Introduction

Nonlinearity often appears in systems of interacting components as they increase in size and complexity1,2. This phenomenon is observed in various real world systems, including social3, economic4, and biological5,6 systems, where intricate nonlinear interactions introduce a high degree of complexity7. In systems comprised of a multitude of interacting elements, the behavior often becomes irregular and challenging to predict. For instance, in economic markets, individual stock values may exhibit complex fluctuating patterns of behavior, and the aggregate of these patterns can give rise to collective booms and busts. These patterns, observed in a multitude of nonlinear systems across contexts or scales, challenge our ability to develop an understanding of system behavior, but also hold the potential to lead to profound insights8. Such insights are crucial because recognizing and understanding patterns across systems can lead to enhanced predictability of system behavior, more effective system design, and the development of improved control strategies9.

The emergence of dynamic complexity can be attributed to interactions among a system’s constituent elements and/or to the nonlinear behaviors exhibited by these elements10,11. Such complexities typically give rise to nonlinear collective behaviors that defy direct proportionality to the individual behaviors of the entities, thus imparting unique characteristics to the system12,13,14. In these systems, the conventional characterization of input-output relationships does not hold15. The central challenge for system description lies in identifying universal properties so that it is not necessary to develop distinct characterizations for each individual element or system7,16,17. While universality may arise at the aggregate level due to the effects of interactions among elements, it can also manifest at the individual level when the behaviors of individual entities can be succinctly described by a parameterized mathematical function18.

Many investigations of non-linear systems have focused on the premise that individual entities of the system adhere to specific patterns19. When there is noise in the dynamics of individual entities, predicting their behaviors over the short term may become challenging. However, their longer term dynamics may be still be well characterized by statistical dynamic functions20, especially when the aggregate behavior of the system does not exhibit abrupt shifts21. Studying the dynamics of all entities of a system can unveil emergent patterns that might not be evident when examining individual entities in isolation. These patterns can assume various forms, from predictability to chaos, simplicity to complexity, reflecting the inherent nature of such systems22.

In this context, complex systems often exhibit emergent behaviors–global patterns that arise from local interactions and are not reducible to the sum of their parts11,23. A hallmark of these systems is non-ergodicity, where statistical properties observed across an ensemble of entities do not match the temporal dynamics of any given individual. This distinction implies that population-level averages may obscure significant variation and critical transitions within individual trajectories. Furthermore, these systems often display non-monotonicity, where the progression of states is irregular and marked by phases of acceleration, stagnation, or reversal, defying simple linear or unidirectional models24. Together, these features–emergence, non-ergodicity, and non-monotonicity–pose significant challenges for conventional modeling approaches that rely on assumptions of homogeneity, equilibrium, or smooth progression. Instead, they call for analytical frameworks capable of capturing variability, phase transitions, and feedback-driven dynamics that unfold over multiple scales of time and organization. Our approach addresses this gap by using the sigmoid model as a mesoscopic tool: it captures a broad range of nonlinear dynamic behaviors (e.g., growth, decline, saturation) of entities while abstracting over micro-level noise. By fitting sigmoid curves to individual trajectories, we bridge the complexity at the entity level with macro-scale regularities, enabling comparative insights across systems and domains.

In these systems, an entity’s activity grows and then diminishes, exhibiting phases of acceleration and deceleration, akin to a bell-shaped curve. The initiation of activity sets in motion a self-activating pattern that increases activity over time. However, due to various internal or external constraints, an inhibitory pattern eventually arises, slowing down the activity, eventually leading the entity to end its activity. The activating and inhibiting processes give rise to unique behaviors for each entity that include stochastic variation. To analyze and understand such behaviors, we employ a sigmoid function, which we demonstrate to be effective as a universal description of the aggregate over time of growth and decline. This nonlinear function25,26 captures phenomena that commence gradually, undergo acceleration, and eventually reach saturation, creating an “S”-shaped pattern. Sigmoid functions have previously been utilized to model diverse phenomena, ranging from the growth of biological populations27 and the spread of infectious diseases28 to the adoption of new consumer innovations29. Leveraging real world historical data, a sigmoid analysis allows us to predict both the time when an entity is likely to saturate (ending activity) and the total amount of its activity. The parameterized trajectories for each entity can be visualized, enabling us to gain insights into its lifepath, providing both quantitative prediction for the near future and characterization of the long term behavior.

