Applying Graph Theory in Ecological Research

340 Pages·2017·5.448 MB·english

by  Mark Dale

Applying Graph Theory in
Ecological Research
MARK R. T. DALE
UniversityofNorthernBritishColumbia
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107089310
DOI:10.1017/9781316105450
(cid:2)C CambridgeUniversityPress2017
Firstpublished2017
PrintedintheUnitedKingdombyTJInternationalLtd.PadstowCornwall
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
ISBN978-1-107-08931-0Hardback
ISBN978-1-107-46097-3Paperback
Contents
Preface pagex
1 GraphsasStructureintheEcologicalContext 1
Introduction 1
1.1 GraphsasStructure 3
1.2 GraphsandEcologicalRelationships 10
1.3 GraphsandLocations:SpatialandTemporal 11
1.4 NetworksandDynamics 17
1.5 GraphsandData 18
1.6 EcologicalHypothesesandGraphTheory 27
1.7 StatisticalTestsandHypothesisEvaluation 29
1.8 ConcludingComments 35
2 ShapesofGraphs:TreestoTriangles 37
Introduction 37
2.1 AcyclicGraphs 37
2.2 DigraphsandDirectedAcyclicGraphs 41
2.3 WeightedDirectedTrees 45
2.4 LatticeGraphs 46
2.5 Triangles 49
2.6 SmallerThanTriangles:Singletons,IsolatedPairsandWhiskers 50
2.7 HowItLooks 51
2.8 ConcludingComments 52
3 SpeciesInteractionNetworks 54
Introduction 54
3.1 Objects 57
3.2 Properties 60
3.3 GenerativeModels 68
3.4 Comparisons 72
3.5 ConcludingComments 77
4 TrophicNetworks:Structure,PartitioningandDynamics 79
Introduction 79
4.1 TrophicNetworksandDerivedGraphs 82
4.2 TrophicNetworkCharacteristics 86
4.3 ConcludingComments 102
5 SpeciesAssociations,CommunitiesandGraphsofSocialStructure 105
Introduction 105
5.1 GraphsofSocialStructure 107
5.2 ClusterDetectioninGraphsandNetworks 113
5.3 TransitivityandReciprocity 121
5.4 Balance 122
5.5 Change 124
5.6 KeyNodes;KeyEdges 126
5.7 ConcludingComments 127
6 Competition:HierarchiesandReversals 128
Introduction 128
6.1 ConceptsforCompetitionInteractionGraphs 130
6.2 MeasuringCompetitiveOutcomes 135
6.3 ChoosingEdgesandFindingHierarchies 137
6.4 Example:ArabidopsisthalianaEcotypes 141
6.5 ConcludingComments 144
7 Mutualism,ParasitismandBipartiteGraphs 147
Introduction 147
7.1 InternalStructureofBipartiteGraphs 147
7.2 ApplicationsofBipartiteGraphs 158
7.3 ConcludingComments 163
8 TemporalorTime-OnlyGraphs 164
Introduction 164
8.1 PropertiesofTemporalGraphs 170
8.2 TechniquesforTemporalGraphs:TestingSignificance 180
8.3 ApplicationsofTechniques 183
8.4 ConclusionsandAdvice 185
9 SpatialGraphs 191
Introduction 191
9.1 PropertiesofSpatialGraphs 193
9.2 TechniquesforSpatialGraphs:TestingSignificanceandOther
Assessments 201
9.3 ChoiceandApplicationsofTechniques 202
9.4 ConcludingComments 221
10 Spatio-temporalGraphs 222
Introduction 222
10.1 Characteristics 226
10.2 TwoSpatio-temporalProperties 228
10.3 ExamplesofEcologicalApplications 232
10.4 ConcludingComments 251
11 GraphStructureandSystemFunction:GraphletMethods 252
Introduction 252
11.1 GraphsforStructureandDynamicsinEcologicalSystems 257
11.2 GraphCharacteristicsandMethodsBasedonGraphlets 259
11.3 ConcludingComments 269
12 SynthesisandFutureDirections 271
Introduction 271
12.1 ComparisonsandMatching 271
12.2 WhatNext? 276
12.3 ConcludingComments 285
Glossary 286
References 297
Index 328
Appendix 333
Colourplatesarefoundbetweenpages212and213.
