria in paleontology [7] and associated with irreversible decrease of free energy in complex physical systems.

In a number of diverse applications, the statistics of these rare events, called *quakes* to distinguish them from pseudostationary fluctuations, can be described by record dynamics [19, 23], which posits that quakes are statistically independent events uniformly distributed on a logarithmic time scale. Quakes drive the dynamics through a sequence of metastable states. As a consequence, macroscopic variables (e.g., fitness, energy, or magnetization) change at a *decelerating* rate. Evolutionary optimization processes are typically characterized by conservation of the number and quality of their components and interactions.

Evolutionary *expansion* is associated with a growing number of fundamental components and interaction types, and the resulting fitness may improve at an *accelerating* rate. The Cambrian explosion [28], which started approximately 600 million years ago and radically changed the biosphere, provides a unique example of evolutionary expansion in the fossil record. Assuming that similar mechanisms control evolution at all scales, the cultural and technological evolution of human societies features clear expansion phases, which, unlike the Cambrian explosion, can be studied statistically due to ample available data [9, 11, 16, 20]. Also, the onset of biological evolution with the origin of life required an expansion of fundamental components and interaction types as life presumably emerged from pre-biotic geochemical processes. Evolutionary origins and expansion processes of this type are currently studied as components are assembled [17] in the lab to create bottom-up minimal living processes [18]. Further, for machine learning processes to be able to continue to learn new tasks, an inner component expansion is presumably necessary.

The same system may operate in both evolutionary optimization and expansion modes (e.g., at different times and/or in different regions of its configuration space), or it can operate simultaneously in both modes. Based on empirical data, we shall demonstrate how optimization and expansion phases can be identified and quantified, and also give examples where they coexist.

We shall in the following argue that *open-ended evolution* [24] may be obtained in systems with at least sporadic periods of evolutionary expansion.

## 2 Evolutionary Optimization

*e*, decreases as a power law

*c*of a particular technology and the cumulative amount

*y*produced by that technology:

*f*, is also observed to follow a power law as a function of the number of generations,

*g*:

_{i}< 1 for all systems. It should be noted that the value of α is neither universal across systems nor constant within the same system at different sizes or with different environmental conditions. We also note that the difference between a power law with a numerically small exponent and a logarithmic law can be negligible over large time scales, for example, in the fitness evolution

Finally, we note that the independent variable most conveniently producing a simple decelerating dynamical behavior can vary: In the case of biological evolution the number of generations is used, while for technological evolution, the number of produced units, not time, is the relevant variable.

### 2.1 Spin Glass Evolution

Intensely studied since the late 1970s, spin glasses are disordered magnetic systems with exemplary status for the research community working with complex systems of physical origin. A spin glass models, for example, the interactions of magnetic dipoles associated with impurities (e.g., iron or nickel) randomly dispersed in a nonmagnetic host material (e.g., gold). Of the manifold aspects of the fascinating dynamics of spin glasses [25], we only focus on the slow and decelerating magnetic relaxation observed after removal of a magnetic field [21] following a thermal quench into the spin glass phase. Of particular interest is the fact that the behavior can be understood in terms of mesoscopic events—our quakes—which are uniformly distributed on a logarithmic time axis [23]. Experimental thermoremanent magnetization (TRM) data are shown in Figure 1, upper panel. The data are redrawn from Figure 2 of [21], which we refer to for a more complete discussion. In brief, the data are obtained as follows: The sample is thermally equilibrated above the glass transition temperature *T*_{g} in a nonzero magnetic field. At time *t* = 0 the temperature is rapidly decreased to *T < T*_{g} and thereafter kept constant. After waiting a time *t* = *t*_{w}, the magnetic field is removed. The ensuing TRM decay approximately depends on the ratio *t*/*t*_{w}, with a small systematic deviation known as subaging, which we presently gloss over. Here, *t*_{w} = 100 s and the abscissa is the time *t* elapsed from the thermal quench. The important feature for the present discussion is that the TRM decay is nearly logarithmic over at least five time decades. Figure 1, lower panel, shows the number of quakes as a function of time for a spin glass simulation [23], which are also distributed homogeneously on a logarithmic time axis. Clearly, both the number and type of interactions responsible for the observed dynamics in this physical system are constant over time.