We utilize two distinct datasets. The first is a database of industrial customer orders. Individual customer decisions may be simple, but together they form recognizable life cycles. While customer purchasing behavior is inherently complex, as each customer seeks to satisfy individual needs30, we show that the order frequency trajectory follows typical behavior consistent with sigmoid functions. Customers place initial orders and progressively increase their frequency until they gradually reduce their orders and eventually stop. This behavior may reflect changes in market conditions, competition, time and/or resource constraints that limit customer purchasing and give rise to the observed nonlinear behavior. The second data set refers to the legislative process, including the introduction of new bills by parliaments, congresses, or other similar institutions. Individual lawmakers propose and support bills, but collective patterns–such as polarization or tipping points–emerge at higher levels of aggregation. For certain topics, numerous bills may be introduced; some attract sustained interest, while others quickly fade away. Averaging across all legislative activity can obscure these nuanced dynamics. A given issue may initially gain support, then stall, lose momentum, and ultimately be forgotten–revealing a non-monotonic trajectory that cannot be captured by simple aggregate trends. We use texts of proposed laws to extract named entities about acts, environmental agencies, associations, agencies, boards, various codes delineating minimum requirements (e.g., building codes or energy conservation codes), and numerous other named entities. The dynamics of the usage of names of such entities in introduced bills reveal a nonlinear behavior that can similarly be characterized by sigmoid curves. We show that awareness of the dynamics of customer orders and term usage in legislation offers valuable predictive tools for these contexts.

Results and discussion

A sigmoidal curve is defined by three key parameters: inflection point, slope, and amplitude. Here, we show that the evolution of the fitted parameters provides a clear understanding of the current state of activity and can make reliable predictions about future behavior. The results for the primary dataset are presented in this section, while those for the second dataset are available in the supplementary materials. In Figure 1A, we show a sigmoid curve fit to the cumulative time series of a particular customer who initiated their orders in 2006 and ceased ordering in 2008. The customer transitioned from an accelerating phase to a decelerating one early in 2007. The model suggests that the customer is unlikely to place any new orders after 2008. Using the reduced \(\chi ^2\) statistic (Fig. 1B), we find the sigmoidal fit significantly outperforms a linear fit, as seen in smaller \(\chi ^2\) values. In Fig. 1C, the sigmoid fit and its parameters are shown for consecutive years for another representative customer. We see that as a customer continues to grow with new orders over the years, the parameters of its sigmoid fit change. In the following, we show how by tracking the sigmoid parameters we can analyze their behavior and predict future trends.

Fig. 1
figure 1

Sigmoid curve fitting and parameter spaces for customer ordering behavior. (A) Customer time series and fitted sigmoid curve. Red bars represent the number of orders over time for a customer; the blue line is the cumulative time series; and the orange line is the sigmoid curve fitted to the cumulative time series. Sigmoid curve parameters include inflection time (green dot, center), slope of the curve at the inflection time, and amplitude or total orders. (B) Comparison of the linear and sigmoid fits to the customers using the reduced \(\chi ^2\) test. (C) Ordering behavior of a representative customer in different years and the fitted sigmoid curves and their parameters. Parameter spaces for all customers are shown for (D) slope, inflection time, and amplitude, (E) slope, start time, and saturation status, and (F) slope, inflection time, and saturation. The third parameter is shown in color (scale in inset). For the customers that are early in their growth, the inflection time and amplitude are not reliable, so we assign them an arbitrary inflection time of 2030 with their last total orders as amplitude.

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To visualize properties of the entire population, we constructed parameter spaces, where dots represent the coordinates of each entity for three parameter values, with color serving as the third dimension. These parameter spaces reveal collective system properties that are not visible when examining individual entity behaviors. In addition to the three primary parameters—inflection time, slope, and amplitude—two other pertinent parameters are the start time of activity and the saturation value, defined as the ratio of the fitted amplitude A to the final number of cumulative orders. In the parameter space defined by inflection time, slope, and amplitude (Fig. 1 D), customers who place orders over an extended duration tend to exhibit a lower slope and are situated in the upper area of this parameter space. Repeat customers, marked by green-yellow dots, occupy this region as they consistently place orders over an extended period. Additionally, a few smaller customers in the purple category also appear in this segment when their orders span a considerable time frame. Conversely, customers who make a single order are characterized by the steepest slopes and are represented as purple dots positioned at the bottom of the graph. In cases where customers are at the nascent stage of their growth and exhibit a steep rise in slope, the inflection time becomes an unreliable metric, and is given an arbitrary value of 2030. In our model, a saturation status approaching 1.0 signifies the point at which customers cease placing orders, while values exceeding 1.0 indicate a likelihood of future orders. In the parameter space defined by start time, slope, and saturation, we observe that customers with lower slopes at any given start time are categorized as unsaturated (represented by red dots), indicating their potential for continued growth (see Fig. 1 E). Furthermore, inflection times for unsaturated customers are either in the recent past, meaning the rate of ordering has begun to slow down, or in the future, indicating customers with the most growth potential (Fig. 1 F).

