Dear Bin Jiang,

Just thinking about long tails and lognormal distributions (and not directly related to your question). Maybe this data could be arranged to a variable that follows power laws? Sometime logarithms could be interpreted as an special case of more general functions (http://ref.scielo.org/6nqmzd)

Anyway, if you are able to redefine the properties of fractals in a way that satisfy your new definition then I don't see why it could not be valid. So i think i must read your paper before I can tell you anything more ;-)

I didn't read your paper. However I must agree with your colleagues in one thing, A fractal is a geometrical structure of all length scales. In other words must be scale free. the power law, and by all means the fractal dimension, is a direct consequence of this definition.

Dear Oscar Sotolongo-Grau, Many thanks for your comments!

You are right that Benoit Mandelbrot defined fractal based on fractal dimension or power laws. This definition is too strict for big data. Following this definition, a set of data values that do not follow a power law but a lognormal distribution is not fractal. However, this set is fractal according to the new definition, since there are far more small values than large ones.

Mandelbrot himself said that “For me, the most important instrument... is the eye. It sees similarities before a formula has been created to identify them.” I believe that the new definition captures fairly well human intuitions on fractal or scaling patterns.

Mandelbrot shifted his thinking from strict fractals such as Cantor set and Koch curve to statistical fractals such as coastlines. To some extent, I further relaxed his definition (on power laws) towards heavy tailed distributions in general. 

Dear Bin Jiang

Thank you for this interesting question.

Let X be the class of fractals that fit the standard definition

Let Y be the class of fractals that fit some new definition.

I am not sure but I would think that a new definition is acceptable if it is shown that either (1) X is a subclass of Y (which does not seem to be the case) ; or (2) each of X and Y is itself a subclass of some yet-to-be-defined class Z. 

Dear H.E. Lehtihet,

Many thanks for your insightful comments!

To confirm, X is indeed a subclass of Y. To be more specific,

Let A be the class of strict fractals such as Cantor set and Koch curve (defined by mathematicians before Mandelbrot)

Let B be the class of statistical fractal such as coastlines (defined by Mandelbrot)

Let C be the class of fractals based on or characterized by the ht-index (the new definition)

A is contained in B, which is further contained in C, i.e, denoted by A⊆B⊆C. It implies that under my new definition, all previous fractals (either strict or statistical) remain fractal.

Under the new definition, a smooth curve is fractal, while it is NOT under Mandelbrot's definition; see the illustration below. In other words, not only a coastline, but also a highway is fractal. Do you agree?

Dear Bin Jiang,

Just thinking about long tails and lognormal distributions (and not directly related to your question). Maybe this data could be arranged to a variable that follows power laws? Sometime logarithms could be interpreted as an special case of more general functions (http://ref.scielo.org/6nqmzd)

Anyway, if you are able to redefine the properties of fractals in a way that satisfy your new definition then I don't see why it could not be valid. So i think i must read your paper before I can tell you anything more ;-)

Dear Oscar, Very valuable suggestion and paper.  Thank you.

Dear Oscar Sotolongo-Grau,

Your comment makes perfect sense to me! The key point is that power laws are not absolute; what are claimed to be a power law could a power law for the head part, and the tail part with exponential distribution. Also fractal is not absolute, but with different degree of being fractal. For example, with respect to the following figure, the fractals have different degree at different stages or iterations. The different degree is captured by the ht-index.

dear Bin Jiang, you are talking  about spatial fractals; what about temporal scale free structures? Do they fit your new definition?

Thank you Bin Jiang for your question Do we need a new definition of fractals for big data? Or must fractals be based on power laws? 

In mathematical terms fractal objects can be described by power-law scaling relations. For fractals observed in nature this cannot generally be true. My studies on fractals in nature reveal that the dynamics of fractal development is much more complex than the simple (or trivial in terms of fractal geometry) power law. Natural observed scaling relations are valid only over a limited range of scales. I observe some duality in the process of fractal expansions and it is also a mystery why some power-laws fail to be existent in chaotic systems and why some power-laws have a unique topological representation whereas some do not have any. Natural fractals do not emerge as a bubble from a soap water inflated through a straw. In fact they undergo processes of subdivisions and mergers. And the standard merger number is known as the Feigenbaum constant 4.669... (F=4.669... is not a mere bifurcation constant but plays other roles in the mechanics). The merger is preceded by the emergence of the 3.5699 border. You should know that 3.5699 is not the general power-law number, still its presence is evident in a big number of chaotic systems. Between 3.5699 and 4.669 there should be one factor related to a subfractal but it is mysteriously  always missing there. Probably because it would disallow the fractal merger. So initially up to the point of F we have at least one mysterious and theoretically unexplained behavior. I would say that for fractal evolution power-law is just an optionality which Mother Nature chooses to use but not always and even when it does, then not in every stage of the evolution. And besides the power-law, Mother Nature chooses to divide and to combine things, and those divisions and mergers take place BEFORE F=4.669..., hence right from the beginning fractal expansions become intrinsically complex, being a combination of three distinctly different processes. And hitherto I managed to talk a bit only about stretching forces.  Therefore next, in order to say something about the squeezing forces,you need to take into account the second Feigenbaum constant 2.5029....and study  the Henon attractor. Now we are dangerously approaching the limits of cognitive abilities as the complexity of stretching is enhanced by the complexity of squeezing (I decided to not mention folding at all). I hope you get the feeling that the raw definition of fractals fails to grasp the idea of fractals in nature and seems to be incomplete in general. I do not pretend to redefine the definition, I am leaving it to others. I reiterate that this incompleteness most probably arises from the fact that mathematicians consider empirical studies dirty science, they prefer living in the world of purity. In a more general sense their reductionism is the culprit.   