### 2.2 *E. coli* Bacterial Evolution

Lenski and Travisano [10] and later Wiser et al. [26] measured the fitness of 12 *E. coli* populations grown in a constant environment over 50,000 generations. The authors find an evolutionary dynamics as expressed in Equation 3 with small α values, which makes the dynamics similar to a logarithmic slowdown. If we identify quakes as the mutations with a fitness impact in this system, the dynamics seem to belong to the same universality class as aging physical systems (see Figure 2), and we may interpret the fundamental interaction types within the bacterial monoculture as being constant over time. This means that both the internal cellular biochemical ecology of interactions and the cell-cell interactions should be of qualitatively similar types over time. This seems to be a reasonable assumption, although we do not provide any detailed specification of the interaction types involved. Like that of spin glass, the evolutionary dynamics of a bacterial monoculture also seems to proceed at a constant pace on a logarithmic time axis, as the number of generations and time can be assumed proportional, *g*(*t*) ∼ *g*_{t}*t*, where *g*_{t} is the generation time. The physical system is minimizing energy, while the biological system is maximizing fitness, and the resulting α's differ.

### 2.3 Photovoltaic Technology Evolution

McNerney et al. [11] compiled and compared performance improvement for a number of technologies in terms of costs per produced unit. For this improvement dynamics, they find power laws that depend on the complexity of the producing technology. Wright [27] was the first to measure technology performance in this manner by studying airplane production costs in 1936, and used the approach discussed in Equation 2.

Below we show the evolutionary dynamics for photovoltaic optimization (see Figure 3), often called technological learning dynamics [9], as a function of accumulated installed photovoltaic (PV) capacity [5, 15]. The latter has been shown empirically to grow approximately exponentially with time [9] (thus *y*(*t*) ∼ *e ^{at}*), while the cost of a PV unit decays exponentially with time. However, using the total number of units produced, or an equivalent measure, highlights that the improvement stems from an optimization process with a characteristic power law decay.

The value of the exponent α for technology improvement has been measured across many different technologies in [9]. For the PV case α ≈ 0.2; see Figure 3. Moore's law indicates that the cost of a given computational power decays exponentially in time [1, 11]. However, if we measure the computational cost per unit produced, which is an exponential function of time, Moore's law becomes equivalent to Wright's law [1, 12].

### 2.4 Simple Theory and Simulation for Generic Evolutionary Optimization

What is the origin of the power laws often describing optimization processes? Consider a complex system consisting of many interacting subcomponents; recall our discussion in the previous subsections. Changing the performance of one component will then also affect the performance of other components. In this way components may be viewed as nodes in a directed network with links from each component to those that depend on it. Thus the relationship between the nodes and links can be characterized by an adjacency matrix.

Such a conceptual model captures the effect of changes in many systems composed of multiple interacting subsystems, for example, a bacterium, or a specific technology. As we shall show below, its behavior can be described analytically by record dynamics, which then explains the origin of the generic power-law decay of the cost-performance function in optimization searches.

*N*components, where the state of each component is

*s*

_{i}, 1 ≤

*i*≤

*N*, with an integer value, for concreteness between 1 and 20. The state of the full system is then described by an

*N*-dimensional column array of integers

*s*

_{i}. The total energy cost (or negative fitness)

*C*associated with a state is

**G**is a matrix with real-valued entries. Lacking information on the nature of the adjacency matrix, we take

**G**to be random, with entries drawn independently from a standard normal distribution of zero average and unit variance.

**s**and accepted if the move decreases the cost function. We note that the range of the latter is known, since the ground state of the system is

**G**with the maximal and minimal interactions weights for

*s*respectively.

