I'm trying to understand the difference between Oracle's graph technology, which apparently has an inherent understanding of topology, and Ayasdi's Topological Data Analysis technology. Are these two completely different and where does one draw the line when distinguishing them? Is Oracle's graph technology operating on a completely different domain (in terms of topology) than Ayasdi's algebraic topological data analysis approach?
Thanks in advance!
Disclaimer: I work at Ayasdi and am not an expert or even very familiar with Oracle's technology.
I think when you're talking about Oracle's Graph Technology you mean this:
http://docs.oracle.com/cd/E16655_01/appdev.121/e17897/toc.htm
Oracle's technology implements a limited topological view - topological objects consist of nodes, edges and faces and are either constructed "by hand" or built automatically from "real world" geo-spatial data. The focus is on representing either graphs such as computer networks or on geo-spatial data. It is not intended to implement simplical complexes in general nor will it build them from a topology specified in some way (like from a metric on points, or from an atlas of open sets).
Ayasdi has a much more general use of topology. We are building simplicial representations of point cloud data - in particular we build the nerve of an open cover (http://en.wikipedia.org/wiki/Nerve_of_a_covering) where we typically use a combination of a metric on the data and some "meaningful" function to create the cover. This is described here:
http://comptop.stanford.edu/preprints/mapperPBG.pdf
and here:
http://www.ams.org/journals/bull/2009-46-02/S0273-0979-09-01249-X/home.html
(But we also have other ways of building useful covers of spaces.)
These simplicial representations are used to understand complex and potentially high dimensional "Big Data" problems by summarizing the important and essential characteristics of the data (The story is a little long to describe here. You can see these videos for more detailed explanation: http://youtu.be/x3Hl85OBuc0 and http://youtu.be/3Z73Wd2T1xE - I apologize a head of time for posting vanity videos)
Disclaimer: I am one of the co-founders of Ayasdi.
When Oracle says topology, they mean network topology. So, if you have some data and construct an ad-hoc network from it (say, you construct a graph in which you connect customers bought the same products), then Oracle's software will allow you to build measures on this graph (such as centrality, page rank, clusters etc.)
Ayasdi's technology on the other hand is focused on building topological spaces in the mathematical sense. So, we take a bunch of data and try to construct a representative simplicial description.
These are two completely different technologies with different use cases. In general, Ayasdi's technology supersedes and does much more than Oracle's in this domain.
