Complexity and Information: Measuring Emergence, Self-organization, and Homeostasis at Multiple Scales (1205.2026v2)

Abstract: Concepts used in the scientific study of complex systems have become so widespread that their use and abuse has led to ambiguity and confusion in their meaning. In this paper we use information theory to provide abstract and concise measures of complexity, emergence, self-organization, and homeostasis. The purpose is to clarify the meaning of these concepts with the aid of the proposed formal measures. In a simplified version of the measures (focusing on the information produced by a system), emergence becomes the opposite of self-organization, while complexity represents their balance. Homeostasis can be seen as a measure of the stability of the system. We use computational experiments on random Boolean networks and elementary cellular automata to illustrate our measures at multiple scales.

• The paper introduces formal, measurable definitions for emergence, self-organization, complexity, and homeostasis using information theory metrics.
• Applying these information-theoretic measures to simulated Boolean networks and cellular automata reveals how complexity relates to system dynamics across scales.
• This framework offers a rigorous, multi-scale toolset for analyzing complex systems across disciplines, improving theoretical understanding and practical design.

Complexity and Information: Measuring Emergence, Self-organization, and Homeostasis at Multiple Scales

The paper authored by Carlos Gershenson and Nelson Fernandez presents a rigorous approach to understanding the concepts of complexity, emergence, self-organization, and homeostasis through the lens of information theory. Their work addresses the frequent ambiguities and misuses of these terms within the scientific community by proposing concise, formalized measures that leverage the mathematical foundation of information theory to better quantify these phenomena.

Key Contributions

1. Formal Definitions via Information Theory:
   • This paper delineates emergence as a measure of the novel information produced by a system that was not present initially. Emergence is calculated as the ratio of output to input information.
   • Self-organization is characterized as the reduction of information entropy within a system, posing it as a negative function of emergence.
   • Complexity is conceptualized as a balance between self-organization and emergence. When applied, it integrates the tendencies of high order and high disorder, offering a numerical representation of this balance.
   • Homeostasis is defined as the ability of a system to maintain information stability over time, inversely related to the Hamming distance between sequential states.
2. Experimental Illustrations:
   • The measures were applied to simulate random Boolean networks (RBNs) and elementary cellular automata (ECA), providing computational examples that illustrate the dynamics of these concepts across multiple experimental scales.
   • The results confirm that systems with critical dynamics exhibit higher complexity compared to those demonstrating merely ordered or chaotic dynamics.
3. Multiscale Approach:
   • By accounting for multiple scales, the proposed measures can accommodate diverse levels of observation, potentially offering more nuanced insights into the behavior of complex systems. This multiscale assessment reveals how the perceived emergence, self-organization, and complexity can change with the scale of observation.

Implications and Speculations

• The integration of information theory provides a robust and unambiguous mathematical framework for exploring the dynamics of complex systems. Such formality can enhance the replicability and comparability of studies, fostering a deeper theoretical understanding across varied disciplines including computational biology, artificial intelligence, and network theory.
• Practically, these measures can inform the design of systems that aim to balance between predictability and adaptability, drawing from nature-inspired systems like neuronal networks, ecosystems, or social networks where such balances are beneficial.
• The specification of these measures opens avenues for exploring ultra-large-scale systems and their dynamics. Studies could further probe real-world applications such as urban planning or climate modeling, where complexity and emergence are prevalent and significant to system sustainability and adaptability.
• Finally, the paper hints at the potential for advancing our understanding of computational universality and the conditions under which systems can achieve a universal computational capacity through observed indicators such as complexity and homeostasis.

In conclusion, this paper provides an important theoretical clarification and practical toolset for analyzing and interpreting complex systems. It sets a foundational benchmark for future research to calibrate, compare, and potentially enhance the measures for different applications ranging from theoretical physics to computational architecture. As researchers extend these concepts and measures to new domains, the theoretical constructs detailed in this work promise to yield significant insights into the nature of complexity itself.