Scaling Laws for Function Diversity and Specialization Across Socioeconomic and Biological Complex Systems

TL;DR

It is found that the number of distinct functions scales with organizational size, approximately as a power law with an exponent of 1/2, which suggests that human organizations, despite differences in their purpose, structure, and culture, may share common mechanisms for creating specializations.

Abstract

Function diversity, the range of tasks individuals perform, and specialization, the distribution of function abundances, are fundamental to complex adaptive systems. In the absence of overarching principles, these properties have appeared domain-specific. Here, we introduce an empirical framework and a mathematical model for the diversification and specialization of functions across disparate systems, including bacteria, federal agencies, universities, corporations, and cities. We find that the number of functions grows sublinearly with system size, with exponents from 0.35 to 0.57, consistent with Heaps' Law. In contrast, cities exhibit logarithmic scaling. To explain these empirical findings, we generalize the Yule-Simon model by introducing two key parameters: a diversification parameter that characterizes how existing functions inhibit the creation of new ones, and a specialization parameter that describes how a function's attractiveness depends on its abundance. Our model enables cross-system comparisons, from microorganisms to metropolitan areas. The analysis suggests that what drives the creation of new functions depends on the system's goals and structure: federal agencies tend to ensure comprehensive coverage of necessary functions; cities tend to slow the creation of new occupations as existing ones expand; and cells occupy an intermediate position. Once functions are introduced, their growth follows a remarkably universal pattern across all systems.

Figures (5)

• Figure 1: The number of distinct functions versus system size (top row) and rank-frequency distribution of function abundance (bottom row) in several complex adaptive systems. (A,D) Bacterial and archaea cells, (B,E) US federal agencies, and (C,F) US metropolitan statistical areas (MSA). The dependence of the number of functions on size measure $N$ in (A) and (B) can be approximated by the power law, $D \sim N^\beta$. Also shown is the best-fit scaling exponent, $\beta$, and its $95\%$ confidence interval in the square brackets. (C) In contrast, logarithmic scaling occurs for MSA's, $D\sim \log N$. In (D--F), grey dots represent all available data points, while select entities are highlighted in color.
• Figure 2: Conceptual diagram of the mathematical model. As each new individual joins the system, it either creates a new function with probability $p$, where $p$ depends on the existing abundance distribution; or joins an existing function with probability $1-p$. When it joins an existing function, the probability of joining a particular existing function with $k$ members, $q_k$, follows a generalized preferential attachment process.
• Figure 3: Summary of the model calibration results for the normalized rank-frequency distributions of (a) bacterial cells, (b) federal agencies and (c) cities, each for a given case of that system. The legends show the calibrated values of $\gamma$ and $\theta$, as well as the particular system in question. Each system was simulated using an initial condition that arose from one of the lesser size organizations in that data set (see SI for further details). Panels (d), (e), and (f) show the respective diversity plots from simulations given the mean values of $\theta$ across all calibrations for that system, starting from the initial condition size used for calibration.
• Figure 4: The parameter space of specialization ($\gamma$) and diversification ($\theta$) estimated for each system across different classes of systems. The vertical dashed line shows $\theta = 1$, the horizontal dashed line shows $\gamma = 1$, and the diagonal dashed line shows $\gamma = \theta$. We selected the federal agencies with $N>1000$ (40 instances), all of the bacterial data we had access to (46 species), and the largest twenty cities.
• Figure 5: Calibrated parameters ($\theta$ and $\gamma$) for the four largest metropolitan areas in the longitudinal dataset, by year. Parameter values show no systematic temporal trends.