Revision as of 21:08, 29 June 2017

Project Description

Here is an example of a directed hypergraph (hyperdigraph)

This project focuses on expanding the theoretical applications of (i,j)-step competition graphs. Specifically, this project examines a generalization of the (i,j)-step competition graph by expanding its definition to directed hypergraphs (hyperdigraphs)--digraphs whose arcs consist of two disjoint (and possibly empty) sets, the tail and head set (see figure). After establishing a definition for our generalization, we will explore its innate attributes concerning its construction and applications. As the project progresses, results are finalized, and the paper is ready for submission/publication, more details will be added.

My weekly log can be found below.

Background Readings

I felt having a list of the background reading I am doing would be helpful. Hence, I have created one below. These readings include the ones Dr. Factor gave me and the ones I have found. Note: if a link is missing, the reading is either from a textbook or I was unable to find one.

• BioMath: The Biology and Mathematics of Food Webs by Midge Cozzens, Nancy Crisler, Randi Rotjan, and Tom Fleetwood
• Chapter 3 from Applied Combinatorics by Fred S. Roberts
• (i,j) Competition Graphs by Kim A.S. Hefner, Kathryn F. Jones, Suh-ryung Kim, J. Richard Lundgren, and Fred S. Roberts.
• The (1,2)-step competition graph of a tournament by Kim A.S. Factor and Sarah K. Merz
• An Introduction to (1,2)-Domination Graphs by Kim A.S. Factor
• The m-step, same-step, and any-step competition graphs by Wei ho
• Connected triangle-free m-step competition graphs by Geir T. Helleloid
• The m-step competition graph of a digraph by Han Hyuk Cho, Suh-Ryun Kim, and Yunsun Nam
• A Characterization of Competition Graphs by R.D. Dutton and R.C. Brigham
• Interval Graphs and Food Webs: A finding and a Problem by Joel E. Cohen
• Using Food Webs In Order To Determine Possible Predictors of Primary and Secondary Extinctions by Kaitlin A. Ryan (a master's thesis)
• The (1,2)-step competition number of a graph by Kim A.S. Factor, Sarah K. Merz, and Yoshio Sano
• On The Hypercompetition Number of Hypergraphs with Maximum Degree at Most Two by Yoshio Sano
• Competition hypergraphs by Martin Sonntag and Hanns-Martin Teichert
• On the hypercompetition numbers of hypergraphs by Boram Park and Yoshio Sano
• Directed Hypergraphs and Applications by Giorgio Gallo, Giustino Longo, and Stefano Pallottino
• On k-hypertournament matrices by Youngmee Koh and Sangwook Ree
• On the Computation of the Competition Number of a Graph by Robert J. Opsut

Project Log

This is my daily log for what I accomplished. Each day includes an estimate on the number of hours I actually worked on the project or in Cudahy. My activities may include updating the wiki page, attending working lunches, meetings with Dr. Factor, working on various projects with Carissa, or configuring printers; that list is not, by any means, exhaustive. Note: the end-of-week reflections are the last things I do on the last day of each week before I leave the office, naturally.

Week 1 (5/30/17-6/2/17)

Day 1 (5/30/17)

• Attended MSCS REU orientation
• Met with mentor, Dr. Kim A.S. Factor, and began discussing project details
• Dr. Factor and I discussed goals: having BioMath: The Biology and Mathematics of Food Webs done by Thursday and reading The (1,2)-step competition graph of a tournament (by Kim A.S. Factor and Sarah Merz) by next Wednesday
• Completed pages 1 through 30 of BioMath: The Biology and Mathematics of Food Webs. I plan on finishing the packet tomorrow and seeing how far I can get through chapter 3 of Applied Combinatorics by Fred S. Roberts

Day 2 (5/31/17)

• Attended library orientation
• Finished the BioMath packet.
• Began reading The (1,2)-step competition graph of a tournament
• Began reading chapter 3 of Applied Combinatorics.

Day 3 (6/1/17)

• Attended a talk on good research practices by Dr. Kim Factor
• Filled out direct deposit form
• Met with Dr. Kim Factor and drafted the milestones and discussed moving our lunch with Carissa to Wednesday
• Reviewed milestones and compared them to the program's google calendar
• Printed additional readings for the weekend
• Continued reading chapter 3 of Applied Combinatorics
• Continued reading The (1,2)-step competition graph of a tournament

Day 4 (6/2/17)

• Uploaded a pdf of my milestones for the summer
• Worked on chapter 3 of Applied Combinatorics
• Continued reading The (1,2)-step competition graph of a tournament
• Began reviewing other papers
• Created project's Wiki page

Reflection: End of Week 1

Concerning my research, this week was relatively uninteresting. That is, I spent most of my time trying to find literature (beyond what Dr. Factor gave me), working through background readings, and finding a good work area. Of those three tasks, I spent most of my time on finding a good work area (it was my top priority); I can happily say I completed this task. I cannot wait to really dig into my research--make it past the background reading.

I accomplished my goals for this week. I attended various meetings, orientations, and finished the BioMath module, among others. I intend on furthering my research this weekend. Specifically, I hope to read multiple papers and complete chapter 3 from Applied Combinatorics. Next week, I'll be meeting Carissa on Tuesday, and we'll finally discuss the material Dr. Factor gave us; I'm excited.

