Isospectral Transformations: A New Approach to Analyzing Multidimensional Systems and Networks

189 Pages·2014·3.52 MB·English

by  Leonid Bunimovich

Springer Monographs in Mathematics
Leonid Bunimovich
Benjamin Webb
Isospectral
Transformations
A New Approach to Analyzing
Multidimensional Systems and Networks
Springer Monographs in Mathematics
Moreinformationaboutthisseriesathttp://www.springer.com/series/3733

Leonid Bunimovich • Benjamin Webb
Isospectral Transformations
A New Approach to Analyzing
Multidimensional Systems and Networks
123
LeonidBunimovich BenjaminWebb
SchoolofMathematics DepartmentofMathematics
GeorgiaInstituteofTechnology BrighamYoungUniversity
Atlanta,USA Provo,UT,USA
ISSN1439-7382 ISSN2196-9922(electronic)
ISBN978-1-4939-1374-9 ISBN978-1-4939-1375-6(eBook)
DOI10.1007/978-1-4939-1375-6
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2014944753
MathematicsSubjectClassification:05C82,37N99,65F30,15A18,34D20
©SpringerScience+BusinessMediaNewYork2014
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection
with reviews or scholarly analysis or material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of
this publication or parts thereof is permitted only under the provisions of the Copyright Law of the
Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.
PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations
areliabletoprosecutionundertherespectiveCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
While the advice and information in this book are believed to be true and accurate at the date of
publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor
anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with
respecttothematerialcontainedherein.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
ToLarissaandRebekah

