# Entropy of natural networks [duplicate]

How does one define the entropy of a natural network (say for example, a river network, or a morphological skeletal network of a lake in the figure below) ? For example, the following report suggests defining it in terms of a topological diameter. But,again it involves defining the generator of the fractal network. Ref: http://www.idrologia.polito.it/~claps/Papers/vecchi_papers/J_Hydrol96.pdf • Link is broken. What is a "natural" network? – N. Steinle Aug 1 '18 at 14:08
• Thanks,I have updated the link. Natural networks are those networks which we see on earth, for example,the Drainage network of rivers,Deltas etc. – vbj Aug 1 '18 at 14:13
• Ah yes, then you want informational entropy, rather than thermodynamic. Equation 1 of the paper you provide defines the entropy. So what is your question then? – N. Steinle Aug 1 '18 at 14:15
• This equation ultimately uses the generator of the fractal, whereas for my purpose I have the network ab initio, I am looking for some graph theoretic measures and topological properties of the networks. I have some idea about the topological invariants but have used it in condensed matter. I would like to know if there are graph theoretic topological measures too. – vbj Aug 1 '18 at 14:18
• Thanks for clarifying. I've updated my answer. It seems there's quite a lot of work in what you're looking for, so I hope I've helped you find the correct rabbit hole ;) – N. Steinle Aug 1 '18 at 15:48

It ultimately depends on whether you want information entropy or thermodynamic entropy. Either way, I would suggest taking a graph theory approach, since the structure of a graph IS a network (or one could say that the structure of your network IS a graph!). That said, it would depend on what kind of graph you want to use to model your network, and then it will depend on what kind of entropy measure you want to try to use for your system.

Here's a nice review of entropy measures on graph networks.

Here's a specific example for a random graph using Shannon entropy.

There are several well-defined measures of entropy for graphs, and the literature is abundant with examples. If you specify a bit more, then you might get better help here. (what is a "natural" network?)

Another option within the graph theory approach is to simply use statistical mechanics on the graph (i.e. microcanonical ensemble or canonical ensemble) and derive the thermodynamic entropy from it. This article has some great examples of how to do so.

EDIT: Per your comment, there do exist topological measures of entropy for graphs. It seems that you have some options: either you use 1) the "graph entropy" definition introduced by Korner on your network, or 2) view your graph topologically, and then use the "topological entropy" definition. It seems you desire to use the latter approach.

I would like to know if there are graph theoretic topological measures too.

This review contains lots of great references that you can explore, and on page six they examine topological entropy of graphs by looking at how "graph maps" behave. If this is unfamiliar to you, consider the famous quote, paraphrased here, by Hermann Weyl: "if you want to understand some mathematical structure, look at its group of automorphisms." And this paper is nice to keep the various ideas of entropy separate.

• Link-only answers are discouraged on this site. It would be more helpful/more useful if you could summarise the relevant information contained in the links you provide. – sammy gerbil Aug 1 '18 at 17:07
• It is not a link-only answer. I have explained briefly what sections/pages are relevant to the OP's interest - their question is a vague one about the existence of something, so it gets a vague answer. If they specify the nature of their problem more, then we can get into more detail. – N. Steinle Aug 2 '18 at 18:36