Predicting System Dynamics of Universal Growth Patterns in Complex Systems

TL;DR

This study demonstrates the applicability of the sigmoid curve to capture the acceleration and deceleration of growth, predicting an entitys ultimate state well in advance of reaching it, and introduces a classification of entities based upon similar lifepaths.

Abstract

Predicting dynamic behaviors is one of the goals of science in general as well as essential to many specific applications of human knowledge to real world systems. Here we introduce an analytic approach using the sigmoid growth curve to model the dynamics of individual entities within complex systems. Despite the challenges posed by nonlinearity and unpredictability in system behaviors, we demonstrate the applicability of the sigmoid curve to capture the acceleration and deceleration of growth, predicting an entitys ultimate state well in advance of reaching it. We show that our analysis can be applied to diverse systems where entities exhibit nonlinear growth using case studies of (1) customer purchasing and (2) U.S. legislation adoption. This showcases the ability to forecast months to years ahead of time, providing valuable insights for business leaders and policymakers. Moreover, our characterization of individual component dynamics offers a framework to reveal the aggregate behavior of the entire system. We introduce a classification of entities based upon similar lifepaths. This study contributes to the understanding of complex system behaviors, offering a practical tool for prediction and system behavior insight that can inform strategic decision making in multiple domains.

Figures (4)

• 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.
• Figure 2: Universality 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).
• 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.
• 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).