Dear Veerendra, there is no one (= unique) definition of complexity. Good science does not work - unlike in the past - with definitions but with problems. We are scientists because of the problems we desalt with, we try to solve, etc.

Regarding a definition of complexity, you probably know Horgan's classical article in SciAm (1995). By the time, Horgan mentions at least 46 definitions. From then on to-date we can safely count much more that 60 definitions.

Now, concerning the problem about measuring complexity there are, as it happens, two general positions, thus:

a) on the one hand there are those who claim that every single complex system has its own measure:

b) on the other hand, there are those who believe that it is possible a general measure of/about complexity. (I can mention authors, schools, etc.).

You can safely say that multiplicity, diversity is one of the features that do characterize complexity, indeed.

In business, there's a pretty interesting company that measures the level of risk associated with complexity in an enterprise, process, or in politics - see at Ontonix dot com . Look at their description of what complexity is under the 'complexity' tab as well.

The term 'Complexity' has become so ubiquitous that it appears in every form of conversation and discourse- from complexity in human relationship to complexity of socio-economic, engineering and biological systems etc. Computer science measures space and time complexity of algorithms with precision. However, algorithmic complexity is very different from the notion of complexity we use when we talk of socio-economic or biological systems etc. It is not always possible and perhaps desirable to 'define' everything formally. It is also true that complexity means different things in different context. However, it is worth while to attempt to discover all possible dimensions / aspects of complexity in a given context  or system. 

Take a look at the article http://www.nature.com/nphys/journal/v8/n1/full/nphys2190.html

by Jim Crutchfield and at that whole issue  of Nature Physics. All the best, Leo.

A very nice book about the subtle relations among complexity and diversity (and robustness (or fragility)) in systems is Scott E. Page's "Diversity and complexity":

http://press.princeton.edu/titles/9208.html

You can also check "The difference", where he goes deeper on this issue.

http://press.princeton.edu/titles/8757.html

Best wishes!

Thanks Marti for sharing the references.

One interesting attempt:

Complexity as thermodynamic depth

Lloyd and Pagels (1988)

http://adsabs.harvard.edu/abs/1988AnPhy.188..186L

There seem to be two broad lines of work on complexity.

One concerns the relationships (mathematical or other) between components of a system that give rise to complex characteristics. I see this as an extension of systems analysis, trying to understand the inner workings of systems that give rise to certain properties and behaviour. It includes efforts to measure complexity and identify the parts of a system that are generating it.

The other approach to complexity takes emergence and related characteristics as its starting point and goes on to consider how to operate in and on such a system, generally a human system, to achieve a desired outcome or enable the system to optimise its own behaviour. This approach makes no effort to quantify complexity or compare the relative complexity of systems but focuses instead on how to distinguish a complex setting from other types and an understanding of the types of strategies that will be useful in each type of setting.

This is the approach championed by David Snowden and documented in his work on the Cynefin framework, see the Cognitive-Edge web site for formal references and a large amount of informal material ranging from Youtube explanations of key points to blogs recording the development of these ideas.

There is no conflict between the two approaches but, while the methods based on the Cynefin framework are only a decade or so old they show signs of being valuable management tools for operational use. The mathematical approach of analysis and measurement may offer valuable insights but seem unlikely to assist non specialists.

If anyone disagrees with any of this I'm very keen to hear about it. The subject is still very new, as far as I can see, and there is a lot yet to be learned.

Dear Stephen, thank you very much for calling our attention on to Cognitive-Edge. I did not know it. I appreciate it.

Just for the sake of conversation, I  think there are notable differences between systems science and complexity science - something that has not been thorough fully worked out in the literature.

However, I' d like to mention that there is a large consensus among the community of complexologists in that the social systems and particularly the human systems are the most complex ones - in any respect. Probably the first to point to this was H. Pagels, a physicist, himself. The distinction - that I do understand - that you make is not tenable any longer, it seems.

This is at the very least a complicated subject in its own right, possibly complex, and I didn't expect my broad generalisations to stand up in absolute terms. If I can build on your observations, based on my understanding of the subject so far, to clarify:

I didn't mean to say systems science and complexity science are the same, just that there are some similarities in the sense that mathematical and other relationships are used to model or otherwise analyse systems to seek to explain or to model behaviour and properties. This is a different response to the existence of complexity to taking the behaviour as a fact and finding ways to work with it.

I wouldn't make a qualitative distinction between the complexity of human and other systems but in quantitative terms it seems that human systems are probably one end of a scale with physical and technological systems at the other. The main point for me is that we should not expect to make too much headway in finding predictive models or methods of control in the truly complex domain of human systems so I for one am not inclined to apply much energy to that.

In the short term, one of the implications of this insight is that attempts to explore complex human systems with predefined frameworks such as conventional questionnaires are almost certain to miss something because we cannot know in advance what all the important cause-effect relationships will be. This means that oblique approaches, such as narrative capture and other methods, are likely to be more effective and efficient. This has significant implications for management research and efforts to bring about change.

