You may find value in a review of the work of Stan Salthe.   See:

http://www.nbi.dk/natphil/salthe/

http://www.amazon.com/Development-Evolution-Complexity-Biology-Bradford/dp/0262513838

in my view a good first approach to Complex System's hierarchy [and maybe a fundamental reading] is: 

Herbert Simon [1962] The architecture of Complexity

Many author's have written later building on Simon's proposals.

Good Luck

Dear Franciso, voilà a nice question, indeed. I would say the following: the past in the reseacrh in complex systems did talk about hierarchies, as it happens. Ricardo's comment is a good indication about it. Several others can be mentioned.

The truth is -and this is a theoretical discussion, of couse- that properly speaking,in complex systems there is no room for speaking about hierarchies. The very idea of "hierrahies" comes from the past. If you wish back from Aristotle.

However, the truth is that in nature there are NOT hierarchies, no matter what. A fined-tuned comprenehnsion of complex systems goes in this new direction. This is why we speak of multicalarity, networks, topology, and the like.

Thank you all for your answers. 

@William. Salthe's conclusions on hierarchy are relevant no doubt; sometimes difficult to digest and definitely he poses many challenges for those of us working on social sciences and trying to build our own discipline-based complexity approach. Long to say in this space though...

@Ricardo. Thanks for Simon's reference. I agree Simon's work is a "must-read", however this time I'm seeking for a step further than conceptualization and into the practical consequences and measures of complexity

@Carlos. Nice to hear from you again. I welcome your challenging argument, so let me simply ask you back this question: does mathematics (or physics, or biology) exist in nature? what do you think? I guess you'd answer NO, too. Sure, they do not exist, they are human constructs to approach and understand nature, aren't they? So it is "hierarchy": an observer's construct to approach complexity. Moreover, who cares whether hierarchy exists in nature or not after all?

By the way, one last comment. What are networks, but hierarchies? It seems as if your "fine-tuned" comprehension of complex systems came back to an idea "from the past".

Dear Francisco, Thank you very much for yur fine and kind replies to each of us. I accept gladly your questions.

The very idea of hierarchies is a human, i.e. anthropological one. Since the late Paleolitics, it appears, we used to think in terms of hierarchies. An idea that, as you know, was seriously reinforced after the Greeks, thanks to the Christians, on to Modernity. This is probably one of the most ancient beliefs. Accordingly, we the (wrongly) thought that there were and are hierarchies in nature - as well as in Heaven (en passant).

mathematics does exist in nature. Not with the levls of abstraction we humans have. But animals do have math - as a number of publications have shown it. The matter is abuot a fiffrenec of quality - namely, the abstraction we have developped.

I must apologize: notworks are not hierarchies. Certainly not complex networks. Let's remember scale-free networks, f.i.

Hi Carlos. Nice also to hear from you again

On the other hand, I must admit that I am somehow intrigued with your comments. 

Of course I am aware that currently those issues you mention [multiescalarity, networks, etc...] are more frequently debated than hierarchies, something probably related and greatly developed in parallel to the growth of one of our biggest creations: the World Wide Web.

While I admit they are interesting approaches, I think they add to the previous points of view, yet do not eliminate them. Somehow a ‘hierarchy’ appears as a more vertical approach and a ‘network’ as a more horizontal approach [something that also can be already suggested by Simon’s ‘large span hierarchies’]. They can perfectly complement one with another.

But your comment suggests that at some point during this process ‘hierarchies’ have been 'scientifically rebutted', which for me it is ‘news’… Is it so? Are no longer armies [CAS] organized as hierarchies? Neither are anthills [another CAS] currently divided into different types of ants with different functions? Companies [also CAS] are no longer organized hierarchically? [last time I worked in one of them –not so long ago- they still ‘believed’ in hierarchies…

I have seen many proofs that hierarchies exist in CAS, yet I have no seen any proof that they do not exist [in fact, your’s is the first comment I receive stating it]. As an expert in Epistemology you may inform me if I have missed something…

I do not try to diminish the relevance of the approaches you mention, but I do not understand the need of denying both the utility and existence of hierarchies. In my view, both types of organization exist and are useful for our analysis and understanding of reality.

