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Suppose that we have a dataset of n elements each element is composed of m sub elements, is there any existing algorithm that we can use to calculate the complexity of this dataset?

I mean by the complexity anything that includes the following elements:

  • Diversity of dataset (number of distinct elements)
  • Distance between each element
  • Correlation between the elements of this dataset
  • Size of the dataset
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  • $\begingroup$ You can determine these things fairly easily in your database, and efficiently so if you accept approximations (e.g.,. APPROX COUNT DISTINCT). You can also define a figure that combines multiple metrics, but how is up to you. $\endgroup$ – Emre Mar 9 '18 at 1:21
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well ... "if n elements each element is composed of m sub elements" sound like you have a set of sets from discrete objects. If this is the case you can go through Combinatorial Analysis like Graph Theory. You can model it in two ways:

1) Using Hypergraphs: You have your elements as hyper-edges of a Hypergraph. This model lets you compute distances, diversities, sizes, and many more graph theoretical measures e.g. look at this, this and this.

2) Simple Graphs: You can create simple complete graphs from each sub-element and connect them to other complete graph (forming cliques) of other sub-elements. See the example bellow:

element1 = [[a,b,c],[b,c,d],[e,f,g]]  
Graph = [(a,b),(b,c),(a,c),(b,c),(c,d),(b,d),(e,f),(e,g),(f,g)]

using this you will have a huge graph (network) of your object on which you can calculate tones of structural and statistical measures.

I assume my thought were naive as your question is not clear and you need to provide more detailed example to get a more accurate answer.

Good Luck!

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