For my project I will studying the various ecosystems of the Hudson River. I will be creating food webs which model the way predators and prey interact in the environment. The food webs will be modeled using directed graphs where the connections will go from the prey to the predator. From these digraphs I will create competition graphs which will be non-directed graphs representing the connections between predators that share a common prey. These graphs will be analyzed using techniques from Graph Theory, including the maximum lengths for paths from a prey to a predator and the competition graphs will be tested for robustness using a method in Graph Theory known as Edge Clique Cover.
I arrived at Rutgers on Tuesday May 31st. Throughout the week there were a couple of orientations to get the students involved so that they may get to know each other. We also met with our mentors and the students that we would be working with. Alma, Cristina, Herman, and I met with Dr. Gene Fiorini and Dr. Urmi Ghosh-Dastidar to discuss the projects. We would have to pick one of the two projects. One which uses graph theory to analyze the Hudson River food web and the other dealing with K Class-0 subgraphs which are part of a graph theory game called pebbling. The rest of the week was spent studying graph theory and getting to know the two choices. We also spent some time getting familiar with Python since it would be used to analyze the data.
In the second week the four of us gave a joint presentation for 20 minutes discussing both projects. During this week we finally concluded which project we would work on. Alma and I decided to work on the Analysis of the Hudson River food web while Herman and Cristina chose to do the project of K Class-0 subgraphs in Pebbling. On Thursday we had a picnic and Friday was cultural day where students presented about their cultures or life experiences at their Countries, Cities or Universities.
This week was spent doing research on the Hudson river's biodiversity and food web. We also had a lot of reading to do on Graph Theory topics which we were not familiar with prior to our research. On Tuesday we had a speaker named Swarup Acharya from Alcatel-Lucent Bell Laboratories which spoke to us about innovations in technology and how to disrupt the market. I found this talk to be very useful for entrepreneurs in the technology age that we live in today. On thursday we had a talk about uniformity which was given by Dr. József Beck from Rutgers University.
During this week I put together the adjacency Matrices that represent the food webs. From these Matrices I was able to create the Competition Graphs using an algorithm which I wrote on MATLAB. The rest of the week was followed by studying concepts of Graph Theory and by writing out proofs so that we know that we understand the concepts that we will be using. Also during this week we visited Telcordia for a seminar where we were given several talks by many different individuals who work there. It was a good experience that allowed us to see how our skills in Mathematics will be used in the real world when we are done with school. That Thursday we had a talk by Dr. Bob Vanderbei of Princeton University who spoke to us about astronomy and how scientist have found techniques to determine whether there are planets orbiting nearby stars. This was a very interesting talk.
This week I was assigned plenty of work to do. I read and learned about the combinatorial Laplacian and incidence matrix and I constructed an algorithm in matlab to create them from the food webs and tested out several proofs. I also read about the normalized Laplacian and weigthed competition graphs. And I read papers on similar studies and learned about many techniques that could be used to evaluate the robustness of the food webs. On Tuesday we had a talk by Dr. Aaron Jaggard of DIMACS and on Thursday we had a field trip to the Cancer Institute of New Jersey where we saw a few presentations and had a small tour of the facilities while we learned about how cancer patients are treated and how mathematics and computer science play a role in cancer research.
During this week I had to perform my military duties which carried on into the next week as well. I was pretty busy traveling and setting up at my new location to get much work done. But I managed to take some time to read some of the work that my partner send me and attempt to create an Algorithm to get the trophic status of all the species. Unfortunately I was unsuccessful. BUt I also found time to review some of my reference articles and began the introduction to the research paper.
During this week the Marine Corps kept me fairly busy but I found some spare time and created an algortithm to get the longest path from one species to another. Using this we can find the trophic status. So I send it to my partner so that we can both try to work out an algortithm for the trophic status. Communication through emails proved to be a lot harder then I thought at first and not much was accomplished. Meanwhile my partner had to complete a presentation and there was very little time left to be spent trying to add new things.
This week I came back to NY and I made it back to Rutgers by Wednesday afternoon. I was behind on my work and still needed to complete my paper and my mentor was away for a conference. With the program coming to an end I had to pack up and leave Rutgers with a lot of work still undone.
This week I worked on calculating the Trophic Status of each of the Species in the four food webs. Since I could not write a MatLab algorithm to do it for me I decided to do it all by hand. Although it was very tedious, the calculations were not difficult. I also removed species from a food web one by one and calculated the change in normalised connectance as the number of links changed and plotted the results in a graph which turned out to be linear. Also using an equation from one of my sources I solved it to find the amount of links that can be removed so that the connectance remains the same using a food web where cannibalism is not considered.