MCM 2014 A: The Keep-Right-Except-To-Pass Rule

In countries where driving automobiles on the right is the rule (that is, USA, China and most other countries except for Great Britain, Australia, and some former British colonies), multi-lane freeways often employ a rule that requires drivers to drive in the right-most lane unless they are passing another vehicle, in which case they move one lane to the left, pass, and return to their former travel lane.

Build and analyze a mathematical model to analyze the performance of this rule in light and heavy traffic. You may wish to examine tradeoffs between traffic flow and safety, the role of under- or over-posted speed limits (that is, speed limits that are too low or too high), and/or other factors that may not be explicitly called out in this problem statement. Is this rule effective in promoting better traffic flow? If not, suggest and analyze alternatives (to include possibly no rule of this kind at all) that might promote greater traffic flow, safety, and/or other factors that you deem important.

In countries where driving automobiles on the left is the norm, argue whether or not your solution can be carried over with a simple change of orientation, or would additional requirements be needed.

Lastly, the rule as stated above relies upon human judgment for compliance. If vehicle transportation on the same roadway was fully under the control of an intelligent system – either part of the road network or imbedded in the design of all vehicles using the roadway – to what extent would this change the results of your earlier analysis?

MCM 2014 B: College Coaching Legends

Sports Illustrated, a magazine for sports enthusiasts, is looking for the “best all time college coach” male or female for the previous century. Build a mathematical model to choose the best college coach or coaches (past or present) from among either male or female coaches in such sports as college hockey or field hockey, football, baseball or softball, basketball, or soccer. Does it make a difference which time line horizon that you use in your analysis, i.e., does coaching in 1913 differ from coaching in 2013? Clearly articulate your metrics for assessment. Discuss how your model can be applied in general across both genders and all possible sports. Present your model’s top 5 coaches in each of 3 different sports.

In addition to the MCM format and requirements, prepare a 1-2 page article for Sports Illustrated that explains your results and includes a non-technical explanation of your mathematical model that sports fans will understand.

ICM 2014 C: Using Networks to Measure Influence and Impact

One of the techniques to determine influence of academic research is to build and measure properties of citation or co-author networks. Co-authoring a manuscript usually connotes a strong influential connection between researchers. One of the most famous academic co-authors was the 20th-century mathematician Paul Erdös who had over 500 co-authors and published over 1400 technical research papers. It is ironic, or perhaps not, that Erdös is also one of the influencers in building the foundation for the emerging interdisciplinary science of networks, particularly, through his publication with Alfred Rényi of the paper “On Random Graphs” in 1959. Erdös’s role as a collaborator was so significant in the field of mathematics that mathematicians often measure their closeness to Erdös through analysis of Erdös’s amazingly large and robust co-author network (see the website ). The unusual and fascinating story of Paul Erdös as a gifted mathematician, talented problem solver, and master collaborator is provided in many books and on-line websites (e.g., Perhaps his itinerant lifestyle, frequently staying with or residing with his collaborators, and giving much of his money to students as prizes for solving problems, enabled his co-authorships to flourish and helped build his astounding network of influence in several areas of mathematics. In order to measure such influence as Erdös produced, there are network-based evaluation tools that use co-author and citation data to determine impact factor of researchers, publications, and journals. Some of these are Science Citation Index, H- factor, Impact factor, Eigenfactor, etc. Google Scholar is also a good data tool to use for network influence or impact data collection and analysis. Your team’s goal for ICM 2014 is to analyze influence and impact in research networks and other areas of society. Your tasks to do this include:

1) Build the co-author network of the Erdos1 authors (you can use the file from the website or the one we include at Erdos1.htm ). You should build a co-author network of the approximately 510 researchers from the file Erdos1, who coauthored a paper with Erdös, but do not include Erdös. This will take some skilled data extraction and modeling efforts to obtain the correct set of nodes (the Erdös coauthors) and their links (connections with one another as coauthors). There are over 18,000 lines of raw data in Erdos1 file, but many of them will not be used since they are links to people outside the Erdos1 network. If necessary, you can limit the size of your network to analyze in order to calibrate your influence measurement algorithm. Once built, analyze the properties of this network. (Again, do not include Erdös --- he is the most influential and would be connected to all nodes in the network. In this case, it’s co-authorship with him that builds the network, but he is not part of the network or the analysis.)

 2) Develop influence measure(s) to determine who in this Erdos1 network has significant influence within the network. Consider who has published important works or connects important researchers within Erdos1. Again, assume Erdös is not there to play these roles.

3) Another type of influence measure might be to compare the significance of a research paper by analyzing the important works that follow from its publication. Choose some set of foundational papers in the emerging field of network science either from the attached list (NetSciFoundation.pdf) or papers you discover. Use these papers to analyze and develop a model to determine their relative influence. Build the influence (coauthor or citation) networks and calculate appropriate measures for your analysis. Which of the papers in your set do you consider is the most influential in network science and why? Is there a similar way to determine the role or influence measure of an individual network researcher? Consider how you would measure the role, influence, or impact of a specific university, department, or a journal in network science? Discuss methodology to develop such measures and the data that would need to be collected.

4) Implement your algorithm on a completely different set of network influence data --- for instance, influential songwriters, music bands, performers, movie actors, directors, movies, TV shows, columnists, journalists, newspapers, magazines, novelists, novels, bloggers, tweeters, or any data set you care to analyze. You may wish to restrict the network to a specific genre or geographic location or predetermined size.

5) Finally, discuss the science, understanding and utility of modeling influence and impact within networks. Could individuals, organizations, nations, and society use influence methodology to improve relationships, conduct business, and make wise decisions? For instance, at the individual level, describe how you could use your measures and algorithms to choose who to try to co-author with in order to boost your mathematical influence as rapidly as possible. Or how can you use your models and results to help decide on a graduate school or thesis advisor to select for your future academic work?

6) Write a report explaining your modeling methodology, your network-based influence and impact measures, and your progress and results for the previous five tasks. The report must not exceed 20 pages (not including your summary sheet) and should present solid analysis of your network data; strengths, weaknesses, and sensitivity of your methodology; and the power of modeling these phenomena using network science.

*Your submission should consist of a 1 page Summary Sheet and your solution cannot exceed 20 pages for a maximum of 21 pages.

This is a listing of possible papers that could be included in a foundational set of influential publications in network science. Network science is a new, emerging, diverse, interdisciplinary field so there is no large, concentrated set of journals that are easy to use to find network papers even though several new journals were recently established and new academic programs in network science are beginning to be offered in universities throughout the world. You can use some of these papers or others of your own choice for your team’s set to analyze and compare for influence or impact in network science for task #3.

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  • Albert, R. and Barabási, A-L. Statistical mechanics of complex networks. Reviews of Modern Physics, 74:47-97, 2002.
  • Bonacich, P.F., Power and Centrality: A family of measures, Am J. Sociology. 92: 1170- 1182, 1987.
  • Barabási, A-L, and Albert, R. Emergence of scaling in random networks. Science, 286: 509-512, 1999.
  • Borgatti, S. Identifying sets of key players in a network. Computational and Mathematical Organization Theory, 12: 21-34, 2006.
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  • Graham, R. On properties of a well-known graph, or, What is your Ramsey number? Annals of the New York Academy of Sciences, 328:166-172, June 1979.
  • Kleinberg, J. Navigation in a small world. Nature, 406: 845, 2000. Newman, M. Scientific collaboration networks: II. Shortest paths, weighted networks, and centrality. Physical Review E, 64:016132, 2001.
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