Made during Week 2 on Day 3: I was told an hour estimate for each week would be nice. I worked approximately 40 hours this week (I did some stuff on Saturday to make up the missing time from Monday).

Week 2 (6/5/17-6/9/17)

Day 1 (6/5/17)

• Got in around 10:30AM
• Finished reading chapter 3 from Applied Combinatorics
• Created a LaTex document where I am writing my notes on Applied Combinatorics--a link to it is here, please note it is still in progress
• Created a LaTex document for teaching Carissa TeX
• Configured the printer in room 410
• Finished formatting my personal wiki and began a rough draft of the project's wiki
• Confirmed a meeting with Carissa tomorrow, 6/6, for 8:30 AM before the ethics training.
• Left around 3:30 PM, estimate for the day is 5 hours

Day 2 (6/6/17)

• Attended meeting with Carissa at 8:00AM
• Attended ethical research training at 9:00AM
• Handed off Applied Combinatorics to Carissa for reading on Graph theory
• Began reviewing The (1,2)-step competition graph with Carissa at 3:15PM
• Made note to include approximate total number of hours in end-of-week reflection
• Left the building at 7:08PM, will be back in two hours...Didn't make it back to Cudahy, finished at the apartment. Estimate for the day is 13.5-14 hours

Day 3 (6/7/17)

• Got to Cudahy at 9:00AM, discussed various definitions with Carissa
• Lunch with Dr. Factor around 12
• Received The (1,2)-step competition number of a graph by Factor, Merz, and Sano for reading and began reading it
• Began compiling definitions, lemmas, propositions, and theorems with Carissa for an examples pdf
• Made edits to the background section for this project's wiki, will finish them tomorrow
• Left at 6:00pm, estimate for the day is 9 hours

Day 4 (6/8/17)

• Got to Cudahy at 10:30
• Finished reading Kaitlin Ryan's master's thesis
• Attended working lunch
• Finished reading The (1,2)-step competition number of a graph by Factor, Merz, and Sano
• Began creating examples for our examples pdf (they will be added tomorrow)
• Drafted potential research questions
• Left lab at 4:40, time estimate is 6 hours

Day 5 (6/9/17)

• Arrived at 9:00AM
• Finished making examples pdf with Carissa
• Began formulating definitions for generalization
• Began presentation for the meeting with Dr. Factor on Tuesday
• Began file with BibTeX citations on background readings and paragraphs detailing what they were about
• Finished formatting background section for project's wiki page
• Left around 4:20, approximate time for the day is 7 hours

Reflection: End of Week 2

During this week, I worked on reading papers, teaching Carissa LaTeX, and formatting definitions. I met Carissa on Tuesday and our working partnership has been going well. We are good at catching each other's errors and talking through our confusion, if any, regarding various concepts. We made significant progress on finishing the wiki, setting up where our research will go, and are getting ready for our presentation with Dr. Factor on Tuesday. Our research is going well, overall, but could be moving faster.

For my personal research, I have some working ideas. I am turning over a few ideas for various generalizations regarding this project. Dr. Factor would like some working definitions, and I intend on giving her some on Tuesday. Over the weekend, I will work on cementing my some ideas into something more concrete and read other papers concerning hypergraphs. I am very excited at the prospect of proving theorems and actually working on math. If she wants it, I intend on helping Carissa find a general direction to go with her research question, if she doesn't want to join me. That all being said, I only have ideas and no actual thought supporting them; I have what my professor, Sean Raleigh, would call a "highdea."

My hour estimate for this week is 41 hours.

Weekend 2 (6/10/17 & 6/11/17)

• Began formulating definitions in LaTeX for Dr. Factor

Week 3 (6/12/17-6/16/17)

Day 1 (6/12/17)

• Arrived around 10:00AM
• Finished presentation with Carissa
• Went through presentation with Carissa
• Edited examples pdf with Carissa
• Finished reading hypergraph, characterization, and other papers
• Left around 4:00 (will come back), approximate hours for today is 6 (so far)
• Worked on TeXing up generalization, total hours is around 9

Day 2 (6/13/17)

• Arrived around 10:00am
• Worked on generalizations and example to show Dr. Factor
• Met with Dr. Factor with Carissa at 1:00pm
• Began editing paper
• I now have to defend definitions by Thursday (if possible)
• Left lab at 4:00, hours for the day are 6

Day 3 (6/14/17)

• Not feeling well, didn't get in until 11:00am, may go home early--(Note:feeling better)
• Working on generalization edits, needs serious work
• Started and finished ethics training quizzes
• Looked over the remaining modules
• Left office at 6:00pm, approximate time is 7 hours

Day 4 (6/15/17

• Began working at 8:30, got in at 11:00
• Met with Dr. Factor at 11:00, was late because I thought it was the working lunch
• Discussed scanning notes over to Dr. Factor, discuss them on Tuesday
• Attended working lunch at 11:30
• Working on edits
• Modified milestones for relevance
• Left around 4:30, approximate hours for the day: 8