Also in another dimension, I agree with Francisco, hierarchies as a conceptual tool have proven their utility in many scientific fields ranging from Fuzzy Logic [Fuzzy Signatures are currently widely used in hospitals for diagnosis complicate diseases as SARS]; Decision theory [Analytical Hierarchy Process is widely used, …], and many others...

Just to highlight their great utility, 100% of the current approaches to modelling sustainability [I think we can agree it is currently one of our more important concerns] adopt a hierarchy to model Human Society's sustainability [and I believe we also agree that any Socio Ecological System is a complex system].

Kind Regards

Dear Ricardo, as you know there are numerous ways to grasp a complex system. CAS, originally pointed out by M. Gell-Mann is one of them. Historically it is a good concept. However -I must apologize for this - it falls short in properly understanding the complexity of a system. The argument more or less goes as follows:

Adaptation is a very good Darwinian way to understand evolution. However, as we all know, adaptation is closely related to selection. Now, selection is but, according to Darwin himself, only one way to grasp evolution. Many works have been done in order to complement, if you wish, selection; whence, adaptation. This means that there are far more robust concepts in grasping complexity than CAS. (By the way, that bunch of cyberneticians out there, love CAS…).

As to sustainability: I am a radical one. Sustainability is the most recent "happy face" of capitalism. You now where the trap lies? Speaking in heavy economics: in sustainability the production function remains untouched. Hence, it is exactly the same thing as classical liberalism, neoliberalism, and the like.

@ Carlos

It is sometime now already since I decided to get back to use simply the designation Adaptive Systems instead of Complex Adaptive Systems.

And in relation to sustainability, of course the power structures implied in capitalism try [as always has been] to use the term in their best interest. As adaptive systems they just try to sustain themselves, and in order to do so they also adapt to the new terms, transforming them for their own interest [survival, sustainability…].

Currently existing powers maximize their probability to endure the more they prevent society from changing [his kind of thought comes from long time ago...], and in order to do so in a 'capitalist consumer society' sustainability 'needs urgently to be translated' into ‘how can we sustain our production/consumption patterns as long as possible. Otherwise somebody could translate it into the ‘need to change production/reduce consumption’, shaking the whole foundations of society and current power structures….

But other approaches are possible [and not even radical at all].

If you have the time, you may like ‘A systemic approach to sustainability’ [though there are some references to the ‘past’ i.e., Simon, Von Bertalanffy...]. But I must advise you also may not like it.

@ Manuel

In the referred article there are some basics in relation to hierarchically understanding of systems which are later mathematically sustained with a few rules for consistency.

Book A mathematical Theory of Sustainability and Sustainable Development

Dear all, although discussion is interesting I'd suggest to further answer the original question, which is if a measure of hierarchy in a complex system can be provided. Thanks.

Dear Ricardo, systems approach? I pass.

You are right, Francisco. I have already said something about hierarchies and complexity. To me, the core question remains about how to measure a complex system.

Finding hierarchies is a graph theoretic problem.

For example, there should be trees within a network.

Or topological sorting of nodes: there

should be no circles, or at least,

no significant ones. Circles should be

breakable by treating one or more edges

as not very relevant.

Related literature:

http://www.eurekalert.org/pub_releases/2008-05/sfi-npd050108.php

http://www-personal.umich.edu/~mejn/papers/cmn08.pdf

Regards,

Joachim

@ Joachim Pretty interesting article. 

@ Francisco et al.

From the perspective of measuring information content of networks, I suggest:

Chapter 5. Quantitative Measures of Network Complexity. Danail Bonchev and Gregory A. Buck [2005] provides some interesting insights.

Another interesting article focused on measuring the Information contained in networks is ' Information Theory of Networks. Matthias Dehmer [2011]. It can be related to hierarchies as a particular type of graphs.

From the perspective of 'interpreting' hierarchically structured information, I suggest the approach made within fuzy logic usually designated as 'Fuzzy Signatures'.

There are many short articles explaining them in the internet. A complete PhD Thesis from 2008 can be found as 'Fuzzy Signatures: Hierarchical Fuzzy Systems and their applications'.  Author: Balapuwaduge Sumudu Udaya Mendis.

