algebra for temporal network measures · 1 an algebra for temporal network measures lucia falzon...

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ANALGEBRAFORTEMPORALNETWORKMEASURESLucia Falzon

Human & Social Modelling and AnalysisDefence Science & Technology

10 October 2017Networks Course, University of Adelaide

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SOCIALNETWORKANALYSISANDAPPLICATIONS

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SocialNetworkAnalysis(SNA)

•  Asocialnetworkissimplyanetworkconsis6ngofasetofactorsbetweenwhomrela6onal6esaredefined(Wasserman&Faust,1994)

•  Itcanbemathema6callyrepresentedasagraphwhosenodesandlinkscorrespondtoactorsandrela6onal6esrespec6vely

•  SNAu6lisesmathema6calgraphtheoryandsta6s6cstorepresentandanalysethestructural,sta6s6calandposi6onalproper6es

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SomeSNAInsights§  Therela1onalperspec1ve(e.g.Emirbayer,1997)highlightsthe

importanceof:–  embeddednessand–  emergentstructure

§  Showspromiseforunderstanding–  individualand–  systemicleveloutcomes

§  Forusefulsocialapplica6ons–  Recruitment,socialisa6on,norms,influence,power,socialcapital,

knowledge,opportunity,resilience,efficiency,effec6venessetc…

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Sometypicalanalysisques6ons*

–  Whoiscentralandperipheralinthenetwork?–  Whatarethedifferentroleswithinthenetwork?–  Whatsubgroupsexistinthenetwork?–  Whataretheimportantcommunica6onspathsandmethodsof

communica6ng?–  Whichindividualscouldbeinfluencedtobesteffectthenetwork?–  Whatistheoverallstructureofthenetwork?

* AdaptedfromMcAndrew,D.TheStructuralAnalysisofCriminalNetworksinCanter,D.andAllison,L.J.(Eds),“TheSocialPsychologyofCrime:Groups,TeamsandNetworks”,DartmouthPublishingCompany,UK,2000.

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…andsometypicalmeasures

§  Individualmeasuresinclude–  degreecentrality-thenumberofdirect

connec6onsanactorhastotheothernodesinthenetwork;

–  betweenness–theabilityofanactortoactasabrokerorconduit;and

–  closeness–theaveragenumberoflinkssepara6nganactorfromallothernodesinthenetwork;

–  eigenvectorcentrality–thedegreeofconnec6vitytohighlyconnectedactors

Thesemeasureshelpdeterminethekeypeopleinthenetwork.

§  Networkmeasuresmightdescribetheclusteringnatureofthenetwork;theredundancyofitsconnec6ons;anditscohesiveness.Theycantellusaboutanetwork’svulnerabilitytodisrup6ons.

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GroupsandFac6ons

•  Dense,complexnetworkwithhighlyoverlappingcliquestructure;

•  169cliquescomposedof185dis1nctnodes:

–  50ofsize>=4;–  13ofsize>=5;–  5ofsize6;

•  Uselessrestric1vek-corestodeterminecohesivegroups;

•  Useop1misa1ontechniquetoiden1fyfac1ons

–  Bytrialanderrorchosetopar11onintosixfac1ons;

Fromthis…

…tothis

Anetwork’svisualrepresenta1onmightbedifficulttointerpretbutformalanaly1caltechniquescanhelpusmakesenseofthenetworkstructure.

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Rela6onalstructures:Sta6cvs.Dynamic

§  Sta6cnetworksdescriberela6onshipsthatareuniversallypresentorabsent

–  e.g.,parenthood,marriage,friendship

§  Dynamicnetworksdescribeinterac6onsthatoccuroversetsofintervalsor6me-stamps

–  e.g.,short-termpartnerships,transac6ons,informa6onexchange

Time-stampeddataprovideopportuni6estore-constructtheunfoldingofcoordina6onprocessesover6meandtomodelexplicitlythedura6onandpath-lengthsofspecificsequencesofexchanges.

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Developingmeasuresfornetworksofsocialinterac6ons

Socialnetworktheorypositstwoimportantaxiomsonwhichmostnetworkmeasuresarebased(Robins,2015):

–  networkstructureaffectscollec6veoutcomes–  loca6onswithinnetworksaffectactoroutcomes.

