**Project Description**

This project will study the rapid flow of information in large social networks. This is important in "viral marketing" and comes out of early work done on the spread of epidemics and social network theory. A graph is built with nodes representing individuals in a population and an edge between two nodes if they have some form of communication. Each node has a choice of two behaviors, an old behavior and a new behavior, and an incentive (payoff) for matching behaviors. Each node plays the coordinated game with each neighbor and the payoff for a node is the sum of the individual game payoffs. A sample problem is to determine the k most influential people in the network, an NP-hard problem. We seek variations on the model, such as weighting the edges with other payoffs (e.g., with marketing strategies of price reductions) to better determine the most influential players (often those to start marketing to).

Presentation 1 - Friday, June 11

Presentation 2 - Friday, July 16

**Project Log**

**Week 0:** Traveled to Rutgers University in Piscataway, NJ and attended orientation.
There are approximately 40 students from all across the United States, as well as seven from Charles University in Prague, Czech Republic.
Read miscellaneous notes from a graduate course in Game Theory as well as Jon Kleinberg's "Cascading Behavior in Networks: Algorithmic and Economic Issues."
Met with our advisor to discuss possible directions that the project could take. Looked for other relevant papers to read in preparation for the first presentation.
Set up web page.

**Week 1:** Read multiple papers on social network theory and diffusion models for networks.
Reviewed basic concepts from graph theory, as we will be using graphs to model the networks under investigation.
Individuals in the network are represented by nodes (or vertices), while edges connecting two nodes indicate that they are related, and hence exert some form of influence on one another.
Worked with project partners to prepare Presentation 1, which we gave on Friday.

**Week 2:** Read papers on graph theory, focusing on graph connectivity.
Of particular interest were critically connected graphs and corresponding cut strategies resulting in a disconnected graph or a clique.
Began playing with simple types of graphs (cycles, trees, grids, etc.) to observe diffusion for various infection / conversion rates on these different structures.

**Week 3:** Wrote a program to simulate diffusion on a graph given user specified initial condiditons.
Developed a game theoretic aspect for modeling cascading behavior on a path.
Played with variations of the path game given different payoff matrices to determine the impact on the game's mixed strategy Nash equilibrium.
Sought to generalize observations apparent in the path cascade models.

**Week 4:** Developed a model of national elections as an application of viral marketing.
Considered implications of viewing the campaign cycle as a two-player game played by the Democrats and Republicans.
Wrote code to simulate developments in the game, thereby forecasting election results, based on each party's marketing choices.

**Week 5:** Expanded upon my program to comprehensively model different strategies.
Developed complete payoff decision tree diagrams for small versions of the game.
Found the equilibrium strategy sequence in these cases and compared it to the actual campaign developments and subsequent result of the election.
Compared and contrasted differences in payoffs produced by different game variations.

**Week 6:** Made final touches on work completed thus far.
Worked on a draft of a paper to submit to my advisor documenting the work I had done.
Prepared and gave Presentation 2.

**Week 7:** Departed for Prague, Czech Republic to spend time with DIMATIA at Charles University and attend the Midsummer Combinatorial Workshop.

**Resources / Works Consulted**

This is not an exhaustive list of materials viewed, but intends to capture the most relevant books, papers, articles and online resources I came across over the course of the project.

Dixit and Skeath. __Games of Strategy.__

Jackson, Matthew. "Diffusion on Social Networks."

Kempe, David, Jon Kleinberg and Eva Tardos. "Influential Nodes in a Diffusion Model for Social Networks."

Kleinberg, Jon. "Cascading Behavior in Networks: Algorithmic and Economic Issues."

Lopez-Pintado, Dunia. "Diffusion in Complex Social Networks."

Young, H. Peyton. "The Diffusion of Innovations in Social Networks."

**Links**

Rutgers DIMACS REU

University of South Carolina Math Department, Stat Department, and Honors College

Phi Beta Kappa - The Nation's Oldest Academic Honor Society

Pi Mu Epsilon - The National Mathematics Honor Society

Omicron Delta Kappa - The National Leadership Honor Society