Archive for the ‘Group Theory’ Category

Group Explorer 2.2

Sunday, April 20th, 2014

Group Explorer 2.2

From the webpage:

Primary features listed here, or read the version 2.2 release notes.

  • Displays Cayley diagrams, multiplication tables, cycle graphs, and objects with symmetry
  • Many common group-theoretic computations can be done visually
  • Compare groups and subgroups via morphisms (see illustration below)
  • Browsable, searchable group library
  • Integrated help system (which you can preview on the web)
  • Save and print images at any scale and quality

Are there symmetries in your data?

I first saw this in a tweet by Steven Strogatz.

BTW, Steven also points to this example of using Group Explorer: Cayley diagrams of the first five symmetric groups.

Production and Network Formation Games with Content Heterogeneity

Tuesday, September 27th, 2011

Production and Network Formation Games with Content Heterogeneity by Yu Zhang, Jaeok Park, and Mihaela van der Schaar.


Online social networks (e.g. Facebook, Twitter, Youtube) provide a popular, cost-effective and scalable framework for sharing user-generated contents. This paper addresses the intrinsic incentive problems residing in social networks using a game-theoretic model where individual users selfishly trade off the costs of forming links (i.e. whom they interact with) and producing contents personally against the potential rewards from doing so. Departing from the assumption that contents produced by difference users is perfectly substitutable, we explicitly consider heterogeneity in user-generated contents and study how it influences users’ behavior and the structure of social networks. Given content heterogeneity, we rigorously prove that when the population of a social network is sufficiently large, every (strict) non-cooperative equilibrium should consist of either a symmetric network topology where each user produces the same amount of content and has the same degree, or a two-level hierarchical topology with all users belonging to either of the two types: influencers who produce large amounts of contents and subscribers who produce small amounts of contents and get most of their contents from influencers. Meanwhile, the law of the few disappears in such networks. Moreover, we prove that the social optimum is always achieved by networks with symmetric topologies, where the sum of users’ utilities is maximized. To provide users with incentives for producing and mutually sharing the socially optimal amount of contents, a pricing scheme is proposed, with which we show that the social optimum can be achieved as a non-cooperative equilibrium with the pricing of content acquisition and link formation.

The “content heterogeneity” caught my eye but after reading the abstract, this appears relevant to topic maps for another reason.

One of the projects I hear discussed from time to time is a “public” topic map that encourages users to interact in a social context and to add content to the topic map. Group dynamics and the study of the same seem directly relevant to such “public” topic maps.

Interesting paper but I am not altogether sure about the “social optimum” as outlined in the paper. Not that I find it objectionable, but more that “social optimums” are a matter of social practice than engineering.

Group Theoretical Methods and Machine Learning

Saturday, February 19th, 2011

Group Theory and Machine Learning


The use of algebraic methods—specifically group theory, representation theory, and even some concepts from algebraic geometry—is an emerging new direction in machine learning. The purpose of this tutorial is to give an entertaining but informative introduction to the background to these developments and sketch some of the many possible applications, including multi-object tracking, learning rankings, and constructing translation and rotation invariant features for image recognition. The tutorial is intended to be palatable by a non-specialist audience with no prior background in abstract algebra.

Be forewarned, tough sledding if you are not already a machine learning sort of person.

But, since I don’t post what I haven’t watched, I did watch the entire video.

It suddenly got interesting just past 93:08 when Risi Kondor started talking about blobs on radar screens and associating information with them…., wait, run that by once again, …blobs on radar screens and associating information with them.

Oh, that is what I thought he said.

I suppose for fire control systems and the like as well as civilian applications.

I am so much of a text and information navigation person that I don’t often think about other applications for “pattern recognition” and the like.

With all the international traveling I used to do, being a blob on a radar screen got my attention!

Has applications in tracking animals in the wild and other tracking with sensor data.

Another illustration of why topic maps need an open-ended and extensible notion of subject identification.

What we think of as methods of subject identification may not be what others think of as methods of subject identification.