Aretheseaxiomss6llapplicabletonetworksofinterac6ons?–  Andpar6cularlytodigitalinterac6ons?

Robins, G. (2015). Doing social network research: Network-based research design for social scientists. Sage Publications, London, UK

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Thetemporalanddynamicnatureofinterac6ons

Someobserva6onsondynamicnetworks:§  Wecannotassumethatallnetworkedges

areconcurrent§  Digital6me-stampsofferfine-grained

informa6onon:–  Thesequen6alorderofinterac6ons–  Dura6on,6mingandtemporal

pagerns§  Takingnetworksnapshotsover6me

obliterateswhathappensinbetweendiscreteobserva6onpoints

Theobjectofanalysischangesfromstructuresofsta1crela1onstodigitaltracesandflowpaXerns–fromfixedstructuretodynamicprocess

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Networkmeasuresbasedondynamicprocesses

Reflect an individual’s importance or ranking based on the structures associated with processes in the interaction network •  Time-ordered path-based measures e.g.

•  Temporal reachability as a measure of the direct and indirect connections in the networks

•  Temporal closeness - shortest distances to all other nodes in the network

•  Temporal betweenness - the proportion of shortest paths that go through a particular node

Paths reflect dynamic trajectories and therefore consider temporal sequence of interactions => ordered in time

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§  Interac6oneventsareboundby6me–theyhaveastartandendpoint§  Two6me-orderedinterac6onsmaybeconsideredasequenceifthe6me

elapsedbetweenthefirstandsecondinterac6onis<=δ(decayvariable)§  Sequencesofeventsdeterminepoten6altrajectoriesthroughwhich

informa6on,resourcesorvirusescanbetransferredfromactortoactor(Moody,2002)

A1me-orderedpathisdefinedasaatemporallyorderedsequenceofinterac1onsthatconnectoneactortoanother.Itinducesatemporaldirec1on.

Time-orderedpaths

Moody, J. (2002). The importance of relationship timing for diffusion. Social Forces, 81(1), 25– 56.

A C

ED

B3

2

6

1

2

8 7

4

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Exactlyone1me-orderedpathconnectsAtoEthroughB,CandDwhenδ <= 3

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Definingtemporal“shortest”paths

§  Geodesic:fewestnumberofhopsorshortestpathdura6on?

§  Respec6ngsequen6alordering:calcula6ngtheshortestpathisnotrecursive

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UNCLASSIFIED

§  Considerssequenceintheconstruc6onofwalksthroughthenetwork

§  Forexample,Moody’s(2002)networkrepresenta6onofdiseasetransmission§  Rulesforpoten6altransmission§  Previouscontacthasimplica6onsforallfutureinterac6ons

§  Includesconsidera6onofmaximumlapsed6me(δ)betweeninterac6onsinasequence

Animportantaspectoftemporality:Sequenceofevents(Kontoleon,Falzon&Panson,2013)

Temporal direcSonaffectsreachability

Transi6vetriadsintradi6onalSNAOnly1triad

Butifweconsider6meorder...

3possibletriads

Kontoleon, N., Falzon, L., and Pattison, P. (2013). Algebraic structures for dynamic networks. Journal of Mathematical Psychology, 57(6), 310 – 319. Moody, J. (2002). The importance of relationship timing for diffusion. Social Forces, 81(1), 25– 56.

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ANALGEBRAICFRAMEWORKFORNETWORKSOFTIME-ORDEREDINTERACTIONS*

*Kontoleon, N., Falzon, L., and Pattison, P. (2013). Algebraic structures for dynamic networks. Journal of Mathematical Psychology, 57(6)

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TheBooleanalgebraforsta6cnetworks*Considerasetofnetworktypes(e.g.friendship,collabora6onetc.)onafixedsetofnodeseachrepresentedbya(Boolean)adjacencymatrix

Thesetofbinaryrela6onsonthesetN,withtheopera6onsunionandrela6onalcomposi6onisapar6allyorderedsemiring*P.E.Panson.Algebraicmodelsforsocialnetworks.CambridgeUniversityPress,NewYork,1993.

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Walk-andpath-basedmeasuresinsta6cnetworks

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Temporalintervalsforsocialinterac6ons

Usefulrela6onalrepresenta6onsinthedynamiccontextareonesthatenableustotraceflowsofmaterialandabstractresourcesthroughthenetwork.

