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I've read an article about a comparison between human brain and artificial neural networks. It's said that human brain contains $\approx 10^{11}$ neurons and each neuron is connected to $\approx 10^4$ others! Is it possible? What kind of graph would it be?

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Nice question Gigili, welcome back +1 –  B. S. Jan 3 '13 at 9:57
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up vote 13 down vote accepted

The network you are describing is called the connectome. The study of real-world networks, such as brain networks, is a enormous task, largely due to the sheer size of the network. However, there are other complicating factors:

  • There are different ways to interpret what constitutes an "edge" (e.g. physical connectivity, electrical signalling, or chemical signalling).

  • Different people have different connectomes. Brains change over time (particularly during child development).

  • There are different technologies and methods for mapping the connectome, which don't always agree.

However, there are some things that can be said about the connectome. For example:

  • The organisation of these networks (e.g. clustering), allows for robustness and flexibility.

  • They display "broad scale" degree distribution (like scale-free, but there is a kind of cut-off on the maximum degree).

I'm sure there are many other things that can be said about connectomes that I'm not familiar with. They tend, however, to be rather general in flavour.

References:

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A graph with a lot of vertices ... Unless you provide further information about the local, or global behavior, nothing much else can be said.

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Not even the fact that it is a connected graph or not? –  Gigili Jan 3 '13 at 7:56
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No, because the person could have a stroke in the Left Brain. Note that dolphins actually have 2 distant brains, and one can sleep while the other stays awake. –  Calvin Lin Jan 3 '13 at 7:57
    
Umm, not very interesting! Thank you for your answer anyway. –  Gigili Jan 3 '13 at 7:59
    
I suspect what you are interested in is Computational Neuroscience. They do a lot more, with the graph theory aspect, by imposing certain conditions on the connectedness, local ergodicity, etc. –  Calvin Lin Jan 3 '13 at 8:02
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