@Arturo Tozzi

dear Bin Jiang, you are talking about spatial fractals; what about temporal scale free structures? Do they fit your new definition?

Fig 4 shown in my question is both spatial and temporal: 2 minutes, 5 minutes, 1 hour, and 12 hours, and temporal fractals are reflected in space. This paper has more details on the temporal aspect.

Jiang B. and Miao Y. (2014), The evolution of natural cities from the perspective of location-based social media, The Professional Geographer, xx(xx), xx-xx, DOI: 10.1080/00330124.2014.968886, Preprint: http://arxiv.org/ftp/arxiv/papers/1401/1401.6756.pdf

@Paul Koszarny

... a combination of three distinctly different processes. stretching forces, ... squeezing forces...

Thank you for insightful comments! What are the third process or forces? Do you have any references on these including the Feigenbaum constant 4.669...ideally for non-mathematician?

Why are there fractals in nature? apart from the classic explanations (self-organized criticality, exponents, and so on), could it be a matter of decreasing free energy in the systems or  in the structures?

Dear Arturo, Very good question!

As to why fractals in nature and society, there are many classic explanations offered by physicists. I am not in a good position (as a geographer) to answer the question, so I prefer to leave it to physicists. However, I tend to think of some natural forces or self-organized processes that create fractals. For example, drop a piece of regular glass forcefully into stone ground, and it will end up with numerous pieces that are fractal: (1) each broken piece has an irregular shape, and (2) there are far more small pieces than large ones.

Dear Arturo, fractals emerge from nonlinearities and feedback loops in open systems with a floating entropy. We cannot make an assumption that the energy is decreasing for some reason as the energy comes from the sun in natural systems. In financial systems the energy is in the form of capital inflow. Why do fractals emerge and not something else? It is a difficult question in fact. Assuming an exponential growth is taking place in a system, why it always ends somewhere and does not evolve to infinity? Take a fern leaf, one could imagine it growing more and more until we would not be able to discern it as a whole from the arising complexity. It never happens so. Chaotic systems (deterministic chaos) have the ability of creating self-organized structures and soon some chaotic borders emerge at some universal distances from the initial perturbation. If one knows the energy of the perturbation, one can calculate those borders before they emerge. The energy of perturbation is contained in the trigger. For example it the size of the trigger is 10, then the standard Feigenbaum border and a critical place is at 4.669 times 10=46.69. The catastrophic (exponential) expansion 14.208 yields 142.08 for the same trigger.  Nikkei225 crashed exactly to the 14.208 border in 2011 and bounced back from the invisible line. Fractals emergence is due to the existence of stretching, squeezing and folding forces present in the system. Topologically they cover all possible forces which names may remain completely opaque to us. Please note that squeezing forces and dissipative forces are not the same thing. Dissipative forces will always be there, squeezing forces may be locally absent. Fractals remain contained in their bounded phase space due to those forces mentioned above and also due to the spontaneous emergence of triggers in  opposite directions. So in nature we deal with complexities with a number of active triggers in opposite directions even when the interactions between the agents are quite simple. What we eventually  see is a beautiful balance of linearities and nonlinearities and the intermittance of order and chaos, simplicity and complexity which ensure an overall and structural stability. Fractals are tokens of structural stability, self-organization and well behaving holistic systems which once they have come into being, will strive to survive and fight for their existence by reorganizing themselves when they face new threats or feel more stressors.  What are those triggers? They may emerge as a pre-planned development (willed) or due to some randomness or both. Recall that randomness is our unknowledgable part of nature.

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Dear George,

Association to that definition is essential, just as Mandelbrot's definition (of statistical fractals such as coastlines and mountains) links to the older definition (of strict fractals such as Koch curve and Canter set). But I do not think we need to stick to measuring the length or topological dimension. Instead we should stick to whether there are far more small things than large ones. The notion of far more small things than large ones underlies all definitions (of strict fractals, statistical fractals, and the fractals emerged from big data). What do you think?

Thanks and cheers.

Bin

Back to the question, I have some fresh thoughts. The new definition of fractal aims not only for understanding structure or statics, but also for dynamics; not only for understanding complexity, but also for creating living structures. 

https://www.researchgate.net/publication/312044463_How_Complex_Is_a_Fractal_Headtail_Breaks_and_Fractional_Hierarchy

https://www.researchgate.net/publication/309428627_The_third_definition_of_fractal

Article How Complex Is a Fractal? Head/tail Breaks and Fractional Hierarchy

Presentation The third definition of fractal