Best regards to all

PS: I have also proposed some issues on how to measure meaningful information contained in hierarchies, including some personal proposals of information aggregation. The aim is to provide a measure of the concept which characterization is intended through the hierarchical representation [it may be organization, but also others]. Hence it links both information & Logic approaches. You can check results at

http://vixra.org/abs/1411.0052

I think the question is too general because it depends on the type of hierarchy under consideration. If for example are you talking about networks the usual way of measuring the hierarchy is the centrality. And there are different types of centrality... At the end the criteria depends on the objective.

For some type of hierarchies, it could be useful the Schreiber's transfer entropy.

Dear Francisco:

Somehow I read again your question and I am a bit confused and intrigued. It seems to me that your question refers to ‘how a hierarchy can be measured’ while your explanation of the question refers to ‘how and underlying hierarchy can be detected’.

For me [roughly] the first one relates to ‘hierarchy characterization’ and can be confronted from many existing so-called ‘complexity measures’ [as they mainly try to measure organization] while the second refers more to ‘pattern detection’ [criteria for information aggregation –but not for information measuring-, approaching us to Simon 1955, Shpak 2004,…].

But also as Federico suggested, the goal of the analysis is one of the important issues for providing a good answer. Therefore, can you tell us a bit more of which is your objective?

I am quite interested in the issue at the moment

Best Regards

PS: Just to explain my interest. I am currently finishing a hierarchical operational model for increasing ‘urban sustainability degree’ through urban transformations, for which the hierarchy has to be aggregated at different levels. 

For this model I have developed a number of aggregation formulations based both on Communication/Information theory and Complexity measures [hence, for me the first issue, how to measure hierarchy in relation to my aims is solved].

But I still have more than one possible way of grouping some indicators [i.e., I find that a group of indicators admit more than one hierarchical ordering] something that can modify the outcome.

For instance, one of my concerns is… if we admit correlation between variables should be key for hierarchically grouping [something that can be also sustained on Information Theory as Miller,…] Should the hierarchy be built on ‘current’ or ‘expected’ correlations between indicators if they are different?

@Ricardo

Yours are good points. To make my answer brief and simple I'd say we have two different stages in the approach to a (complex) system here or, at least, this is my own practical approach:

1- when you approach the system for the very first time and you need to characterize the system itself. Here you'd need to answer questions like: is the system complex (whether it fulfils certain properties or conditions)? which is the basic building block of the system? what are the components? which are the relationships among components?... To get the answers you need to describe and measure "things"; the goal being to draw the system architecture. I'd label this stage as "mapping of complexity".

2- when you already have "sufficient" knowledge of your system: you are certain your system is complex and you've been able to draw its architecture, then you may want to unveil its hierarchy. Here again you need to measure "things" and come up with a hierarchical model. I'd label this stage as "system modeling".

My question was mainly focused on this second stage.

Hi Francisco

I find your answer brings up some interesting things. Let me pose a question. What do you mean by a complex system?

The first use I find of the term is in Simon’s 1962 The Architecture of Complexity’ where he proposes: “Roughly, by a complex system I mean one made up of a large number of parts that interact in a nonsimple way. In such systems, the whole is more than the sum of the parts, not in an ultimate, metaphysical sense, but in the important pragmatic sense that, given the properties of the parts and the laws of their interaction, it is not a trivial matter to infer the properties of the whole” [Simon, 1962:468]

As it is the first use, we should consider it as its first definition [at least for my knowledge, if you know a previous definition, please let me know].

Now, there have been a lot of proposed definitions for ‘complex system’ later, something very curious in my view as it is something very unusual since…

• … to my knowledge, Simons definition has never been rebutted, systems as he described do exist, and their analysis is useful in many areas.
• …in any other scientific field, when a scientist propose a new model, he searches for a new word, do not try to change the meaning of an already existing term, which is useful to designate other type of existing scientific objects.

So, let us then stick to the initial definition of Simon.

For me, such definition has a degree of redundancy, since Von Bertalanffy’s definition of system already implies that ‘the whole is more than the sum of the parts, i.e., non-linearity or non-sumativity. But it is quite important and it provides two parameters that still underlie any definition of complex system:

…it is composed by many interacting parts.

…it is not a trivial matter [i.e., it is difficult] to infer the properties of the whole from the parts.

In summary, Simon’s definition is referring to ‘having many parts’ and ‘being difficult to understand’ as the main qualities that characterize a complex system.