Day 5 (6/16/17)

• Did not go into the lab today, worked in the apartment
• Read paper at home
• Approximate hours: 6

Reflection: End of Week 3

This week, I started interacting with mathematical objects. I worked on my generalization a bit, and I stumbled across some other interesting ideas. I am now looking for definitions concerning hypertournaments in lieu of k-hypertournaments. I'm currently looking at the paper Dr. Factor gave me to look over; it is very dense but at an appropriate level. I have a growing interest in hypergraphs and their colorings. I wonder if I could combine the coloring of hypergraphs and my generalization. I feel as if my "highdea" is becoming more concrete. Lastly, Carissa and I have officially split up, in terms of research questions; she, however, if interested, may join my project. I plan on finishing the edits on that paper before Monday, preferably by the end of tomorrow.

My weekly hour estimate is 36 hours.

Week 4 (6/19/17-6/23/17)

Day 1 (6/19/17)

• Began at 10:00
• Finished revising paper
• Determined a "hypertournament" is ridiculous
• Question of (i,j)-hypertournaments arose---just a layered (i,j)-step competition graph, probably
• Working various proof ideas
• Working on ideas for colorings and their relationships
• Left building at 2:45, finish researching at 5:30.

Daily hours: 7.5

Day 2 (6/20/17)

• Began at 10:00
• Met with Dr. Factor
• Given explicit deadlines for Thursday
• Began reconciling definitions
• Began mini presentation
• Began NP proof
• Finished at 5:30

Daily hours: 7.5

Day 3 (6/21/17)

• Began working at 10:00
• Worked on mini presentation
• Worked on definitions
• Worked on NP
• Finished at 8:00

Daily hours: 10

Day 4 (6/22/17)

• Began around 9:30
• Worked on mini presentation
• Attended working lunch
• Met with Dr. Factor, obtained approval on my definitions/resolution of definitions
• Worked on TeXing hypergraphs and directed hypergraphs
• Left around 6

Daily hours: 8.5

Day 5 (6/23/17)

• Began working at 9:00AM
• Worked on building a hypergraph is LaTeX
• Ended at 11:00AM
• Resumed at 10:00PM and finished at 12:00AM

Daily hours: 5

Reflection: End of Week 4

This past week did not feel very productive regarding my research question. I determined my definition is new and interesting enough for pursuit. I also figured out the question regarding whether or not it is NPC is rather simple; hopefully, I'll have a proof by the end of the week. I have also produced enough questions to pursue over the summer. These questions concerning my definition are as follows: is it really just a layering of (i,j)-step competition graphs; is its construction NPC; does it have a universal competition number; are there any algebraic connections to this layering at direct products (another 'highdea').

My weekly hour estimate is 38.5 hours.

Week 5 (6/26/17-6/30/17)

Day 1 (6/26/17)

• Began working at 10:00AM
• Worked on presentation all day (drawing hyperdigraphs in a simply way, finally figured it out)
• Ended at 4:00PM
• Resumed at 9:00PM and finished at 12:00AM

Daily hours: 9

Day 2 (6/27/17)

• Began at 9:00AM
• Polished presentation and sent to Dr. Factor
• Took a gym break at 11:00AM
• Resumed at 1:30PM
• Made suggested edits to presentation
• Ran into BibTeX problems and worked on those
• Poslished presentation, practiced
• Ended at 7:30

Daily hours: 8 hours

Day 3 (6/28/17)

• Began at 11:00AM
• Polished hyperdigraph model
• Practiced presentation with other researchers
• Gave feedback on presentations
• Finished at 10:00PM

Daily hours: 11

Day 4 (6/29/17)

• Began at 8:00AM
• Practiced presentations with Carissa
• Listened to mini presentations from 11:00AM to 1:30PM
• Worked on wiki page
• Mapped out project directions (which aspects I'll tackle first)
• Left at 4:00PM

Daily hours: 8

Day 5 (6/30/17)

Reflection: End of Week 5

Week 6 (7/3/17-7/7/17)

Day 1 (7/3/17)

Day 2 (7/4/17)

Day 3 (7/5/17)

Day 4 (7/6/17)

Day 5 (7/7/17)

Reflection: End of Week 6

Week 7 (7/10/17-7/14/17)

Day 1 (7/10/17)

Day 2 (7/11/17)

Day 3 (7/12/17)

Day 4 (7/13/17)

Day 5 (7/14/17)

Reflection: End of Week 7

Week 8 (7/17/17-7/21/17)

Day 1 (7/17/17)

Day 2 (7/18/17)

Day 3 (7/19/17)

Day 4 (7/20/17)

Day 5 (7/21/17)

Reflection: End of Week 8

Week 9 (7/24/17-7/28/17)

Day 1 (7/24/17)

Day 2 (7/25/17)

Day 3 (7/26/17)

Day 4 (7/27/17)

Day 5 (7/28/17)

Reflection: End of Week 9

Week 10 (7/31/17-8/4/17)

Day 1 (7/31/17)

Day 2 (8/1/17)

Day 3 (8/2/17)

Day 4 (8/3/17)

Day 5 (8/4/17)

Reflection: End of Week 10