_{i}The statistical data on which Figure 4 is based are obtained using 300 independent repetitions of the optimization process associated with Equation 5. Each repetition uses the same 20 × 20 matrix **G**, and a state vector **s** whose elements change during the simulation.

*n*such events occur in the time interval (

*t*

_{1},

*t*

_{2}) is

*μ*

_{0}is a constant. Record dynamics posits that the transformation

*t*→ ln

*t*captures all the memory of the process and makes it effectively memoryless. A way to check this for the above process is to construct the time series

*τ*

_{k}= ln

*t*

_{k}− ln

*t*

_{k−1}= ln(

*t*

_{k}/

*t*

_{k−1}), where

*t*

_{k}is the time of the

*k*th event. See the left-hand panel of Figure 4.

*τ*

_{k}are the waiting times between independent events on a logarithmic time axis, whenever these are a Poisson process, their probability density function (PDF) has the exponential form

*μ*

_{0}values read off the pre-factor and the argument of the exponential function.

*C*(

*n*) over the probability that

*n*quakes occur in the interval [1,

*t*] (see Equation 7):

*μ*in Equation 8. The above arguments give a generic power law decay with a small negative exponent

*μ*

_{0}α for the cost function of the optimization process. Recall that

*μ*

_{0}is the intensity of the quaking process (quakes per unit logarithmic time), and α gauges the relative improvement that follows from a simple quake. Figure 4 summarizes the properties just discussed: The left-hand panel depicts the probability distribution of the logarithmic waiting time between two quake events, and the insert shows that successive quakes are uncorrelated, which we take as a proxy for statistical independence. The exponential form shown in the main picture implies that quaking process is a Poisson process. The right-hand panel shows that the average cost function as a function of time is a power law with a small exponent, as expected.

Note that evolutionary optimization processes are frequently observed, generated, and reported in the artificial life, evolutionary biology, and machine learning literature.

### 2.5 Natural Parameterized Independent Variables for Evolutionary Dynamics

Physical systems such as spin glasses and colloids usually have fixed constituents, and time *t* is the natural independent variable for their evolution processes. The same is true for species extinction dynamics [13]. For bacterial monocultures it also seems reasonable to assume fixed cellular constituents (internally and externally), and *g*(*t*), the number of generations, is the natural independent variable for such processes. Here *g*(*t*) ≌ *g _{t}*

*t*may be approximated with a simple linear function of time, where

*g*is the generation time.

_{t}In the case of technological improvement, the natural independent variable is the number of produced units, *y*. Empirical studies show that the *number* of produced units increases approximately exponentially in time, so *y*(*t*) ∼ *e ^{at}*. It is less clear for this technology evolution example (1976–2016) that the

*quality and number of the interacting constituents*underpinning the production and improvement also can be considered constant over time, as in the previous physical and biological examples. However, they might as a first approximation be considered constant, as was done in [11] to estimate the effect of design complexity in technology improvement.

*I*(

*t*), which can also contain information about changes of the underpinning components and interactions. Recall for example Equation 4 for fitness optimization, which can then be rewritten as

Consider now human cultural and technological evolution. The human population has increased dramatically since the Paleolithic age, when humans were hunters and gatherers—an increase generated by an equally dramatic change of the cultural and technological setup of human societies. Yet, it is reasonable to assume that on a small scale—single humans, families, and communities—the same attempt to improve one's life condition has been carried out throughout. The dichotomy between optimization and expansion can be captured by a suitably defined *I*(*t*), as we shall see in the next section.

## 3 Evolutionary Expansion

### 3.1 Human Cultural and Technological Evolution

The evolution of human technology is different from the previous examples, because qualitative changes and expansions of human-human and human-technology interaction patterns are commonplace. This means that an expansion of component numbers as well as new components and interactions can be expressed by *I*(*t*), although we shall not specify the details of these growing constituents and interactions.