Preface
Applicationsofgraphtheoryhavebeenproliferatingthroughoutecologyoverthepast
severaldecades,whetherexplicitlyrealizedorimplicitintheapproachesused,andnot
onlyinthecaseswhichfallclearlyintothepopularcategoryofnetworks.Thereasons
for this increased interest are as diverse as the areas of research. A basic impetus is
that graphs and graph theory are about structure and provide the methods to analyze
structure as abstracted from almost any ecological (or other) system. The second rea-
sonisthegreatpopularityofnetworkstudiesandnetworktheory,originallyappliedto
socialrelationships,communications(includingtheInternetasaprimeexample),trans-
portationandthespreadofdisease.Itisanobvioussteptotakenetworkconceptsand
models from these sources and see how well they apply to ecological systems. Such
networkstudiesareobvioussourcesofinspirationforinvestigationsofecologicalinter-
actions of all kinds (such as predation, competition, mutualism and facilitation) using
themethodsdevelopedforthoseothersystems.Athirdreinforcementforgraphtheory
applications arises from the growing sophistication of ecologists in analyzing spatial
dataortime-ordereddataorthecomplexitiesofspatio-temporaldata;and,onceagain,
methodsbasedongraphtheoryprovidetherightmixofsimplicityofconceptbutflex-
ibility in application to provide valuable insights that would otherwise be impossible.
Putting together interaction networks and spatio-temporal data brings a researcher to
thechallengesandrewardsofstudyingtheinterplayofformandfunction(or“pattern
and process” or “structure and dynamics”) in ecological systems in which both form
andfunctionchangethroughtimebyreciprocalinfluencesandeffects.
Thebookisorganizedinanorderthatreflectsthisrangeofsources.Firstisanintro-
duction tothinking withgraphs based on the theme of graphs and structure (Chapters
1 and 2). There are then several chapters on ecological interaction networks, first in
general (Chapter 3), followed by more specific topics: predation (Chapter 4), social
structure (Chapter 5), competition (Chapter 6) and mutualism (Chapter 7). The next
threechaptersareaboutlocationalgraphs,inwhichthenodeshavepositionsinoneor
more dimensions: time only (Chapter 8), space only (Chapter 9) and spatio-temporal
(Chapter 10). Chapter 11 describes approaches to studying the dynamics of networks
inthecontextofthereciprocaleffectsofformandfunction,focussingonthefascinat-
ingandpromisingmethodsbasedongraphlets.Thelastchapter(Chapter12)attempts
to draw together a number of the themes that emerged throughout the book and pro-
videasynthesisofthecommonthreads;italsotakesontheriskytaskofmakingsome
predictionsaboutfuturedirectionsanddevelopmentstobeexpectedinthisfield.
Theworkingtitlestartedoutas“SmartThingsEcologistsCanDowithGraphThe-
ory”;andthatisagooddescriptionoftheintention.Thebookisnotprimarilyanintro-
duction to graph theory developed for ecologists; it is intended to make researchers
awareofthewiderangeofpossibilitiesfortheirownresearchprojects,evenwhen(or
especially when) they have yet to be fully tried out in ecological systems. A prime
example is the many forms of analysis based on graphlets that are recently developed
andappliedinotherbiologicalsystems(e.g.protein-proteininteractions)butnotyetin
ecology.Thegoalistoprovideenoughbackgroundthattheresearcherknowshowand
wheretostartandwheretofindsomeexamplesthatwillprovideinspirationandsupport.
Thetreatmentsofthevarioustopicsareveryheterogeneous;somehaveagoodrangeof
examplestobecited(e.g.foodwebsortrophicnetworks;mutualism),butothershave
virtuallynone.
Myowninterestingraphtheoryasausefulapproachtoansweringecologicalques-
tionsrelatedtostructurestartedwithmyMScresearchmanyyearsago,andIowealarge
debttomythen-supervisor,TonyYarranton,whosuggestedtheareaandencouragedmy
exploration of the field. I owe thanks to John Moon, who helped me understand some
of the more formal aspects of graph theory and its application (look at his Topics on
Tournaments,ifyouhavenotalready:agreatexample).Inacknowledgingpeoplewho
have helped with this book, I thank the following for reading chapters, sometimes as
theydeveloped:AlexAravind,TanBao,ConanVietch,JCCahillandBrendanWilson.
