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Complex Adaptive systems tend to organize themselves into hierarchical, nested and 'scale-free' systems.

Complex systems exhibit important scalar dynamics from two perspectives. First, they are often built up from nested sub-systems, which themselves may be complex systems. Second, at a given scale of inquiry within the system, there will be a tendency for the system to exhibit some form of power-law (or scale-free) dynamics in terms of how the system operates. This simply means that there will be a tendency in the system for a small number of elements within the system to dominate: this system domination can manifest in different ways, such as intensity (earthquakes) frequency (citations) or physical size (road networks). In all cases a small ratio of system components (earthquakes, citations, or roads) exert a large ratio of system impact. Understanding how and why this operates is important in the study of complexity.

Nested Features

To understand what we mean by 'nested', we can think of the human body. At one level of magnification we can regard it as a collection of cells, at another as a collection of organs, at another as a complete body. Further, each body is itself part of a larger collection - perhaps a family, a clan or a tribe - and these in turn, may be part of other, even larger wholes:  cities or nations. In complex systems we constantly think of both parts and wholes, with the whole (at one level of magnification) becoming just a part (at another level of magnification). While we always need to select a scale to focus upon, it is important to note that complex systems are open - so they are affected by what occurs at other scales of inquiry.

When trying to understand any given system within this hierarchy, the impact of subsystems typically occurs near adjacent scales. Thus, while a society can be understood as being composed of humans, composed of bodies, composed of organs, composed of cells, we do not tend to consider the role that cells play in affecting societies. Instead, we attune to understanding interactions between the relevant scales of whatever system we are examining.  Depending on the level of enquiry that we choose,  we may look at the same entity (for example a single human being) and consider it be an emergent 'whole',  or as simply a component part (or agent) within a larger emergent entity (one body within a complex society).

Various definitions of complexity try to capture this shifting nature of agent versus whole, and how this alters depending on the scale of inquiry. Definitions thus point to complex adaptive systems as being hierarchical, or operating at micro, meso, and macro level.  In his seminal article The Architecture of Complexity, Herbert Simon describes such systems as  'composed of interrelated sub-systems, each of the latter being, in turn, hierarchic in structure until we reach some lowest level of elementary subsystem'.

Why is this the case? And why does it matter?

Simon argues that, by partitioning systems into nested hierarchies, wholes are more apt to remain robust. They maintain their integrity even if parts of the system are compromised. He provides the example of two watch-makers, each of whom build watches made up of one thousand parts. One watchmaker organizes the watch's components as independently entities - each of which needs to be integrated into the whole in order for the watch to hold together as a stable entity. If one piece is disturbed in the course of the watchmaking, the whole disintegrates, and the watchmaking process needs to start anew. The second watchmaker organizes the watch parts into hierarchical sub-assemblies: ten individual parts make one unit, ten units make one component, and ten components make one watch. For the second watchmaker, each sub-assembly holds together as a stable, integrated entity, so if work is disrupted in the course of making an assembly, the disruption affects only that component (meaning a maximum of ten assembly steps are lost).  The remainder of the assembled components remain intact.

If Simon is correct, then natural systems may preserve robustness by creating sub-assemblies that each operate as wholes. Accordingly, it is worth considering how human systems might benefit from similar strategies.

Scalar Features

Simon's watchmaker is a top-down operator who organizes his work flow into parts and wholes to keep the watch components partitioned and robust, creating a more efficient watch-making process. What is noteworthy is that self-organizing  systems have inherent dynamics that appear to push systems towards such partitioning, and that this partitioning holds specific structural properties related to mathematical regularities.

A host of complex systems exhibit what is known as Self Similarity - meaning that we can 'zoom in' at any level of magnification and find repeated, nested scales.  These scale-free hierarchies follow the mathematical regularities of Power Laws distributions.  These distributions are so common in complex systems, that they are often referred to as 'the fingerprint of self-organization" (see Ricardo Solé).  We find power-law distributions in systems as diverse as the frequency and magnitude of earthquakes, the structure of academic citation networks, the prices of stocks, and the structure of the World Wide Web.

