第一句问候No matter what de?nition or theoretical framework is used,network sci-ence is inherently and essentially mathematical at its core;there is plenty for applied mathematicians to do.Most networks are suf?ciently complex that
Guest Editorial187 simply relying on visualization produces erroneous intuitive conceptions—or,worse yet,complete misunderstanding.De?ning,computing,and mea-suring well-de?ned properties can counter those misguided perceptions and improve network modeling and analysis.Indispensable roles for math-ematical modelers in network science are
?working with social scientists to build explicative and empirical models,?creating appropriate measures for important applications,and
??nding appropriate properties and formalizing their measurement sys-tems and calculations.Like many other mathematical modeling con-structs,these properties can be classi?ed as structural or functional;lo-cal,global,or regional;discrete or continuous;dynamic or static;and deterministic or stochastic.
The network science world needs mathematicians to help sort out these characteristics and do even more.
Signi?cance
Network science has become a major global research thrust(with fund-ing potential equal to or exceeding that of mathematics and many sciences) in the research agenda of governments,societies,militaries,businesses, and organizations.Its publication and citation qualities and quantities are signi?cant.Just look at the remarkable citation record of the works by someone such as Albert-LászlóBarabási to see this subject’s in?uence in science.New societies,new conferences,and new journals with a network science theme are emerging at a dazzling 067aad14c281e53a5802ff30work science is reaching the popular press and also entering the business world with tremendous fanfare.
Mathematicians’Engagement
One could argue that all the Mathematics Awareness Month themes since1997,when the theme was“Mathematics and the Internet,”have been related to networks in some way.The theme for2004,“The Mathematics of Networks,”established a?rm connection,and I still use the myriad networks on that year’s poster as examples to students of the variety and beauty of networks.However,while mathematicians are certainly aware of network science,I still do not see much real engagement by the U.S. mathematics community.Recent meetings of the Mathematical Associa-tion of America and the American Mathematical Society show only minimal mathematically-connectednetwork research.The Society for Industrial and
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Applied Mathematics and the Institute for Operations Research and Man-agement Science and their members are slightly more engaged,although “when it comes to the research agenda now popularized by network sci-ence,[operations research]has been an underutilized resource”[Alderson 2008].
In my opinion,mathematics is a vastly underutilized and unfortunately often missing part of network science.In a comment on Alderson[2008], Nagurney adds that“it is not just the network topology and associated sta-tistical aspects of networks that matter but?ows that must be incorporated into network modeling as well as behavior”[Nagurney2008].Alderson and colleagues also wrote about mathematics and its engagement in Inter-net research that“surprising little attention has been paid in the mathemat-ics and physics communities...in the Internet research arena”[Willinger et al.2009].Of course,there is much more to network science than the In-ternet,but it is a signi?cant network that most of the world confronts many times and in many ways every day.
Family Tree
It may be worth looking at a family tree for network science.While these relationships are subjective and incomplete,one can see in Figure1a ?ow that brings together many elements of mathematics.The mathematics community should not miss out on an opportunity as rich and stimulating as network science.Ultimately,the mathematical elements of this discipline will be accomplished somehow and by someone.I suggest that this work be done by mathematicians—and the sooner,the better.
Network Science for Undergraduates Does network science extend to the undergraduate curriculum?Based on their interest in this year’s ICM problem,I believe that undergraduates would respond in the af?rmative.In some institutions,network science is even making inroads in establishing undergraduate programs,and courses and programs traditionally offered at the graduate level are entering the undergraduate realm.My mathematics students tell me that they want to learn network science.As analysis of social media and online games are be-ginning to be seen to be parts of network science modeling,student interest in network science is growing in leaps and bounds.The bottom line is that network science is highly popular with students.I have taught three differ-ent network science courses over the past four semesters and am designing a fourth for the Fall semester of2012.I usually have to turn away students or add more sections.This past Spring,I team-taught social network analy-sis with a sociologist and enjoyed the interdisciplinary modeling aspects of
Guest Editorial189 this exciting 067aad14c281e53a5802ff30st year’s award-winning publication for INFORMS
was a network science book by Easley and Kleinberg from Cornell written for undergraduates[Easley and Kleinberg2010].Network science is on the
map of undergraduate education.