Fig. 2
figure 2

Similarity in customer ordering behavior illustrated by the progression of parameter distributions over time. (A) Number of customers N(Y, J) that place more than J orders over different periods from 1999 to year Y (colors, legend in (B)). The black arrow represents the change in the breakpoints over the years. (B) Rescaling of the J-axis and N-axis by \(Y^{\beta _1}\) and \(Y^{\beta _2}\), respectively. Parameter spaces for customer slope, start time, and amplitude are shown for the periods from 1999 to the years (C) 2000, (D) 2004, (E) 2008, (F) 2012, and (G) 2016, with amplitude in colors (key, upper right). The red line on each graph represents the slope at the breakpoint in aggregate distributions from (A).

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Shifts and fluctuations in the distribution of the target variable pose challenges for predicting the collective behavior of a system. In contrast, when the entities within a system exhibit consistent behavior, it enhances our ability to make more reliable predictions, both at the collective level and for individual behaviors11,31,32. In this regard, power law distributions hold great significance, because they unveil a fundamental regularity in system properties at various scales. In a system with a power law distribution, transitions between phenomena at various scales remain consistent regardless of the specific scale under consideration. This self-similarity forms the foundation of power law relationships33,34. Analyzing the distribution of entities based on their key parameters over various time durations provides a valuable approach for assessing the consistency in system dynamics. By plotting these distributions and observing their trends over time, fitting a power law can serve as an effective method to describe the system’s enduring patterns and stability.

In our customer dataset, we charted the distributions of customers based on their order patterns over varying timeframes, spanning from 1999 through the subsequent years. (Fig. 2 A). The distributions count the number of customers N(J) that place at least J orders. The distributions follow a power-law \(N(J)\sim J^{-\alpha }\). They include two regimes, with different \(\alpha\) exponents. The regimes and the breakpoint between them shift toward larger numbers of customers N and orders J over time, as shown by the black arrow. In the first regime, \(\alpha _1 = 0.49\pm 0.02\), and in the second regime, \(\alpha _2 = 1.38\pm 0.08\), both of which are consistent for all years. The difference in the exponent of the two regimes reveals that the proportion of larger customers decreases more slowly in the first regime than in the second regime. If Y represents the number of years we aggregate the data, i.e., \(Y=1,2,3,...,18\), then by scaling the axes by a specific power of years, respectively, all the curves collapse into a single curve described by \(N(Y,J)/Y^{\beta _1} \sim f(J/Y^{\beta _2})\), with \(\beta _1=0.52\) and \(\beta _2=0.60\) (Fig. 2B). The scaling function f(u) demonstrates that the customer distribution is universal and can be generalized to future years.

The rationale behind the growth in these distributions and the identification of breakpoints becomes apparent when we observe the evolution of the parameter space encompassing start time, slope, and amplitude across the years. (Fig. 2 C-G). Over time, as the number of customers and their lifespans grow, we observe an increase in the quantity of points within the parameter space, which shift towards smaller slopes and larger amplitudes. To identify the average slope of customers at the breakpoints (red lines in Fig. 2 C-G), we leverage the joint distribution of slopes and amplitudes. From figure 1E, one can find that active customers (red dots) are in the upper portion of the parameter space across all years. Slopes with values less than that at the breakpoint (i.e., points above the red line in Fig. 2 C-G) encompass customers who have been active across all years. Conversely, larger slopes reveal a gradual decline in customer numbers in recent years, primarily because the most recent customers have not had sufficient time to accumulate numerous orders. This disparity in the exponents of the power law behavior underscores the distinction between the two regimes. As the red line progressively shifts upwards over time, it captures an increasing number of customers with higher order counts situated beneath it. This signifies a burgeoning in the first regime of the distributions, reflecting the expansion of overall business activity.

