# A Unit of Complexity?

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Upon extending a pseudoscientific theory of mine for a science fiction novel, I came upon a rather unforseen problem. I realised that I needed a unit with which to measure the complexity of physical systems. As with most of my struggles, I immediately consulted Google, but could find nothing more helpful than this.

The problem is, quantifying complexity is made very complex (pardon the pun) by principle, as complexity is generally defined by the observer. If you define complexity as the number and intricacy of connections in a system, how do you define which of those connections are more complex? Perhaps the reasons I can't find a unit of complexity is because complexity isn't practically measurable - even if you could assign specific values of complexity to every possible connection (or perhaps go for the much simpler approach of assigning all fundemental connections equal values), it's still practically impossible to go and count all connections present in, for example, a tree-ecosystem.

However, I'm sure it should be possible to formulate a unit of complexity to use in theory. I assume information theory could be used as a backbone for measuring complexity (as systems can be interpreted as information), but I have near zero expertese in that field. Perhaps the complexity of a system could even be defined by the size of the equation needed to describe it.

But, before I get too far ahead of myself, is anyone aware of a unit of complexity currently in existence? If there truly isn't one, on what basis should/could I create a fictitious unit of complexity for theoretical purposes?

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This is imagined. Try the log of (the number of nodes times the number of synapses/neuron) or just log(total synapses), resulting in units of "plex".

For example, from Wikipedia —

The human brain has a huge number of synapses. Each of the 1011 (one hundred billion) neurons has on average 7,000 synaptic connections to other neurons.

Log(1011 neurons × 7,000 synapses) = 14.8.

The average human brain is a 14.8 plex. Einstein may have had twice the number of synapses, or a 15.1 plex.

The nematode C elegans has a 3.7 plex. Poor little C elegans.

Fruit flies have a 7 plex.

Honeybees have a 9 plex.

Mice have an 11 plex.

Rats have an 11.7 plex.

source

Edited by ewmon
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Is "plex" an officially recognised unit, though? If so, is it only for the complexity of a brain?

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No, I hijacked "plex" and invented the definition I gave here.

The term "plex" is found in complex, duplex, etc and also as ply, multiply, four-ply, plywood, etc as well as plait, pleat, etc.

E plex < L plicare, to fold, to weave < PIE *plek, to plait, to interweave

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There are some notions of the measure of complexity, but there is no general measure that I know of. For example in the context of fractals, fractal dimension, upper and lower Minkowski–Bouligand dimension (upper is the same as Kolmogorov dimension), correlation dimension are all used.

Edited by ajb
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I suppose I'll go with a fictional unit, then, to describe the complexity of order in a system. Perhaps I'll call it ewmon's "Plex", or something similar.

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Naw, try calling it Oudtshoorn's Scale of Complexity, which sounds very mathematical, or something like that.

Better yet, if you think complexity is so interesting (and it is), try digging into some real systems (such as animal brains) and develop a useful measurement of complexity. You could become rich and famous by developing it. (Well, you can't have everything!)

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More research must thus be done. Thanks.

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It seems to me you need to consider not only the number of connections, but the complexity of each connection. That would relate, broadly, to the complexity of the equaiton defining the relationship across the connection.

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• 1 month later...

Hello SPACE_LEMON;

I think you need to relate your "complexity unit" to a capacity framework. As to how much a system could handle before failing. Or more general, one with a scale measuring overall poor design. To me complexity is a measure of failing to maintain or to stay within the parameters of an efficient system. As seen in most multi-generational systems like cities, large factory complexes, some theories, and most societies.
Complexity usually requires more people, energy, room, downtime and may include excessive noise, vibration, and overall dissatisfaction of all those involved. The "unit of measure" could be anything loosely related to these examples.
A lack of quality?
Maybe just a calibrated scale moving away from Simple. It almost always takes a highly intelligent person to find a complex solution to a difficult problem. While occasionally an average person finds the simple solution that is its superior.
Maybe it should be measured in I.Q.? ARC

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