Ross Ashby does not differentiate between variety and complexity and says complexity is not an inherent property of objects and it depends on the observer. Some researchers differentiate between complexity and difficulty. Many a times we use difficulty and complexity synonymously. "Difficult" is an object which, by means of an adequate computational power, can be deterministically or stochastically predictable. Complex, on the other hand, is an object which can not be predictable because of logical impossibility or because its predictability would require a computational power far beyond any physical feasibility, now and forever (Biggiero, 2001). This article "Sources of complexity in human systems" by Lucio Biggiero can be found at: link.springer.com/article/10.1023%2FA%3A1009515211632

I am busy with AI a bit, although Latvian AI foundation does not exist anymore.   May be the complexity of human mind can be measured by the number of fields, accessible for consciousness. Fields can be defined as different areas of human activities, using similar math, physical models and tools. Each field can be characterized by the number of probability distributions, necessary for actions in the field. Probability distributions, as we know, can be characterized by information content. Of course, it is not simple to connect these notions under one formula and create a reasonable definition of complexity of human mind. But the human mind is not simple too. 

I do agree with Dr. Grey on that, in order to capture complexity, we must understand the inner workings of systems that give rise to certain properties and behavior (holism, emergence, compositionality), not forgetting about the contexts/surroundings of their workings that also help identify the influencing factors and causalities. But I also agree with Dr. Rai and his quoting the work of Ashby, who argued that complexity is not an inherent property of objects and it depends on the observer. Interestingly, in our research in smart adaptive cyber-physical systems and applications in human/social contexts these are somewhat intertwined, as shown in the enclosed figure.

A measure of complexity is not a simple case. Just considering a very specific family of engineered systems (smart adaptive cyber-physical systems), rather than all physical, biological, human and social systems, we must talk about an aggregate complexity, in which the constituent complexities change on their own way, but they also strongly influence each other due to their tight coupling. In general, complexity of open CPSs increases with: (i) the number of (real or potential) functional components (hardware, software and cyberware), (ii) the intricacies of the distinct functional components, (iii) the heterogeneity of the structural components, (iv) the multiplicity of scales of structural components, (v) the number of explicit and implicit connections between them, (vi) the intrinsic nature of the connections among components, (vii) the overall (compositional) degrees of freedom of a system to adapt itself, and (viii) the enablers and/or inhibitors of self-organization. As an outcome of the interplays of these factors, complexity of these engineered systems may appear in five forms: (i) static complexity (the number and relationships of components that do not change with time), (ii) dynamic complexity (the number and relationships of components that change with time), (iii) self-organizing complexity (open systems reorganize themselves to different systems), (iv) evolving complexity (open systems evolve through time into different systems) and (v) co-evolving complexity (two-way interplay between the changing system and its environment). These may appear together and interaction – and this is the very reason why it was proposed to characterize CPSs based on their aggregative complexity. Also important, in case of many engineered systems, the increase of complexity goes together with the increase of intelligence, adaptability and autonomy, but also of uncertainty and unpredictability. Anyway, Bar-Yam, Y. proposed that complexity is the amount of information that is necessary to describe a complex system.

Dear Mr. Veerendra,

            Complexity is a subjective term and often used to compare two things (e.g. like tall is relative term). The purpose of research for industrial engineering or software engineering is to invent tools and processes for reducing complexity. Things that were complex few years ago no longer complex (or less complex due to inventions of tools or processes).

     Usually, complexity may be reduced by increasing the degree of these 3 factors (i) increasing the degree of division of labor, (ii) increasing the degree of specialization and (iii) increasing the degree of automation.

Best Regards,

Raju

It this true: When we reduce complexity without reducing a complex system to a non-complex state, we reduce the chance of emergence?

Dear Prof. Horvath, you are saying "When we reduce complexity without reducing a complex system to a non-complex state we reduce the chance of emergence?". Somehow, I feel it is other way around. "When we reduce complexity without reducing system to a non-complex state, we actually increase the chance of emergence?". Please think again. There is so much to learn from you Sir. thanks

There are existing many definitions of complex systems. Each of them has its own advantage and disadvantages. This multiplicity is arising from the fact that we have no unified theory of complexity. Mathematically, it is a very challenging task as it requires working with massively parallel systems.

Our own, human, thinking is inherently linear & sequential. We are very underperforming in true massively parallel thinking.

With respect to measuring complex systems, we have again a huge problems with finding the right ways. Nevertheless, there is existing a broad number of complexity measure existing & origination either in statistical physics or Information (Shannon) entropy.

Just to name a few

* Approximate entropy

* Sample entropy

* Permutation entropy

* Multiscale entropy

* Boltzmann & Shannon formulas of entropy

* etc.

Citations and details provided in the paper bellow.

When entropy measures applied to any complex system, it is done in such a way that all minuscule details in evolving CS are sampled by complexity measure. This measure is serving as a projection of a complex system down to much lower dimensionality. It is typically a kind of distribution of some measured feature(s) of the original CS. Surprisingly, in many cases, such brutal simplification works.

Ludwig Boltzmann did exactly the same in his kinetic theory of gases. He sampled the original system with astronomical numbers of freedoms into just distributions. This is trick used in many variants up today. It was used even in the bellow provided paper.

***

An example of how to work with permutation entropy and AI to discern hidden information in biosignals is provided in the paper Torsades de Pointes arrhythmias on the prediction of arrhythmias from ECG recordings.

Application of Machine Learning and Complexity in Medicine: Prediction of Drug Induced Arrhythmias in Rabbit Model up to One Hour Before Their Onset

https://www.researchgate.net/publication/340460356_Application_of_Machine_Learning_and_Complexity_in_Medicine_Prediction_of_Drug_Induced_Arrhythmias_in_Rabbit_Model_up_to_One_Hour_Before_Their_Onset_early_version_requested_due_to_COVID-19_unfinished_m