Now, the above definition is quite interesting, so let us review it briefly:

The first issue is that it relates to two ‘fuzzy qualities in the sense of Fuzzy Sets Theory [Zadeh, 1965]. And there are no crisp limits for fuzzy qualities.

There is no way that we set up an exact limit differentiating a complex from a non-complex system because ‘fuzzy qualities’ do not have exact limits; sometimes they have ‘fuzzy limits’, and sometimes they do not have limits at all. In logical terms, it approaches us to the sorities paradox.

The second issue is that both qualities are greatly ‘subjective’:

The quality of having many parts can differ greatly from one scientific field to another. When we review ecological systems, a number of ten elements is usually not considered as many. But when we review conceptual systems’, a number of ten interacting elements at one level is considered as a maximum number of elements that we are able to value at the same time [Miller, 1951; Saaty, 2003…]

So, there is no way to set a universal parameter of ‘how many elements are many’ and ‘how many elements are few elements’; it has to be established for each particular case.

But also the quality of ‘being difficult to understand’ is greatly subjective; since a person may find trivial what another person finds greatly difficult to understand.

And third, both qualities are ‘independent’; one may change in one direction while the other changes in a different direction.

This is in part related to something mentioned above [the difference in number of elements required to consider complex a real system and to consider complex a conceptual system]

But it is also related to something already present in Weaver’s 1948 ‘Complexity and Science’. The author proposes that as we increase the number of elements, we move from ‘simple problems’ [as problems with very few elements for which we can get to determine a solution] to ‘disorganized problems’ [as problems with a large number of elements for which we can propose approximate solutions based on statistical averages].

So, the complicate [nontrivial or difficult to understand] systems have an intermediate number of elements; enough to prevent us from [or at least hampering] modeling them but not enough to allow us to review them in terms of statistical averages.

Difficulty of understating [non-triviality] modifies in a different way than number of elements. First it increases as systems' number of elements increase, but later it decrease as systems' number of elements keep increasing.

Now, in my view, even when later definitions of complex system sometimes propose different approaches than Simon’s, all of them share these two qualities as main characteristics of so called ‘complex systems’.

So, when you speak about ‘being certain’ that your system is complex I believe that there is no way to acquire such certainty: it is not possible building on current definitions to set a line dividing complex from non-complex systems.

On the other hand, in relation to your queries regarding system’s hierarchies, you may find useful M. E. J. Newman, ‘Modularity and community structure in networks’

Kind Regards

Ricardo

Ps1. If you want to take an epistemological view to concepts ‘complex’ and ‘complexity’ you may like the attached link

Ps2. There is no way to set a crisp limit for differentiating complex and noncomplex systems in the view of Simon’s definition and so called Complexity Sciences, while from an epistemological point of view, there is no need of it, since every system is complex [see Morin's work].

Chapter The definition of Complex and Complexity

Dear Ricardo, no: the very first time when "complex sytems" are introduced as such is that paper you know: Weavers' paper from 1948 publiched in SciAm.

Dear Carlos

I made a full check through Weaver's text [with adobe Acrobat 'find' function] and I can only find the term 'complex problems' but not the term 'complex system'

Are you sure Weaver uses such term? 

Kind Regards

Dear Ricardo, I'm not really keen on the idea of  a "unified theory of complexity", if that means a theoretical framework aimed at making complexity normative, namely, what is and what is not complexity, or how researchers should or should not tackle complexity. I don't think that's even possible or involve any major practical consequences. As a mere intellectual (and spiritual) exercise is interesting, though.

On the contrary, I believe complexity, while being an inherent property of some systems, requires different approaches when coping with different disciplines. In my opinion this is even more valid as you get closer and closer into your discipline own basic building block-bbb (or "complexity-unit").

I'd dare to differentiate between a "macro-level" perspective of complexity, for which a unified vision of complexity might still be possible, and a "micro- level" perspective of complexity, that is, the complexity intrinsic to your discipline bbb (i.e complexity of strings in physics vs complexity of individuals/families in sociology). BBBs are different, so are their micro-level behaviors, so is complexity...

Regarding the question of how can someone be "certain of a system being complex", I mostly agree with you: there is not a universal way to draw the line between complex and non-complex systems and it will never exist. This btw makes sense with the discipline-based micro-level discrimination of complexity. In other words, every discipline at a given moment of time determines what is and what is not complexity, thus taking into account the state-of-the-art of knowledge, tools and trends. No doubt that "complexity" is a dynamic notion, which evolves at the same rate of human knowledge; not in vain complexity is a human approach to the unknown.