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Rela6onalcomposi6onforapar6cularapplica6on

Thisyieldsthelargest6meintervaloverwhichapoten6altransmissionfromimightreachjbytravellingalonga6me-orderedpathorwalkinthenetwork.

*J.Moody,Theimportanceofrela6onship6mingfordiffusion,SocialForces,81(2002)25-56

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Time-orderedpathconstruc6on:matrixmul6plica6on

Let A = (aij) be an n xn “relational interval matrix” of directed graph G(A).

Each i,j entry in A is the set of time intervals during which i is related to or interacting with j.

Each element in aij is an interval, (u,v) ∈ [0,ω]. For aij = {(t1,t2 ) ,… ,(tr-1,tr ) } and ajm = {(u1,u2 ) ,… ,(us-1,us ) },

aij*ajm = {(max(t1,u1 ) , u2 ) ⊕ (max(t2,u1 ) ,u2 ) ⊕ … ⊕ max(tr-1,us-1 ) ,us ) } defines matrix multiplication A*A. The product of this “multiplication” operation is a set of disjoint

intervals.

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Exploi6ngthealgebraicstructure:Dioids(Gondran&Minoux,2008)

A dioid is defined as a canonically ordered semiring (E, ⊕, *) for which the monoid (E, ⊕) is ordered by the (canonical) order relation defined as:

a ≤b ⇔ ∃ c ∈ E such that b=c ⊕ a. So if E is the set of relational intervals endowed with ⊕ and the partial

order ≤ defined previously we can show that ∀ a,b ∈ E, the order relation above holds.

Moreover, ∀ a,b,c ∈ E, a ≤ b ⇒(a ⊕c ) ≤ (b ⊕c ) also holds and therefore

(E, ⊕) is an ordered monoid.

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Proper6esofdioids(Gondran&Minoux,2008)

A(let-)dioid(E,⊕,*)hasthefollowingproper6es:1.  (E,⊕)isacommuta6vemonoidwithiden6tyelementε;2.  (E,*)isamonoidwith(let-)iden6tyelemente;3.  Thecanonicalpre-orderrela6onrela6veto⊕sa6sfies

a≤bandb≤a⇒a=b;4.  ∀a∈E,a*e=e*a=a(foraletdioid,weonlyrequiree*a=a);5.  Theopera6on*isrightandletdistribu6ve(onlyletdistribu6verequiredfor

aletdioid).

Wecanshowthatthesetofintervalsetswiththeunionandcomposi6onopera6onsasdefinedpreviouslyisaletdioidwithεdefinedtobetheemptyset,{}(orthesetcontaininganullinterval),andebeingtheletiden6tyforintervalsetcomposi6one={(0,0)}.

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Whydioids?(Gondran&Minoux(2008)

§  Theclassicalalgebraicstructures(groups,rings,fields)arenotequippedwithappropriateopera6onsfortheconstruc6onof6me-orderednetworkpaths.

§  Therequiredopera6ons(e.g.min(a,b)ormax(x,y))donotalwayshaveinverses,precludingtheuseofsophis6catedstructures.Howeverthecanonicalorderingguaranteesalevelofuniqueness.

§  Semiringanddioidstructureslendthemselvesquitenaturallytoconnec6vityandop6malpathproblemsingraphs.

§  Theflexibilityofdioidsandsemiringsprovidesausefulsetofmachineryfordevelopingacomprehensivesetofprac6calalgorithms.

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Moregeneralcomposi6onsfor6me-orderedwalks

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§  δactstoconstrainopportunityoftransmissionbylimi6ngtheoffsetoftheintervalatwhichatransfermayoccur

§  Bysenngδ=ω(max6meinobserva6onperiod),indica6ngslowdecay,wegetMoody’scomposi6onruleasapar6cularcase

§  Speedytransmissions(e.g.hotgossip)aremodelledbysenngδasaverysmallvalue

§  δcouldbeapropertyofthetypeofrela6onaswellaswhatisbeingtransmiged

§  Differebttypesofflowhavedifferentcharacteris6cs*

δ-composi6on-ageneralisa6onofMoody'sexample

*S.Borgan,Centralityandnetworkflow,SocialNetworks,27S.(2005)55-71.

