Scientists often model networks as graphs, possibly with weighted edges. I will first say why I think this is inadequate: multi-person interactions in a network require higher-dimensional simplices. After recalling the basics ideas of simplicial sets, I will discuss fuzzy sets and fuzzy simplicial sets, which I think are appropriate for modeling networks. Then I will give a metric realization functor, which is a certain left adjoint that is nicely comparable to the usual geometric realization functor. Applying this metric realization functor, we obtain an "uber-metric space" (my temporary name for a metric space in which d(x,y) can be 0 or infinity, for x \not= y). One could then analyze these metric spaces using persistent homology, or analyze the corresponding fuzzy simplicial sets using techniques we heard about from Matthew Kahle in 2008. |
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