At each time t (year), we calculated the probability of having a customer that starts ordering at time \(t_0\) and reaches ln(slope) \(m'\) with ln(amplitude) \(A'\), \(p(t_0,m',A')\). See Supplemental Materials for details and Figures. The dependence of our equation on t is hidden in the start time, \(t_0\), parameter. As the slope and amplitude parameters are not independent, we define the probability as,

$$\begin{aligned} p(t_0,m',A') =p(A'|m',t_0)p(m'|t_0)p(t_0). \end{aligned}$$
(1)

We found that customer start times, \(t_0\), are uniformly distributed, i.e.,

$$\begin{aligned} p(t_0) ={\left\{ \begin{array}{ll} \frac{1}{t_{0b}-t_{0a}} & t_{0a}\le t_0\le t_{0b} \\ 0 & otherwise \end{array}\right. } \end{aligned}$$
(2)

where \(t_{0a} = 0.32\) and \(t_{0b} = 1\) represent the first and last months in scaled cumulative time series at \(t=2017\). The value of \(t_{0b}\) is always equal to 1, and \(t_{0a}\) depends on the time t at which we do the analysis. While the probability of having large slopes (small customers) is independent of their start time, we find that for smaller slopes it is a function of \(t_0\), \(m'_a = 0.5+e^{1.5t_0^{4}}\) (see Figure S4A). Thus

$$\begin{aligned} p(m'|t_0) ={\left\{ \begin{array}{ll} \frac{0.57}{6.5-m'_{a}} & m'_{a}<m'<6.5 \\ \frac{0.03}{9.1-8.9} & 8.9<m'<9.1\\ \frac{0.40}{9.4-9.3} & 9.3<m'<9.4. \end{array}\right. } \end{aligned}$$
(3)

We did not attempt to fit the data in the ranges of ln(slope), \(6.5< m' < 8.9\) and \(9.1< m' < 9.3\), where there were too few points to permit a reliable assessment of the distribution. We observe that the ln(amplitude) \(A'\) displays a normal distribution as a function of ln(slope) \(m'\), as

$$\begin{aligned} p(A'|m',t_0) = {\left\{ \begin{array}{ll} \frac{1}{\sqrt{2\pi } \sigma _1}e^{-\frac{(A'-\mu _1)^2}{2\sigma _1^2}} & m'_{a}<m'\le 6.5 \\ \frac{1}{\sqrt{2\pi } \sigma _2}e^{-\frac{(A'-\mu _2)^2}{2\sigma _2^2}} & 6.5<m'\le 9.4 \end{array}\right. } \end{aligned}$$
(4)

where the parameters for the first regime are given by \(\mu _1=a_0m'^2+a_1m'+a_2\) and \(\sigma _1=b_0m'+b_1\), with \(a_0=0.13\), \(a_1=-2.12\), \(a_2=9.86\), \(b_0=-0.20\), and \(b_1=1.93\), and the parameters for the second regime are \(\mu _2=1.0\) and \(\sigma _2=\ln (1.5)\). Note that the dependence of this equation on \(t_0\) is hidden in the value of \(m'_a\).

Fig. 3
figure 3

Lifepaths of five representative customers. Each customer (legend, inset in (A)) started in a particular year and left the system in 2009. (A) Cumulative orders per year. Sigmoid parameters are calculated each year from the monthly time series of orders, including (B) inflection time, (C) ln(slope), (D) expected customer leaving time, and (E) expected total orders. (F) ln(slope) versus inflection time; closeness of points indicates a customer has already passed their final inflection time.

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The dynamics of the sigmoid curve parameters enable prediction of future system dynamics. We can assess the predictive accuracy of the sigmoid fit by selecting already saturated entities and examining their gradual changes. Furthermore, we can analyze how successfully the sigmoid parameters predict their behavior.

To illustrate visually the activity dynamics of entities, we chose five representative customers that entered the system at different times and left the system in 2009, and we tracked the evolution of their sigmoid parameters from prior years to their “final” values in 2009 and beyond (Fig. 3). The five selected customers include large customers who placed many orders each year (C-1 and C-2), medium-sized customers who placed a few orders each year (C-3 and C-4), and a small customer who placed a few orders in one year (C-5) (Fig 3 A). We examined the evolution of inflection time (Fig. 3 A) and slope (Fig. 3 B). Because each customer left the system in 2009, the sigmoid parameters in 2009 and afterwards represent the aggregate behavior of the customer. Until the final value of the inflection time is reached, the predicted inflection time moves forward to later times and the slope tends to smaller slopes. After passing the final inflection time, the sigmoid function predicts the same parameters from then on, i.e, inflection time and slope do not change after that year. Therefore, the stabilization of parameters is a sign that entity activity has passed the final inflection time. We also observe a stabilization in the evolution of expected customer leaving time (Fig. 3 D) and expected number of orders (Fig. 3 E). However, for some customers (see C-1), the sigmoid fit shows a peak in inflection time (Fig. 3 B), expected leaving time (Fig. 3 D), and expected total orders (Fig. 3 E) one year before the final inflection time. This occurs for customers that are growing rapidly. Furthermore, we can track the evolution of each customer as a trajectory through the parameter space of inflection time and slope (Fig. 3 F). In these trajectories, large distances between consecutive points indicate a customer is still in the first regime of the sigmoid curve (before the final inflection time), while closeness of the points reveals the entity has passed the inflection time and will leave the system in the near future. If the ordering behavior of an entity changes for a period of time or the entity activity ceases temporarily, the trajectory moves backwards and forwards, providing a signal that the entity behavior may warrant further attention.