I thank Marie-Josée Fortin, especially; she read all the chapters, and some more than
once!Fordatausedinexamples,therearemanytobeacknowledged,includingTanBao
andJCCahillfortheArabadopsiscompetitiontournamentmaterialandGordThomas
fortherichdatasetonSaskatchewanweedcommunities.IthankNSERCCanadaand
UNBCfortheirsupportovermanyyears.
I greatly enjoyed writing this book, and discovering all the exciting material I had
notknownwasveryrewarding.Itismyhopethatthereaderswillfindtheworkequally
rewardingandthatitwillhelpcreatepathwaystomorethatisuseful,morethatisnew
andmorethatissurprising.
1 Graphs as Structure in the
Ecological Context
Introduction
Ecology is the study of organisms in the context of their environment, including both
abiotic effects and interactions among organisms. Ecologists, like other scientists, are
lookingforpatternsinthesephenomenathatcanbeusedreliablytomakepredictions,
andthosepredictionscanextendthefindingstootherorganisms,toecologicalsystems
notyetstudiedormerelytosimilargroupsoforganismsindifferentplacesoratdifferent
times.Thosepredictionsmayalsorefertohowasystem’sformorstructuredetermines
itsfunctionanddynamicsandhowfunctionanddynamicsconstrainormodifystructure
andform.
Alongbutnotexhaustivelistofthekindsofproblemsecologistsstudymightinclude
thefollowing:
(cid:2)
thefateofindividualsasdeterminedbyneighboursandenvironmentalconditions
(cid:2)
the interactions of individuals in a social structure and their effects on population
dynamics
(cid:2)
themovementofindividualsthroughtheirenvironmentandtheirreactionstoit
(cid:2)
thedynamicsofpopulationsandcommunitiesinfragmentedhabitats
(cid:2)
theflowofenergyandthepopulationandcommunityeffectsofpredationintrophic
networks
(cid:2)
the effects of competition, both intra- and inter-specific, on survival, growth and
reproduction
(cid:2)
the dynamics of species interactions, such as mutualism, commensalism and para-
sitism
(cid:2)
thedeterminantsofspeciescompositionofmulti-speciescommunitiesinislandsys-
tems
Almostallofthesecanbeapproachedinatheoreticalorabstractedway,orquiteexplic-
itly with locations in time or space, and almost all of these are studied in the context
of a system of some sort and usually in the context of that system’s structure. In fact,
explicitreferencesto“structure”ariseinalmosteverystudyofecologicalsystems,from
behaviourtotrophicnetworksandfromindividualstocommunityinteractions.Theterm
“structure”usuallyreferstohowsystemsareputtogetherortotherelationshipsamong
units that determine how they work together. Structure, like pattern, suggests some
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2 1 GraphsasStructureintheEcologicalContext
Figure1.1 Agraph.Thebasicgraphconsistsofnodes((cid:129))andedges(──)joiningpairsofnodes.
Nodescanhavelabels,weightsorlocations.Edgescanhavedirections,signs,weights,
functionalequationsorlocations.
predictabilityinthewayaphenomenonisorganized,eveniftheprocessthatgivesrise
toithasarandomoriginorstochasticcomponent,suchasthefatesofindividualorgan-
isms.Evenstructuresgeneratedbyfullyrandomprocessesmayhavepredictablechar-
acteristics, as we will see in Chapter 3. Graph theory is the mathematics of the basics
ofstructure(objects andtheirconnections), providing arichtechnical vocabulary and
aformaltreatmentoftheconceptsandoutcomes.Becauseoftheimportanceofunder-
standingandquantifyingstructureinallecologicalsystems,graphtheoryhasimportant
contributionstomaketoabroadrangeofecologicalstudies,includingtrophicnetworks
(Kondohetal.2010),mutualisms(Bascompte&Jordano2014),epidemiology(Meyers
2007)andconservationecology(Keittetal.1997),wherethegraphsdepictfunctional
connectionsamongorganismsorphysicalconnectionsamongspatiallystructuredpop-
ulations(Grantetal.2007).
The graphs that are the focus of graph theory are deceptively simple mathematical
objects,eachconsistingofasetofpointswithasetoflinesjoiningtheminpairs.The
pointsarecallednodes,representedbydotsinadiagram(Figure1.1),andthelinesare
edges,representedbystraightorcurvinglinesinadiagram,althougharangeofterms
canbefoundintheliterature(seeHarary1969;West2001).