We assume we can use the accumulated wealth production—the gross domestic product per capita per year, GDP(*t*)—as a proxy measure for human fitness over time. The growth of GDP(*t*) is mainly the result of technological evolution of both the physical technologies (e.g., hammer and nail, steam engine, computers, Internet) and the social technologies (e.g., governance, institutions, laws, education, religion, myths, social norms). Examples of areas within which qualitatively different and increasing interaction components and types have emerged include communication, transportation, production, energy, education, governance, and religion, and are thus within both the physical and the social sphere. Over the last two centuries new interaction types for communication include the introduction of the telegraph, the telephone, TV, and the Internet; for transportation they include the introduction of the railroad, the automobile, and the airplane; and for governance they include widespread democracy with universal voting rights. All of these and many other new technologies have generated radical societal changes and increased the overall fitness per capita. As mentioned, we assume human fitness evolution to have both an optimization and an expansion component.

In the previous section we learned to express fitness evolution as a function of *I*(*t*), because *I*(*t*), and not *t*, is expected to be the natural variable for the ongoing optimization process for any given set of interactions. To quantify the *difference* between a baseline evolutionary optimization and the evolutionary expansion for GDP(*t*) over a period of time, we may empirically find an expression for *I*(*t*) by using the same ansatz as in Equation 12 and detrend the time series for GDP(*t*) over that period of time.

*t*) time series, we get

*P*(

*t*) is estimated to be a best fit third-order polynomial, and the goodness of the fit does not change significantly on using higher-order polynomials. Using the ansatz from Equation 12 we get

*I*(

*t*), and in the previous section we learned that α is expected to vary across and within systems. This means that α cannot be uniquely derived from data in this manner, and needs to be supplied empirically from elsewhere—for example, as an average across all the physical and social technologies. However, we are only aware of work that investigates physical technologies in a quantitative manner (see, e.g., [9, 11]).

Alternatively, *I*(*t*) or rather ln[*I*(*t*)] may be estimated empirically as a piecewise differentiable function with discontinuities linked to identified salient events in human history. One could for example, divide the series into periods starting with 1270–1400 where the Black Death killed a large fraction of the population, followed by relative stasis (no growth) until the impact of the colonial period is seen ≈1650; the onset of the Industrial Revolution ≈1795 till the Depression ≈1930; and the rest up to the present day. Or one could choose slightly different years as the basis for the piecewise linear approximation of ln[*I*(*t*)] corresponding to piecewise different exponential expansion periods for *I*(*t*). The resulting piecewise linear approximation of ln[*I*(*t*)] may now be fitted with a third-order polynomial, which is depicted in the inset of the lower panel of Figure 5. The normalized GDP, given by GDP(*I*(*t*))/GDP_{0}, is shown in the lower panel of Figure 5, and it is seen to follow the power law *I*(*t*)* ^{α}*, α ≈ 0.18, which is given by the dashed line.

Expansion of *I*(*t*) over time is of course not limited to complex ecosystems and sociotechnical systems. Even in a simple protocellular system, events that increase the physicochemical complexity could increase the number and quality of interactions and thus *I*(*t*). We have previously demonstrated this in self-assembly of dynamical hierarchies [17], and we have proposed how to expand *I*(*t*) for a protocellular system once it has obtained the ability of evolutionary optimization [18].

## 4 Discussion

In the evolutionary biology, machine learning, technology development, and artificial life literature, evolutionary *exploitation* and evolutionary *exploration* usually are used to distinguish different aspects of an overall *optimization* dynamics, while evolutionary *expansion*, as we have defined it, rarely occurs.