Foreword
Thisbookprovidesanewapproachtotheanalysisofnetworksand,moregenerally,
to those multidimensional dynamical systems that have an irregular structure of
interactions. Here, the term irregular structure means that the system’s variables
dependoneachotherindissimilarways.Forinstance,all-to-allornearest-neighbor
interactionshavearegularstructure,becauseeachvariabledependsontheothersin
asimilarmanner.
In practice, this structure of interactions is represented by a graph, called
the network’s graph of interactions or the network’s topology. Depending on
the particular network, this graph may be directed or undirected, weighted or
unweighted,withorwithoutloops,withorwithoutparalleledges,etc.Ineachcase,
thetechniquesprovidedinthisbookcanbedirectlyappliedtothesenetworks.
It is worth mentioning that although these methods are fairly new, they have
already proven to be an efficient tool in some classical and more recent problems
inappliedmathematics.Here,thesetechniquesarepresentedasawayinwhichto
viewandanalyzereal-worldnetworks.
One of the major goals of this book is to make these methods and techniques
accessibletoresearcherswhodealwithsuchnetworks.Withthisgoalinmind,we
notethatthecomputationsrequiredtoimplementthesetechniquesareremarkably
straightforward. In fact, they can be carried out using any existing software
sophisticatedenoughtoperformelementarylinearalgebra.
In terms of the book’s content, we note that each of the results is given with
a mathematical proof. However, with the hope that this book will be read and
used as well by nonmathematicians, the text is written so that those interested in
applications can safely ignore these arguments and use the stated formulas and
techniquesdirectly.Still,westressthefactthatonlyabasicunderstandingoflinear
algebraisneededtounderstandtheproofs.
Thedefinitionsandresultswepresentaremotivatedandaccompaniedbymany
examples, which both nonmathematicians and mathematicians should appreciate.
vii
viii Foreword
Thebookalsocontainsalargenumberofexamplesandfiguresdepictingthegraphs
associatedwithparticularnetworksaswellastheirvarioustransformations.
Almost all of these examples deal with directed graphs. The reason is that
directed graphs are more general objects than undirected graphs. However, the
theory developed here works just as well for undirected graphs. This is important,
forinstance,inthestudyofrealnetworks,sincealargenumberofthosenetworks
haveanundirectedgraphstructure(topology).
Because real-life networks are often large and have a complicated structure,
it is tempting to find ways of simplifying them in terms of both their size and
complexity.Whatisimportant,though,isthatsomebasicorfundamentalproperty
ofthenetworkbepreservedinthisprocess.Yetsuchanattemptseemsdoomedto
failure.Thesearerealnetworks,sowedonotknowmuchifanythingaboutthem,
including whichcharacteristic(s)weshould retain.Moreover, therearepotentially
manywaysinwhichanetworkcouldbereduced.Hence,thereisfirsttheproblemof
choosing which way the network should be reduced and second determining what
the reduced network tells us. Thus many objections are immediately raised if one
wantstoreducethesizeofanetwork.
From this point of view, our goal of reducing a network may seem overly
ambitious. In fact, one could ask how it is possible even to represent an arbitrary
network. The universally excepted answer is that this can be done by drawing a
graphwhosevertices(nodes)correspondtothenetworkelementsandwhoseedges
(links)correspondtothedirectedinteractionsbetweentheseelements.
Equivalently, one can represent a network by a matrix A with entries A . In
ij
thisrepresentation,A isthestrengthorweightofthedirectedinteractionbetween
ij
theithandjthnetworkelements,whereA D 0iftheseelementsdonotinteract.
ij
Suchamatrixiscalledtheweightedadjacencymatrixofanetwork.Ifthenetwork’s
interactionstrengthsarenotknown,thenonzeroentriesofthematrixaresetequal
to1,andAiscalledthe(unweighted)adjacencymatrixofthenetwork.Inpractice,
knowledgeofanetwork’sadjacencymatrixisoftenthemostonecanhopetohave.
Itiswellknownthataverybasiccharacteristicofamatrixisitsspectrum,i.e.,
its collection of eigenvalues including multiplicities. One of the main questions
we address in this book is whether it is possible to reduce a network to some
smallernetworkwhilepreservingthenetwork’sspectralproperties.Phrasedanother
way, this question could be stated as whether it is possible to reduce the size of a
network’sadjacencymatrixwhilemaintainingthenetwork’seigenvalues.
Theimmediateanswertothisquestionis,ofcourse,no.Infact,whilepresenting
these results, we have had audience members protest that what we hope to do is
impossible.Indeed,aseveryoneknows,thefundamentaltheoremofalgebrastates
thatann(cid:2)nmatrixhasneigenvalues,whileasmallermatrixhasfewer.
However, our claim is that it is possible to reduce a matrix and preserve the
matrix’sspectralproperties.Inthisbook,weshowthattheanswertoourquestion
becomesyesifoneconsidersalargerclassofmatrices,namelymatriceswithentries
thatarerationalfunctionsofaspectralparameter(cid:2).Thatis,itispossibletoreduce
an n(cid:2)n matrix with scalar entries to a smaller m(cid:2)m matrix with functions as
entries and maintain the matrix’s spectrum. We refer to this process as isospectral
matrixreduction.
Foreword ix
At this point, the reader may think that by isospectrally reducing a network’s
adjacency matrix we are, in fact, shifting the complexity of a network’s structure
(topology)tothecomplexityofitsedgeweights.Wepausetoreassureourreaders
that we have considered this idea and that many facts and results in this book
demonstrate that such is not the case. However, before moving on, we stress just
onefundamentalfactregardingisospectralreductions.
The structure (topology) of an isospectrally reduced network does not depend
on the strengths (weights) of the initial unreduced network. It depends only
on the network’s structure. The structure of the reduced network will be the
same regardless of the strengths of interactions in the initial network. Therefore,
the isospectral reductions we consider really capture some hidden but intrinsic
informationregardingthestructureofanetwork.
This approach to analyzing networks is based on ideas and methods from the
theory of dynamical networks, which is a part of the modern theory of dynamical
systems. The first dynamical networks addressed in this theory were the so-called
coupled map lattices (CML). CMLs were introduced in the mid-1970s, almost
simultaneously, by four physicists in four countries. The mathematical theory of
CMLwasbegunin[7],inwhichthefirstprecisedefinitionsofspace-timechaosand
acoherentstructureweregiven.Nowadays,thetheoryoflatticedynamicalsystems
isarespectedpartofcontemporarydynamicalsystemstheory(see,e.g.,[13]).
A number of remarkable findings of the late 1990s demonstrated that real
networkshaveverycomplicatedtopologies[3,19,20,23,24,30].Thefirstthought
was that the ideas of dynamical systems theory and of statistical mechanics could
beappliedtosuchsystems,ashadbeendonein[7,12]forCMLs.However,infinite
latticeshaveagrouptranslationproperty,whichismissingifagraphofinteractions
hasanirregularstructure.Anapproachtodealingwiththisirregularstructurewas
eventuallydevelopedin[1,4],inwhichthefollowingwasobserved.
Every dynamical network has three features: (i) the individual dynamics of
the network elements, e.g., a single isolated neuron in a neural network; (ii) the
interactionsbetweentheelementsofanetwork;and(iii)thestructure(ortopology)
of a network. In this framework, we assume that a network’s structure does not
changeovertime,sothatithasafixedstructureofinteractions.However,aswelater
pointout,thetransformationsconsideredinthisbookcouldbeusefulforstudying
networksthatdohaveastructurethatevolvesovertime.
Observe that features (i) and (ii) of a network are dynamical systems. Thus, it
is customary to deal with such systems by analyzing the combined influence of
(i)and(ii),asisdoneinotherspatiallyextendedsystems.Perhapsthemostpopular
exampleoftheseisreaction–diffusionsystemsinwhichnonlinearreactionspushthe
systemtowardschaoticbehaviorwhilediffusionhasastabilizingeffect.However,
the question is what to do with (iii), which is clearly a static characteristic of the
network.
Asisshownin[1],thetopologyofanetworkcanalsobetreatedasadynamical
system generated by considering all infinite paths on the network’s graph of
interactions. This, together with the ideas from the theory of spatially extended
systems,formsthebasisofourapproach.