Further, complex systems tend to 'tune' themselves to what is referred to as Self-Organized Criticality: a state at which the scale or scope of a system's response to an input will follow power-law distribution,  regardless of the intensity (or scope) of the input. While not fully understood, it is believed that systems organize themselves this way because it is a regime in which systems are able to maximize performance while simultaneously using the minimum amount of available energy. When system are poised at this state they also have maximum connectivity with the minimum amount of redundancy. It is also believed that they are thus the most effective information processors in this regime.

Why Nested and not Hierarchical?

The attentive surfer of this website content may notice that in the various definitions of complexity being circulated, the term 'hierarchical' is used to describe what we call here 'nested scales'. We have avoided using this term as it holds several connotations that appear unhelpful. First, a hierarchy generally assumes a kind of priority, with 'upper' levels being more significant than lower. Second, it implies control emanating from the top down. Neither of these connotations are appropriate when speaking about complex systems. Each level of nested orders is both a part and a whole, and causality flows both ways as the emergent order is generated by its constituent parts, and steered by those parts as much as it steers (or constrains) its parts once present. We hope that the idea of 'nested scales' is more neutral vis-a-vis notions of primacy and control, but still captures the idea of systems embedded within systems of different scales.

To learn more about these phenomena, see

Power Laws

Self-Organized Criticality


Urban Applications:

See also:


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    CAS tend to organize to a 'critical state' where, regardless of the scale of a given input, the scale of corresponding output observes of a power-law distribution.

    Strike a match and drop it in the forest. How big will the resulting fire be? The forest is dry but not overly so... vegetation is relatively thick. Will the fire burn a few trees and then flame out, or will it jump from branch to branch, burning thousands of acres to the ground? Read more and see related content for Self-Organized Criticality →

    'Scale-free' networks are ones in which identical system structure is observed for any level of network magnification.

    Complex systems tend towards scale-free, nested hierarchies. By 'Scale-free', we mean to say that we can zoom in on the system at any level of magnification, and observe the same kind of structural relations. Thus, if we look at visualizations of the world wide web, we see a few instances of highly connected nodes (youtube), many instances of weakly connected nodes (your mom's cooking blog), as well as a mid-range of intermediate nodes falling somewhere in between. The weakly connected nodes greatly outnumber the highly connected nodes, but the overall statistical distribution of connected vs unconnected nodes follows a power-law distribution. Thus, if we 'zoom in' on any part of the network (at different levels of magnification), we see similar, repeated patterns.

    'Scale Free' entities are therefore fractal-like, although scale-free systems generally are about the scaling of connections or flows, rather than scaling of pictoral imagery (which is what we associate with {{fractals}} HANDLEBAR FAIL or objects that exhibit Self Similarity . Accordingly, a pictoral representation of links in the world wide web does not 'look' like a fractal, but its distributions of connections observes mathematical regularities consistent with what we observe in fractals (that is to say, power laws).

    A good way to think about this is that, while both scale-free systems and fractals follow power laws distributions, not all power law distributions 'look' like perfect fractals!

    At the same time, sometimes the dynamics of scale free networks align with the visuals we consider to be fractal-like. A good example here is the fractal features of a leaf:

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    Complex System behaviors often exhibit power-laws: with a small number of system features wielding a large amount of system impact.

    Power laws are particular mathematical distribution that appear in contexts where a very small number of system events or entities exist that, while rare, are highly impactful, alongside of a very large number of system events or entities exist that, while plentiful, have very little impact. Power laws arise in both natural and social system, in contexts as diverse earthquake behaviors, city population sizes, and word frequency use.

    'Normal' vs 'Power Law' Distributions

    Complex systems are often characterized by power law distributions. A power law is a kind of mathematical distribution that we see in many different kinds of systems. It has different properties from a well known distribution  - a 'bell curve' 'normal' or 'Gaussian' distribution.

    Let's look at the two here:

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