Source-linked AI summary
Complexity and Information: Measuring Emergence, Self-organization, and Homeostasis at Multiple Scales
Carlos Gershenson, Nelson Fernandez
TL;DR
The paper addresses ambiguity surrounding concepts used in complex-systems research. It proposes information-theoretic measures and illustrates them computationally, concluding that emergence, self-organization, complexity, and homeostasis can be related to information production and system stability.{
Problem
Ambiguity and diversity in definitions and measures of complexity and related concepts have hindered progress toward clear meanings.
Method
The paper proposes abstract measures based on information, distinguishing input information from output information produced through computation.
Results
Random Boolean networks show regime-dependent relationships between H and C, with ordered networks having consistently high H and chaotic networks having consistently uncorrelated H.
Takeaways & Limitations
The proposed measures frame emergence as information produced by a system and provide abstract measures of emergence, self-organization, complexity, and homeostasis.
Takeaways & Limitations
The paper identifies the information used by the measures as an unresolved discussion question.
Abstract
from arXiv · showhide
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.
1 Departamento de Ciencias de la Computaci´on
This section identifies the Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas and its Universidad Nacional Autónoma de México affiliation.
- The listed institutional affiliation is the Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México.
- The listed address is A.P. 20-726, 01000 México D.F., México.
2 Centro de Ciencias de la Complejidad
This section lists the Universidad Nacional Autónoma de México as an institutional affiliation.
- The listed institution is the Universidad Nacional Autónoma de México.
3 Laboratorio de Hidroinform´atica, Facultad de Ciencias B´asicas
This section lists the Universidad de Pamplona, Colombia, as an institutional affiliation.
- The listed institution is the Universidad de Pamplona, Colombia.
4 Centro de Micro-electr´onica y Sistemas Distribuidos,
This section lists the Universidad de los Andes in Mérida, Venezuela, and identifies the paper's focus on complexity and information-related concepts.
- The listed institution is the Universidad de los Andes, Mérida, Venezuela.
- The keywords are complexity, information, emergence, self-organization, and homeostasis.
1 Introduction
Complex-systems concepts have accumulated diverse and sometimes ambiguous meanings, motivating a unified information-theoretic framework. The paper proposes general measures and illustrates them across multiple scales using Boolean networks and cellular automata.
- Diverse notions and measures of complexity and related concepts have hindered a unified framework and encouraged non-scientific misuse.
- The paper proposes general information-theoretic measures of emergence, self-organization, homeostasis, and complexity.
- The measures encompass several earlier definitions as particular cases while remaining formal, precise, and accessible to readers without extensive mathematical training.
- The framework is extended to multiple scales and illustrated with experiments on random Boolean networks and elementary cellular automata.
2 Information Theory and Complexity
The paper frames information as a scale-dependent description of uncertainty and uses it to formalize complexity, emergence, self-organization, and homeostasis. These concepts depend on the observation scale, system dependencies, and the stability of system dynamics.
- Information theory: Shannon information is maximal at P(0) = P(1) = 0.5 and zero when the future binary value is certain.
- Complexity: Complexity can mean the information required to describe a phenomenon at a particular scale, so changing the observation scale can change the measured complexity.
- Complexity: Several approaches treat complexity as a balance between ordered and chaotic dynamics rather than as information alone.
- Emergence: Emergence compares information across scales or system levels, with higher values associated with many dependencies between components.
- Self-organization: Self-organization can be measured as reduced information, but its classification may change with the scale, state-space partition, and observer-assigned meaning of states.
- Homeostasis: Homeostasis concerns maintaining dynamics near attractors and keeping essential variables within a viability zone despite perturbations, without requiring an immobile state.
3 An abstract proposal
The paper proposes information-theoretic measures that formalize emergence, self-organization, complexity, and homeostasis, and extends them across observational scales. Emergence tracks information production, self-organization tracks information reduction, complexity balances them, and homeostasis tracks maintained information.
- Emergence: The proposal uses information transformation to formalize emergence as novel or produced information relative to input information.With input information Iin and output information Iout = f(Iin), emergence is proportional to their information ratio and is unitless.
- Self-organization: Self-organization measures information reduction: it is positive when Iin > Iout and negative when the process generates more information.Its units are bits, and for random inputs with Iin = 1 it has a simplified form.
- Complexity: Complexity is defined as the multiplication of emergence and self-organization, becoming high only when their opposing tendencies are balanced.This treats complexity as a balance between order, represented by S, and variety or chaos, represented by E.
- Homeostasis: Homeostasis is the opposite of information change, with high H indicating maintained information and system stability.It is derived from normalized Hamming distance between input and output and is unitless; high sensitivity to initial conditions corresponds to uncorrelated H.
- Multi-scale measures: Normalized information generally decreases as the base increases in short strings because finite-size effects leave many large-scale values unrepresented.The measures are illustrated using a 32-bit string converted to different bases and a long pseudorandom string at different scales.
- Multi-scale measures: Across scales, E, S, and C behave similarly, whereas H differs because Hamming distance measures the percentage of differing symbols.For uncorrelated states, the lowest expected H decreases with increasing base b as 1 2b.