Fig. 4
figure 4

Validation of sigmoid fit predictions. Selected customers are those who entered the system in various years but all left in 2009. (A) Year a customer is expected to leave, according to the sigmoid fit, per year. Colored lines represent customers. The black dashed line represents the mean for all customers, and the black dash-dotted lines above and below represent the mean plus or minus the variance, respectively. Expected years should converge to 2009, the year all selected customers left. (B) Mean of expected time minus real time per year; error bars are variance. (C) Percentage of entities whose leaving time is predicted each year within one-year (blue line) or two-year accuracy (orange line).

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To validate predictive performance of the sigmoid fit, we picked a sample from the customers that had already left the manufacturing company. We selected the 364 customers who entered the system at any time but left in 2009. We examined the evolution of the expected leaving time of each customer calculated from the sigmoid curve fittings (Fig. 4 A). Each colored line represents the expected leaving time of a single customer, and the black dashed line and dot-dashed lines show the mean and mean±variance, respectively. While variance is very small in the first few years, it increases to around 2–3 years in the years approaching 2009. Fig. 4 B shows the mean of the difference between the expected time and real time in each year; error bars are the variance. In the first few years, when the history of orders is short, the expected leaving time is far earlier than 2009, and the mean difference is around 8 years. As we get closer, the mean leaving time increases towards 2009, and two years before 2009, the mean difference becomes less than a year. Our method is able to predict the leaving time for 56% of customers within a one-year difference in 2008 and for 82% in 2009, and within a two-year difference for 58% in 2007, 74% in 2008 and 85% in 2009 (Fig. 4 C). Overall, we find the sigmoid fit predictions are accurate, with the exception of some fast-growing large entities, which the sigmoid fitting overevaluates and gives a chance to return, and long-lived small entities, which the sigmoid fitting underevaluates and assumes have already left.

In conclusion, the sigmoid curve has been used to model the nonlinear growth of systems with a finite lifetime. Our analysis shows that sigmoid dynamics can effectively represent the behavior of individual entities within such systems. This model is particularly applicable to cases where entities experience an initial onset, followed by acceleration (an engaging phase), deceleration (a disengaging phase), and eventual inactivity. Depending on the distance between the inflection point and the onset time–and the temporal resolution at which the system is observed–this model enables early prediction of cessation. Our case studies on customer engagement and the U.S. legislative system demonstrate that stop times can be forecasted months or even years in advance. These insights can support analysts and policymakers in identifying dynamic trajectories and tailoring intervention strategies accordingly. Additionally, the parameter space derived from the behavior of individual entities offers a compact representation of the aggregate properties of the broader system.

However, it is important to recognize the limitations of this modeling approach. Models, by nature, are simplifications of reality. The sigmoid curve captures dominant trends, but real-world systems are influenced by noise, discontinuities, and contextual factors that may not conform to smooth or symmetric patterns. External shocks, feedback loops, and multi-scalar dynamics can introduce deviations that the model cannot fully anticipate. Although the sigmoid function serves as a valuable analytical tool, it should be interpreted as one lens among many to understand the behavior of complex systems, not as a definitive representation. We encourage future research to explore hybrid or adaptive models that integrate these complexities more explicitly. Non-symmetric sigmoid models or multilevel sigmoid fitting procedures may offer ways to accommodate more complex or nuanced system behaviors.

Methods

Data

We conducted an analysis of entity behaviors in two distinct datasets. The first dataset, from a medium-sized U.S. manufacturing company, recorded orders from 6,065 customers during the time period spanning from November 1999 to September 2017. The number of orders per customer varies from 1 to over 10,000 orders over a period of 18 years. This dataset captures the company’s customer interactions from the moment they initiated data recording, even though the company had already established a customer base prior to this period. Each customer is identified with a unique customer ID. The data include the year, month, day, minute and second of each order. Our analysis is not sensitive to smaller time scales so we aggregated orders by month.

