Would you please help me to identify some applications of set theory, in particular sumset theory, in networks such as in social and biological networks? Please suggest some useful reading too/
Dear Sir,
if I may ask another question: is there a mathematical theory that couldn't be related to any other ?
To give some answer to yours, I suggest to consider as example Ramsey theory and its applications. At the beginning of "Ramsey theory on the Integers", a book by Landman and Robertson, you will find the following definition: Ramsey theory is the study of preservation of properties under set partitions. When defined on graphs (networks), Ramsey Theorem is formulated for colored graphs. If we focus on graphs colored with 2 colors (edge coloring), the theorem is as follows: Let $k,l \ge 2$. The exists at least positive integer R = R(k,l) such that every edge coloring of a complete graph of R vertices with colors red and blue, admits either a complete red subgraph of k vertices or a complete blue subgraph of l vertices.
A direct application of this theorem in social graphs is illustrated by the party problem stated as follows: At a party of at least 6 people, there must be either 3 people who have all met or 3 people who are mutual strangers. You can easily (even by brute force) prove the previous statement to be true for 6 or more people and false for 5 people. Thus you would have proven that R(3,3) = 6. You can then generalize to compute R(3,4) (= 9) and other values. You can also generalize for more than 2 colors, for example R(3,3,3) = 17.
Anyway, this is quite pure... a less theoretical relationship could be found in community detection in social/biological graphs/networks. Algorithms for such partitioning often use measures like normalized mutual information to compare with ideal partitioning; or if the latter is not known, use maximization of some other measures like modularity (that has many definitions from structural to spectral), and surprise (that plays more on combinations). All this stuff is quite related to set theory and you will find their definitions on wikipedia.
Now when you say sumset theory, you may ask yourself what is the right sum operation ? and why would it be relevant to problems you will deal with ? Theory is here to support practical problems with right models and solutions. I am convinced that for every practical problem there is an adequate theory that gives efficient solutions but not all theories are good for all problems.
I hope this would help you.
Dear Sudev Naduvath
I will suggest you this link
Connected Dominating Set: Theory and Applications - Springer
http://www.springer.com/mathematics/applications/book/978-1-4614-5241-6
- Badges
- Science topic
- Similar topics
- Mathematics
- Applied Mathematics