In a classical biological example, optimal foraging behavior is seen as a balance between seeking new feeding locations (explorations) and utilizing known feeding locations (exploitation) [4]. In such a case the foraging animal is optimizing its already existing behavioral possibilities, which include both exploration and exploitation, without expanding its inner components and interactions. In rare cases the exploitation can become a behavioral expansion—for example, an animal community or population can socially learn a new foraging behavior, as seen for orcas hunting seals on small icebergs by collective wake making [2]. Another classical example of optimizing an appropriate balance between exploitation and exploration comes from organizational learning [8]. Reference [8] examines how exploration and exploitation can jointly influence firms' performance in the context of their approach to technological innovation. In this case, however, there is not a clear distinction whether an adopted technological innovation is part of an optimization or an expansion process. In machine learning (ML), the tradeoff between optimization and exploitation can be defined more precisely, as for example, in ML-assisted design of experiment optimization for cell-free protein expression; for details see [6]. Here exploitation can be defined as proposed new *hill-climbing* experiments, which are new experiments that provide the most likely improvement of protein expression yield, given what is already known about the system. In this ML setting the experimental space is predefined by a number of experimental parameters, for example, salt composition and concentration, pH, and buffers, as well as a fixed set of chemical and biological components. Exploration in this system can be defined as proposed new experiments that test the least-known regions of the experimental space, given what is already known about the system. These exploratory experiments can provide valuable new information about unknown parameter regions of the experimental space. Thus, the ML-assisted design of experiments in [6], including the hill-climbing and the exploratory experiments, specifies an evolutionary optimization process in the sense in which we define the concept. An evolutionary expansion process for the system discussed in [6] would mean an expansion of the experimental space, for example, through an increase of the number of experimental parameters.

In this article we propose, and empirically demonstrate, a new approach for quantitatively characterizing evolutionary processes using an optimization ansatz: All evolutionary processes have an optimization aspect and can be described as decelerating power laws of suitable functions of time *I*(*t*). The form of *I*(*t*) describes in turn the evolutionary expansion aspect that is present in some cases, including open-ended evolution.

For simple optimization processes—for example, the relaxation of a disordered magnetic material (spin glass), or improving the fitness of a bacterial monoculture—*I*(*t*) is proportional to the physical time *t* and the biological generation time *g*. In these cases the spin glass sample is constant and the bacterial population size is approximately constant. For technological optimization, the natural independent variable *I*(*t*) is the number of produced units, which turns out to grow exponentially with time for both photovoltaics and the computational power of transistor networks. In these cases, *I*(*t*) can be explained by an exponentially expanding market, which is an externality to the given technology. The underlying dynamics for the above optimization examples is explained by a simple theoretical framework based on record dynamics. An illustrative simulation is also presented.

Finally, we examine the per capita GDP as an indicator of human fitness, but lack a microscopic theory for *I*(*t*) used in the other examples. What causes GDP per capita to expand as *e*^{P(t)}, corresponding to *I*(*t*) ∼ $e1\alpha Pt$, where *P*(*t*) is a polynomial in time and α is a small positive parameter, remains therefore unexplained. We can, however, empirically observe that human fitness accelerates faster than the fitness of any single given physical technology even if we assume the aggregate of all technology markets to grow exponentially, and we can quantitatively express the evolutionary expansion *I*(*t*) for an open-ended evolutionary process: human technological and cultural evolution. To develop a microscopic theory for *I*(*t*) is a well-defined open scientific challenge, whose solution may shed light on the nature of evolutionary expansion processes as well as open-ended evolution.

## 5 Conclusions

Strikingly, systems of very different nature feature similar evolutionary traits: The cumulated number of salient evolutionary events, or quakes, grows as a power law for complex physical, monocultural biological, and single technological systems when using the natural independent variable of the process. The first example can be understood as an energy minimization process while the latter two can be understood as optimization processes. These processes occur in their respective complex configuration spaces with given components and interactions, which as we argue, all lead to power laws.

In the expansion mode of evolution, new agents and types of interactions play a role. We propose that evolutionary expansion is superimposed on optimization and that the full process can be treated as optimization by introducing a suitable interaction variable *I*(*t*) whose time development describes the rule changes that take place during the expansion. While a microscopic theory for *I*(*t*) is needed, we have shown how to empirically extract its time dependence from detrended evolutionary human GDP data. Our method is generally applicable and should be useful in the analysis of other types of open-ended evolution data.