Graphs are about connections and the pattern of connections. In a diagram of the
mostbasicgraph,thepositionsofthenodesonthepageandthelengthsandshapesof
theedgesjoiningthemhavenomeaning;theyareplacedforconvenienceandclarity.It
is the set of connections made by the edges that determines the graph’s topology. The
nodes usually represent components or units of organization, and the essence of the
graphliesinwhatisconnectedtowhat:reallyverysimple!Inthisway,thegraphisan
abstract description of structure or topology because the edges show the relationships
amongorganizationalcomponentsthatthenodesrepresent.
Graphs and graph theory lend themselves extremely well to applications in many
areas of science because there is a wealth of mathematical knowledge that has been
developedovertheyearsfromstudyingthesesimplecomponents.Graphtheoryinves-
tigates all aspects of combinations of nodes with edges joining them; and “all” is no
exaggeration.Whatiscontinuallyimpressiveaboutgraphtheoryisthewaythatitcango
fromwhatseemssimpleandintuitivetoverysophisticated(and,yes,difficult)results;
advancesinrecentdecadeshavereallychangedthefield,andithasimportantlinks(pun
intended)tomanyotherbranchesofmathematics,suchasalgebra,numbertheoryand
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1.1GraphsasStructure 3
topology. An obvious example is the application of graph theory to understanding the
propertiesandvulnerabilitiesofinformationnetworksliketheInternet.
A second reason for the great value of graph theory for ecologists is the flexibility
oftheapproachformeaningfulapplicationstoarangeofecologicalphenomena.This
isaccomplishedbyincludingdifferentcharacteristicsinthegraphsbeyondthesimple
nodesandedges.Theseincludethefollowing:
(cid:2)
nodelabelsthatidentifythenodeasanindividualandidentifiablecomponentofthe
system,suchasaspeciesname;labelsmakeadifferencewhencountingthenumber
ofdifferentstructures
(cid:2)
nodeweightsthatrecordqualitativeorquantitativecharacteristicsofthecomponents,
suchasrelativeabundance
(cid:2)
nodelocations:thenodesmayhavespatialortemporallocations,suchasthetimeand
placeofasinglepredationevent;temporallocationallowsthepossibilityofnodesthat
comeintoexistenceorceasetoexist
and
(cid:2)
directionsfortheedgessothatAtoBisdistinctfromBtoA
(cid:2)
signsfortheedges,indicatingpositiveornegativeinteractionsbetweenthenodes
(cid:2)
weightsfortheedges,orequationsdescribingfloworfunction
(cid:2)
locationsfortheedges,spatialortemporal,dependentonthelocationsoftheirend-
nodes;temporallocationallowsedgestocomeintoexistenceorceasetoexist
For example, nodes could represent identifiable landscape patches of known locations
inaparticularyear,withtheirareasasweights;theedgescouldbemovementcorridors
withweightsrelatedtohowfrequentlyorhoweasilytheroutescanbeusedfordispersal.
Thisintroductorychapterdescribestheconceptsandterminologythatformthefoun-
dations of a tour through graph theory and the smart ways to use it for understanding
ecological phenomena. This tour illustrates the assertion that these graphs are about
structureandthepatternofrelationshipsthataretheessenceofstructure.Asubtledis-
tinctionhereisthatdespitethefactthat“graph”and“network”havecometobealmost
synonymous,“graphtheory”isstillmoreaboutstructureand“networktheory”ismore
aboutfunctionandflow.
1.1 GraphsasStructure
Thebranchofmathematicsthatweknowasgraphtheoryhasarisenfromanumberof
different sources, developed to solve problems in diverse fields. The most famous of
theseisEuler’ssolutionin1736tothe“Königsbergbridgeproblem,”whichconcerned
walkingroutesaroundtwoislandsinariverwithsevenbridgesoverit.Byconvertingthe
questionintoageneralproblemaboutgraphs,itcouldbeshownthataclosedroutethat
crossedeachbridgeexactlyoncewasimpossible(Euler,ascitedinBiggsetal.1976).
This solution is usually cited as the beginning of graph theory, although Tutte (1998)
has suggested that the discipline might date back to ancient times and the study of
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