- Multi-scale measures: The measures can change substantially with scale because information itself may change when binary strings are grouped into larger bases.For example, 10101010 has I1 = 1 but I2 = 0; long random strings instead maintain high entropy across scales, with Ib ≈1.
4 Experiments
Computational experiments apply the proposed measures to random Boolean networks and elementary cellular automata, examining their behavior across connectivity and observation scales. The results associate low connectivity with ordered dynamics, medium connectivity with high complexity, and several cellular-automaton patterns with scale-dependent changes in emergence, self-organization, complexity, and homeostasis.
- Random Boolean networks: Random Boolean networks contain N Boolean nodes whose fixed lookup-table functions depend on the states of K interacting nodes.Their randomly generated structure and function remain fixed during the dynamics.
- Experimental setup: The experiments evaluate emergence, self-organization, complexity, and homeostasis from node time series and network-level information measures.For random Boolean networks, 1000 systems run from random initial states; for cellular automata, 50 instances of selected rules are evaluated.
- Random Boolean networks: Low K produces low emergence and complexity but high self-organization and homeostasis, whereas high K produces high emergence and low self-organization and complexity.These patterns correspond respectively to ordered dynamics with few changes and chaotic dynamics with high variability.
- Random Boolean networks: For 2 < K < 3, emergence and self-organization balance, producing high complexity; homeostasis also balances H = 1 with H ≈0.5.This connectivity range is consistent with critical dynamics in finite random Boolean networks.
- Scale effects: Increasing observation scale can shift cellular-automaton measures: several rules reduce emergence and increase self-organization or complexity, while homeostasis may become highly correlated or anticorrelated.For the broader experiments, the maximum C increases moderately with scale, E slightly decreases, and low H values decrease with scale.
5 Discussion
The discussion interprets the information-theoretic measures across scales, showing how emergence, self-organization, complexity, and homeostasis distinguish ordered, chaotic, and complex dynamics. Multiscale profiles and measure combinations reveal patterns that single-scale measures can miss, while results depend on representation choices and finite-size effects.
- Emergence across rules: Rule 30 produces maximum information and emergence through pseudorandom behavior, while rule 0 loses incoming information after one time step.Rule 1 produces substantial information, and rule 110 produces complex patterns with interacting gliders on a regular ether.
- Multiscale ECA profiles: Rule 0 remains unchanged across scales, whereas rule 1 resembles it at higher scales after showing high E and low S and H at b = 1.Rule 110 decreases E and H while increasing S and C with scale; rule 30 maintains high E and low S, H, and C.
- Complexity as balance: Complexity is maximal when emergence and self-organization balance at S = E = 0.5, while extreme order or disorder produces low C.The measure is presented as simpler to calculate and explain than statistical complexity, despite similar behavior.
- Multiscale interpretation: Multiscale profiles provide more insight than individual scales because some class II rules have high E at one scale but minimal E at higher scales.The authors describe the resulting cross-scale changes as informative about system dynamics.
- Scope and measurement choices: Interpretation depends on finite string length, scale, input assumptions, and representation: ECA results differ when strings are read vertically, horizontally, or diagonally.The discussion also notes that the simplified Iin = 1 assumption affects the emergence experiments.
6 Future Work
The authors propose extending the measures beyond the present simulations and elementary cellular automata to real systems, additional computational models, alternative complexity profiles, and networked systems. They also identify analytical validation and broader conceptual comparisons as future directions.
- Applications to real systems: The authors plan to apply the measures to ultra-large-scale systems and time series from real systems, including cases with non-random Iin.They intend these applications to contrast the usefulness of the measures.
- Analytical validation: Future work includes analytical solutions for random Boolean networks to contrast with the existing computer-simulation results.
- Comparison with other measures: The proposed complexity measure could be compared with Bennett’s logical depth and Fisher information.The authors identify potential relationships with both measures as open questions.
- New multiscale profiles: The authors propose extending the multiscale approach to new complexity profiles that combine Bar-Yam’s profile and the σ profile.They also suggest studying how information changes meaning with context across scales.
- Broader system classes: Further extensions target autopoiesis through a life ratio and computing networks with potential applications to complex-network studies.The life ratio is described as relating information produced by a system to information produced by its environment.
- Research scope: The authors note that many research questions exceed their capabilities because of another finite-size effect and invite broader community investigation.
7 Conclusions
The paper proposes information-theoretic measures of emergence, self-organization, complexity, and homeostasis, illustrating them with computational experiments on discrete dynamical systems. The measures extend across scales and characterize complexity as a balance between emergence and self-organization, while homeostasis captures information stability over time.
- The paper proposes abstract measures of emergence, self-organization, complexity, and homeostasis based on information theory to clarify these concepts.
- The measures were illustrated with computational experiments on random Boolean networks and elementary cellular automata and can produce different results at different scales.
- Emergence measures information produced relative to information received, whereas self-organization measures the input-output information difference.
- Under random inputs, emergence becomes the opposite of self-organization, and complexity represents their balance.
- Homeostasis measures the stability of information over time.
- The paper relates these measures to studying complexity, emergence, self-organization, and homeostasis across phenomena described in informational terms.