Worldwide Interest
The ICM data show there may be differences among nations in per-ceptions or interests in interdisciplinary modeling and network modeling. The U.S.has always been slightly behind the rest the rest of the world in ICM/MCM R interest ratio,as measured by the proportion of teams who choose the ICM problem rather than one of the MCM 067aad14c281e53a5802ff30ually, there is about half as much interest in the ICM from U.S.teams compared to teams from the rest of the world;this year,that ratio dropped to about one-third.I do not know why this is so or if this phenomenon has any real signi?cance.Perhaps American students have a more disciplinary focus on mathematics or just haven’t been exposed to as many network or interdis-ciplinary ideas.Whatever the reason,I personally hope that all students (high school,undergraduate and graduate)in every nation have the oppor-tunity to study some aspects of networks,and that the mathematics they learn in doing so goes to excellent use.
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Exhortation
Mathematicians,let’s not miss this opportunity.Take another look at network science and see where you can contribute.Talk to colleagues in other disciplines and form teams to learn,study,research,teach,and engage in this enjoyable and important?eld of network science. References
Alderson,David L.2008.Catching the“network science”bug:Insight and opportunity for the operations researcher.Operations Research56
(5)(September-October2008):1047–1065.067aad14c281e53a5802ff30/
content/download/255771/2414490/file/networks.pdf. Easley,D.,and 067aad14c281e53a5802ff30works,Crowds,and Markets:Reasoning About a Highly Connected World.New York:Cambridge University Press. Nagurney,067aad14c281e53a5802ff30ment on Alderson[2008].Operations Re-search(Online Forum Commentary)56(5)(October–November2008)on-line commentary.067aad14c281e53a5802ff30/content/download/ 255775/2414506/file/nagurney.pdf.
National Research 067aad14c281e53a5802ff30work Science.Washington,DC:Na-tional Academies Press.
Willinger,Walter,David Alderson,and John C.Doyle.2009.Mathematics and the Internet:A source of enormous confusion and great potential.
Notices of the American Mathematical Society56(5)(May2009):286–299.
067aad14c281e53a5802ff30/notices/200905/rtx090500586p.pdf.
Guest Editorial191 About the Author
Chris Arney graduated from West Point and served
as an intelligence of?cer in the U.S.Army.His aca-
demic studies resumed at Rensselaer Polytechnic In-
stitute with an M.S.(computer science)and a Ph.D. (mathematics).He spent most of his30-year military
career as a mathematics professor at West Point,be-
fore becoming Dean of the School of Mathematics and
Sciences and Interim Vice President for Academic Af-
fairs at the College of Saint Rose in Albany,NY.Chris then moved to RTP (Research Triangle Park),NC,where he served for various durations as chair of the Mathematical Sciences Division,of the Network Sciences Di-vision,and of the Information Sciences Directorate of the Army Research Of?ce.Chris has authored22books,written more than120technical arti-cles,and given more than250presentations and40workshops.His techni-cal interests include mathematical modeling,cooperative systems,pursuit-evasion modeling,robotics,arti?cial intelligence,military operations mod-eling,and network science;his teaching interests include using technology and interdisciplinary problems to improve undergraduate teaching and curricula.He is the founding director of COMAP’s Interdisciplinary Con-test in Modeling(ICM)R .In August2009,he rejoined the faculty at West Point as the Network Science Chair and Professor of Mathematics.
192The UMAP Journal33.3(2012)
Editor’s Note
About This Issue
This year we had5,000(!)participating teams in the two contests com-bined;the18(!)Outstanding papers ran to over500manuscript pages. Editing and publishing all the Outstanding papers,which we once did,is simply not possible any more.
Hence,as in2010and2011,we are able to present in the pages of the Journal only one Outstanding entry for each of the MCM and ICM problems. The selection of which papers to publish re?ected editorial considerations and was done blind to the af?liations of the teams.
All of the18Outstanding papers appear in their original form on the2012 MCM-ICM CD-ROM,which also has the press releases for the two contests, the results,the problems,unabridged versions of the Outstanding papers, and some of the 067aad14c281e53a5802ff30rmation about ordering is at http: //067aad14c281e53a5802ff30/product/cdrom/index or at(800)772-6627.
