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Vol.33,No.3 012
Table of Contents
Guest Editorial
Network Science:What’s Math Got to Do with It?
Chris Arney (185)
Editor’s Note
About This Issue (192)
MCM Modeling Forum
Results of the2012Mathematical Contest in Modeling
William P.Fox (193)
A Close Look at Leaves
Bo Zhang,Yi Zhang,and TianKun Lu (205)
Judges’Commentary:The Outstanding Leaf Problem Papers Peter Olsen (223)
ComputingAlong the Big Long River
Chip Jackson,Lucas Bourne,and Travis Peters (231)
Judges’Commentary:The Outstanding River Problem Papers Marie Vanisko (247)
Author’s Commentary:The Outstanding River Problem Papers Catherine A.Roberts (253)
Judges’Commentary:The Giordano Award for the River Problem Marie Vanisko and Richard D.West (259)
ICM Modeling Forum
Results of the2012Interdisciplinary Contest in Modeling Chris Arney (263)
Finding Conspirators in the Network via Machine Learning Fangjian Guo,Jiang Su,and Jian Gao (275)
Judges’Commentary:Modeling for Crime Busting
Chris Arney and Kathryn Coronges (293)
Reviews (305)
Guest Editorial185 Guest Editorial
Network Science:
What’s Math Got to Do with It?1
Chris Arney
Dept.of Mathematical Sciences
067aad14c281e53a5802ff30itary Academy
West Point,NY10996
david.arney@067aad14c281e53a5802ff30
Introduction
This year’s ICM R problem involved network science,or more precisely, a component of network science—social network analysis.My post-contest re?ections have led me to believe it is time for the mathematics community to engage in this emerging subject to build a rigorous mathematical foun-dation for this important science and to join in performing mathematical modeling and interdisciplinary problem solving.
Some people call network science a“new”emerging discipline,yet,as we know,mathematicians have been developing graph(network)theory for centuries,and scientists and engineers have been modeling networks for decades.What is new is that the traditional techniques have been re-placed by an entirely new arsenal of mathematics,science,and modeling associated with networks.
Others call network science the“new”operations research in that it con-nects quantitative concepts and elements from several disciplines such as mathematics,computer science,and information science with the qualita-tive models from sociology and other social sciences.By its very nature, network science is interdisciplinary and involves emerging areas of science such as complex adaptive systems,cooperative game theory,agent-based modeling,data analytics,and social network analysis.
1With both appreciation and apologies to Tina Turner and her emotional song“What’s Love Got To Do With It”and full tongue-in-cheek realization that unlike“love,”mathematics is certainly not a second-hand emotion.
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186The UMAP Journal33.3(2012)
From another,perhaps simpler,perspective,you could merely call net-work science a form of applied mathematics—applied dynamic graph the-ory with additional data elements and attributes.Or perhaps from a mod-eling perspective,it is simply modeling with a highly structured,entity-linked complex adaptive system framework.
I do not pretend to know exactly what network science is,or where it ?ts in with today’s scienti?c world,or what it will become.However,I believe that much of the strength of network modeling is in its ability to embrace the complexity of the real world.For me,that makes network science an important and empowering form of interdisciplinary modeling and problem solving—worthy of ICM problems and much more.
In particular,I hope that the mathematics community does not ignore 067aad14c281e53a5802ff30work science needs the engagement of the mathematics community to produce its underlying framework and to invent new mathematics and computational techniques for analysis of its complex structures,develop-ment of its synergistic processes,and organizing of its overwhelming data. Likewise,mathematics needs network science to establish the relevance of mathematics in the modern information-based world.As the ICM teams discovered,network science is exciting,relevant,enjoyable,and modern—elements that mathematics currently desperately needs to bolster its future place in society.
Mathematical Elements
What are the mathematical elements of network science?One way to de?ne a network is to establish its
?components(nodes,links,data,processes);
?properties(dynamic,functional,layered);and
?applications(logistics,?ow,transportation,Internet,metabolic networks, social networks,organizational networks—perhaps there are just too many categories to list!).
Another way is to use the concept of a mathematical graph(the nodal-link structure)with its nontrivial topological features and then classify the various types of graphs that occur(random,scale free,small world,scale rich)and the data(often heavy-tailed)that need to be mined and analyzed.
A foundational research management report on network science offered a layered approach of network roles—physical,communicative,informa-tional,biological and social/cognitive—that connect together to produce the overall web-like network framework[National Research Council2006].
