Embedded network data

In June, I saw a neat presentation by mathematician Dr Tiziana Di Matteo on her work summarizing high-dimensional network data.  Essentially, she and her colleagues embed their data as a graph on a 2-dimensional surface.   This process, of course, loses information from the original data, but what remains is (argued to be) the most important features of the original data.
Seeing this, I immediately thought of the statistical moments of a probability distribution – the mean, the variance, the skewness, the kurtosis, etc.   Each of these summarizes an aspect of the distribution – respectively, its location, its variability, its symmetry, its peakedness, etc.  The moments may be derived from the coefficients of the Taylor series expansion (the sum of derivatives of increasing order) of the distribution, assuming that such an expansion exists.
So, as I said to Dr Di Matteo, the obvious thing to do next (at least obvious to me) would be to embed their original network data in a sequence of surfaces of increasing dimension:  a 3-dimensional surface, a 4-dimensional surface, and so on, akin to the Taylor series expansion of a distribution.     Each such embedding would retain some features of the data and not others.  Each embedding would thus summarize the data in a certain way.   The trick will be in the choice of surfaces, and the appropriate surfaces may well depend on features of the original network data.
One may think of these various sequences of embeddings or Taylor series expansions as akin to the chain complexes in algebraic topology, which are means of summarizing the increasing-dimensional connectedness properties of a topological space.  So there would also be a more abstract treatment in which the topological embeddings would be a special case.
M. Tumminello, T. Aste, T. Di Matteo, and R. N. Mantegna [2005]:  A tool for filtering information in complex systems.  Proceedings of the National Academy of Sciences of the United States of America (PNAS), 102 (30) 10421-10426.
W. M. Song, T. Di Matteo and T. Aste [2012]:  Hierarchical information clustering by means of topologically embedded graphs. PLoS ONE, 7:  e31929.

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