Name: | Soham Palande |
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Personal Website | Website |
Email: | palandesoham@gmail.com |
Office: | Virtual |
Home Institution: | Rutgers University- New Brunswick |
Mentor: | Dr. Jie Gao |
Project Paper: | Co-Evolution of Opinion and Network dynamics in Real-world Scenarios |
In this project, we study the co-evolution of opinion and network dynamics in social networks. Specifically, we will study the structural properties of the networks and characterize dynamic models at their limits. Our goals are:
This week I met with Professor Gao and we discussed several areas for the project. Professor Gao briefly discussed her areas of research and we talked about my interests as well. I spent most of the week reading papers in Social Network dynamics and some recent papers by Professor Gao and her PhD students. I am currently brainstorming to figure specific areas that I am interested in. Some areas that have spiked my interest are:
This week i attended a seminar talk by Prof. Saray Shai from Wesleyan University on Topology and Geometry of Urban Road Networks. This was a very interesting talk and was also related to my research.
This week in our meeting I expressed my topics of interest and we finally decided the topic of my project for the summer. I will be working on studying the impact of adversaries on social network dynamics. The goals are to study the structure and properties of networks and chracterize the networks at their limit points (i.e whther the networks end in harmony or polarization). We will also study the properties of networks that are resilient to adversarial attacks.
I also delivered my project proposal presentation this week which went very well. I read several survey papers given to me by Professor Gao to get an overview of the important results and work that has been done. I also attended a meeting with Professor Gao and her collaborators in which they were addressing mathematical problem regarding the evolution of networks in one of their recent works. I have also started reading a book on Graph Representation Learning by William Hamilton which I expect will be very useful in my research project.
This week, I implemented a discrete-time model of the evolution of opinions of individuals in a network. This model was a part of the work done by my mentor and her PhD student. I experimented with several initial states of the opinions of each individual and the weight matrix which represents the influence two individuals in the network have on eachother. The goal was to analyze, rigorously, the intial states that lead to convergence/divergence (or harmony/polarization) of the opinion of the individuals. Since there are a lot of possible cases to analyze, I will be running these simulations throughout the coming weeks even as the focus of my research shifts in other directions.
A major component of the simulations this week was to analyze the final states of the model under the normalization contraint. Previous simulations showed that the opinions either converge (harmony) or, in most cases, diverge to positive/negative infinity (polarization). However, to make the model more representative of the real-world, we chose to observe the evolution under normalization contraints by normalizing the opinion vector representating the opinion of each individual in the network and the weight matrix at each timestep. The results showed that the model lead to convergence in most cases as expected. The same was shown for various normalization techniques- L1/L2 normalization, Frobenius normalization and Infinity normalization.
This week I attended a seminar by Amy Ogan on Human-Computer Interaction and a workshop on Ethics in Research.
For my project, I worked on generalizing the opinion dynamics and influence model from last week to multidimensional vectors. This can be thought of as people having multiple opinions where each opinion corresponds to a differrent issue. For example Opinion 1 could be for/against gun control, Opinion 2 could be for/against free healthcare etc. Each individual has an m-dimensional opinion vector corresponding to opinions on m different issues. For the weight matrix of the network (the weight matrix repesents the influence between two individuals), we consider two cases:
This week I attended a talk by Martin Tancer on the classical Necklace Splitting Problem.
For my project, I coded and ran simulations for the two cases as described in Week 4. Several types of weight matrices were tested- Random Symmetric, Identity, Random Assymetric etc. The results showed that in the case where each entry of the sqaure weight matrix W is an m x m matrix (corresponding to m opinions for each individual, the opinions tend to diverge in multiple directions towards infinity. In the case where the weight matrix is a square matrix with each entry a real number, the opinions tend to diverge in two opposing directions i.e. the opinion vectors either tended to infinity in quadrants 1 and 3 or quadrants 2 and 4.
This week I attended a seminar talk by Misha Tyomkin on the topic Weak Saturation on Graphs and Hypergraphs. I also attended a workshop hosted by Lazaros Gallos on Scientific and Technical writing.
For my project, from the simulations for the two cases I ran last week, my mentor suggested I experiment and document the behavior of the Discrete-time evolution model when the weight matrices (for both case as mentioned in Week 4) are normalized at each iteration. Specifically, I evaluate the model for two types of normalization techniques:
This week was the AI day. It really informative and I realy enjoyed it- especially the industry panel hosted by Chid Apte from IBM Research and Iraj Saniee from Bell Labs.
For my project, I continued with additional simulations. This time I ran simulations for the Discrete-time model using weight matrix W such that each entry in W is a 2 x 2 matrix. However, this time I normalized the opinion vector and weight matrix by dividing by the absolute maximum of the opinion vector and weight matrix respectively. This made sense intuitively since, normalizing by dividing by the absolute maximum would keep the relative magnitudes of the entries in the opinion vector and weight matrix unchanged but would prevent them form diverging to infinity. I ran simulations for a wide range of intial conditions- symmetric/asymmetric matrices (corresponding to directed/undirected networks), negative matrices and positive semidefinite. I also checked for structural balance in all the simulations that I had run so far. Reaching structural balance in all the simulations is an important property of this model and it would be quite surprising if I were find initial conditions that did not eventually lead to structural balance.
This week I attended the Graduate School panel. I really enjoyed this panel and gained a lot of information about graduate school. The panelists were very knowledgeable and insightful.
For my project, this week I moved away from simulations from generated data and used the model with multidimensional opinions and vectors on a real-world benchmark dataset- the Zachary's Karate Club dataset. This dataset was compiled by Zachary over the course of two years during which he witnessed the splitting of a karate club into two factions centered around two individuals. He observed the interactions of all the members outside of the karate club (which forms the intial social network) and the graph corresponding to these interactions can be used to predict the final community membership of each node/member. We set the two nodes around which the split was centered to have opposite opinions i.e. vectors [1, 1] and [-1, -1]. We set all the other nodes to be neutral (having opinion vector [0,0]) and using the opinion vector for the network and the adjancency matrix corresponding to the the graph of interactions among the members. The co-evolution model was able to predict with 100% accuracy the final membership of each node/member.
This week I spent my time consolidating and documenting my results for the final presentation and the paper. I was able to get a good overview of the progress I made and in hindsight we were able to make a lot of progress over the course of these nine weeks.
Update: I delivered my presentation and it went great (I think...). It was great listening to the presentations of my peers listening to a very diverse range of topics and areas of research combining Math and Computer Science.
This work was carried out while the author, Soham Palande, was a participant in the 2021 DIMACS REU program at Rutgers University, supported by NSF HDR TRIPODS award CCF-1934924