Dear Francisco

I understand your reluctance to a unified theory on complexity. So let me explain briefly what it is for, through a very easy example.

Imagine you have to design a city. The drivers have their point of view on how it should be; the pedestrian have different point of view; the guy who has a dog sees it different than the guy who loves skating; the constructor also has his view…. Everybody is working with a point of view that focuses on a particular aspect of the city, while nobody is seeing the city as a whole.

Well, in my view Complexity is currently lacking such overall view. And that is what the unified theory is about. It does not seek to propose detailed solutions for particular issues, but to provide an overall view of the common issues, a set of easy rules [axioms] that allow us to separate consistent from inconsistent proposals.

This means that inside such overall context anyone can still develop his personal proposals and points of view; precisely one of the aims of a unified theory must be to allow particular approaches to build on it.

The axioms are not complicate but simple rules. And their mission is just to provide support for the assertions as well as provide a context against which different proposals can be compared, something currently not possible.

A unified theory does not try to impose a view, but to provide a framework against to test different view. Otherwise, we will continue to be seeing each new author to propose a different approach forever and ever…How can we assess if one approach is better than other?

And it is largely an epistemological issue, since complexity has been born and greatly developed as an epistemological concept. For instance, your term [which I do like much] ‘complexity is a human approach to the unknown’ is a totally epistemic one.

By the way, the etymology of ‘science’ [‘scientia’ or knowledge in latin] leads us to ‘skej’ [Proto Indo European root] as more or less ‘that can be cut’. It seems that those guys long time ago thought that what was able to be cut into pieces [analyzed], could be transformed into science [or knowledge], while what was not able to be ‘divided’ was complex [or unknown]…

Kind Regards

Ricardo

Dear Ricardo, 

I understand what the scope of your work is (an interesting one btw!) However, to be honest, your approach might have some weak connotations. Let me explain them briefly:

1- By establishing a set of rules (or basic axioms) you might make complexity "normative" , that is, "seize" the competence to define what should be or should not be implemented in the city. Some citizens might then come to you and ask:  who the hell are you to decide what proposal is better than others? And they might still be right!. Historically this role belongs to the local authority, because it's the only actor entrusted with the power to enforce "law" among citizens. So, I'd say there is a "representational problem" here regarding the formulation of "unified complexity" and its derivates. A possible solution might be to make citizens participate in an integrative unification of complexity.

2- Your city planning axioms might be valid only for a very short period of time, (maybe weeks or months?). As a complex system, the city changes continuously, therefore you may want to update your rules regularly. Guess the cost of such an effort! So, you might have an "obsolescence problem" here. Besides time scale, you might want to consider the spatial scale of complexity as well, which would hugely increase the "complexity of complexity". 

3- As you disaggregate the city goals into programs, programs into projects, projects into actions.... your "basic" axioms might become inoperative or difficult to apply. I mean, your "rules" might need to become heavily detailed to cope with the lower levels of implementation. The more rules you make, the less practical your approach becomes.

Actually I'm working on a similar problem in the field of the firm, though my approach is different. My analysis of the firm as a complex system has taken me to find a practical solution in the field of network/neural network analysis. As you know this is a field with a strong analytical background, and your work might well fit as a neural network optimization problem, thus aiming at figuring out an algorithm that is able to describe and "predict" the behavior of the actors in the city. These are just some ideas.

I recognized a phi spiral arrangement on mouse corneas and published a report recently:

Promoting convergence: the phi spiral in abduction of mouse corneal behaviors.

In regards to your question, because we are dealing with a situated material system, there should be a clear, connected answer in several years if the problem is interesting enough for further pursuit by all who investigate.  That is, the phi spiral arrangement is recognized to be ~500um and if it's a real logarithmic spiral, it should connect down to the nanoscale.  We can produce atomististic approximations of collagen fibrils, so the question remains how to connect that to the mesoscale and whether the numbers will be consistent in other samples that display similar arrangement.

To illustrate the complexity, one can also think about the surrounding structures that are required for a complete description (that is, the limbus, eye, organism, flow, genetics, cell biology, etc...)  All these things can be given a hierarchical description and a good argument should be connected in relation.