What kinds of transformations preserve network topology? I have been reading a number of "network science" papers where the authors perform transformations on networks that seem to preserve the topology of those networks.  By "topology", I mean a collection of heuristics commonly used to probe the network: modularity, clustering coefficient, and mean path length.  Is there a name for these kinds of transformations and/or some review somewhere of their properties?  I don't have a good mental model for said properties and it would be helpful to have a list of examples or a formal treatment of them.
 A: @ vrume21: Assuming, by the tags and some other hints on your question, that the kind of network you are talking about is related to the classical definition of "graph" (non-empty collection of vertices together with a subset of the relation of the cartesian product of the vertices, meaning, the set of connected vertices), and by some definitions in topology, possibly what you are looking for are transformations which preserve the connectivity. Let me explain:
Connectivity of a network can be studied with Euler's formula or characteristic (not the one Exp[ix] = Cos[x] +i Sin[x], but n-e+f = 2; n = vertices, e = edges, f = faces). This can be found and thoroughly explained, (even with a proof of this Euler's formula) in the book "Introduction to Graph Theory", Douglas B.  West, ISBN 81-7808-830-4. There it is explained how Euler's formula concerns connectivity, being connectivity indeed a topological feature, and since no particular kind of graph is assumed, networks should fall into this way of analysis. Having all said that, you could look for particular transformations in papers of networks with the keyword "Euler Characteristic", or "transformation preserving Euler characteristic", or something of the sort.  For example in the paper in http://www.icmp.lviv.ua/journal/zbirnyk.54/011/art11.pdf (Condensed Matter Physics 2008, Vol. 11, No 2(54), pp. 331–340
"Analysis of urban complex networks", D.Volchenkov), or in Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, January 1964, Volume 2, Issue 4, pp 340-368 "On the foundations of combinatorial theory I. Theory of Möbius Functions" from Gian -Carlo Rota. 