The second data set describes legislative developments related to per- and polyfluorinated substances (PFAS). Commencing from the 2016-2022 period, United States (US) federal and state legislatures have enacted numerous bills prohibiting the use of PFAS in a wide array of products. LegiScan, a legislative tracking service facilitating the monitoring and analysis of legislation at both the state and federal levels in the US, provides extensive data for legislative events. We collected all bills related to PFAS from LegiScan. We used queries such as PFAS OR PFOA OR polyfluor OR perfluor OR ethylene oxide OR microfiber OR microplastic OR halogenated flame retardant OR organohalogen. The results included 228 federal and state bills. We extracted the complete texts of the most recent versions of these bills. Employing natural language processing, we extracted the most significant named entities from the text, and high-frequency keywords frequently associated with PFAS terminology (e.g., product names, action verbs). In total, we identified 3100 relevant terms and their usage.

Sigmoid model and pervasiveness patterns

To elucidate the behavior of each entity, i, we generated time series, \(x_i(t)\), representing the monthly activity (purchasing volume, term presence) of the entity over time. Then, we generated the cumulative time series, \(y_i(t)=\Sigma _{t_{0i}}^{t} x_i(t)\), where \(t_{0i}\) represents the first activity for entity i. While the specific interactive patterns of entities may differ based on particular conditions, a common underlying behavior emerges. Entities initiate their activities at specific times, gradually increase activity over subsequent periods, and then taper off until they ultimately cease activity. This nonlinear pattern—a slow start, acceleration, and deceleration to an end, forms an “S”-shaped curve, as described by Han (1995)35. We fit this behavior to a sigmoid curve defined by three key non-negative parameters: the slope (m), the inflection point (\(t_0\)), and the amplitude (A) in the equation:

$$\begin{aligned} y(t) = \frac{A}{1+e^{-m(t-t_0)}} \end{aligned}$$
(5)

The slope parameter corresponds to the gradient of the sigmoid at the inflection time, akin to Bolton’s concept of relationship depth in the marketing literature36. The inflection time parameter is the moment in time when the activity transitions from acceleration to deceleration. The amplitude parameter A describes or anticipates the total activity by the entity over all time.

The fitting of sigmoidal curves is sensitive to the length of the cumulative time series when the time series is relatively short. To ensure consistency in the fitting process for entities of varying lifespans, we standardized the length of all cumulative series. This standardization involved augmenting each time series with an array of zero values at times prior to the first recorded activity. We added 100 months of zero values at the beginning of each time series. We then rescaled the cumulative time series and t-axes to a range from 0 to 1, enabling comparisons between sigmoid curves and their respective parameters. Entities that initiated activity within the final two time points were excluded from the analysis due to insufficient data. Following these adjustments, we fit a sigmoid curve to the scaled cumulative curve. If the sigmoid curve exhibits an amplitude less than or equal to 1.0, it indicates saturation, suggesting that the entity has either ceased interactions within the observed time frame or will do so imminently. Conversely, an amplitude greater than 1.0 signifies an unsaturated curve, indicating that the entity is likely to undergo more activity in the future. After the fitting procedure, the results are rescaled back to their actual values.

Analyzing the evolution of parameters with a variable ending time of the analysis provides valuable insights into the patterns of behavior and their pervasiveness. This behavior allows us to predict what will happen for the components and the system in the future. To examine the distribution of entities versus their activities, we employ a survival function. This function quantifies the number of entities with a cumulative activity of at least J, denoted as N(J), for the range \(J = {1, 2, 3, ..., J_{max}}\), and is defined as \(N(J) = \sum _{i=J}^{J_{max}} n_i\), where \(n_i\) is the number of entities with cumulative activity i. Our investigation characterizes the distribution of cumulative activity, referred to as entity size. The shape of this distribution reflects intrinsic properties of the system and may evolve over time, as indicated by previous studies37,38,39. In cases where the system encompasses a small number of entities with substantial activity levels and a multitude of small entities (i.e., with small levels of activity), the distribution exhibits skewness, or fat tailed properties. One extreme skewed distribution is a power law distribution40,41. Power law distributions are described by the expression \(N(J)\sim J^{-\alpha }\), where \(\alpha\) serves as a measure of the degree of fat-tailed behavior.