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Analgebraofendomorphisms

*M.Gondran,M.Minoux,Graphs,DioidsandSemirings:NewModelsandAlgorithms,Opera6onsResearch/ComputerScienceInterfaces,Springer,NewYork,2008.

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Nodereachability

Nodejisreachablefromnodeiifthereisasequenceofnodes i=j0, j1, j2, .., jr=jsuchthat,aj0j1*aj1j2*…*ajr-1jr isanon-null interval. Inthis

casewesaythereisapathoflengthrbetweeniandj.LetPijkdenotethesetofpathsoflengthkfromitojandletPij(k)bethesetof

pathsbetweeniandjwithatmostkarcs.WedefineA(k)tobeannxn-matrixinwhicheachentryaij(k)isthesetof

rela6onintervalscorrespondingtoPij(k), i.e.A(k)=I⊕A⊕A2⊕…⊕Ak.

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Pathconstruc6on:definingA*

A*denotesthelimitoftheseries:A(k)=I⊕A⊕A2⊕…⊕Ak.,i.e.ifA(n+1)=A(n),thenA*=A(n)

andIdenotesthe“iden6ty”matrixwithdiagonalentriese={(0,0)}andallotherentriesε,

A*iscalledthequasi-inverseofmatrixA.Its(unique)existencecanbeprovedby

usingtheproper6esofthedioid(E,⊕,*),whereEisthesetofallintervalsetsin[0,ω].

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

eeε

...εe

e

I

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If[Αi,j]k=Φ,thenthereisnok-stepwalkfromitoj» Otherwise,If[Αi,j]kisanon-null6meinterval,thereis

ThereachabilitymatrixA*obtainedusingthisgeneralisedalgebrahasentriesinE, whichcanbeconvertedto0’sand1’stodeterminealldirectandindirectconnec6ons.

Basicruletodeterminereachability

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AMEASUREOFBROKERAGEPROCESSFORNETWORKSOFINTERACTIONS

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Reflectanindividual’simportanceorrankingbasedontheirstructuralposi1onintherela1onshipnetwork§  Degreeoreigenvectorcentrality§  Path-basedmeasurese.g.

–  Reachability:asameasureofthedirectandindirectconnec6onsinthenetworks

–  Closeness:measuringshortestdistancestoallothernodesinthenetwork

–  Betweenness:measuredbycompu6ngthepropor6onofshortestpathsthatgothroughapar6cularnode

Networkmeasuresbasedonposi6on

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§  Trajectoriesofpoten6altransmissionfrom/toego–  Reachability

•  Thepropor6onofactorsthatarereachablefrom/canreachegovia6me-orderedpaths

§  Whereegoliesonthetrajectory–  Betweenness–poten6alforcontrollingtransmission–  Brokerage–bridgingorexploi6ngstructuralholes

Interac6onnetworkmeasures:processandposi6onconsideredsimultaneously

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§  Dataone-mailexchangesfromamedium-sizedorganisa6on–  Emailregularlyusedtoexchangedocumentsandcoordinateac6vity–  Eightmonthsofe-mailcommunica6onsbetween129employeeswho

exchanged591,000e-mails–  Fine-grainedinforma6onabout6mingofinforma6onexchange–  Messagesde-iden6fiedbeforeanalysis–  Wekeptonlyinternale-mails,senttoonerecipient(robustness2,3and

4recipients)andremovedthosesenttoorbyindividualswhowerenotac6veemployeesduringthestudyperiod

–  Duplicatemessagesremoved–  Finaldatasetcontains75,584e-mailexchanges–  The6medecayvariableδwassetat24hoursforallanalysisshownhere

Empiricalexample

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§  Temporalreachability:thenumberofactorsthatareconnectedbyatleastone6me-orderedpaththatoriginatesatego

§  Considera6onoftemporalsequenceinducesatemporaldirec6onofflow

§  Twonodesareunilaterallyconnectedifthereisatleastonetemporalpaththatconnectstheminonedirec6on(Nicosiaetal.,2013)

Therankingorderchangeswhenconsideringsta1candtemporalreachability

Temporalreachability

Nicosia, V., Tang, J., Mascolo, C., Musolesi, M., Russo, G., et al. 2013. Graph metrics for temporal networks. In Holme, P., & Saramäki (Eds.) Temporal Networks: 15–40. Springer..

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Ignoring6me-orderingoveres6matesreachability

25

50

75

100

125

12412948 42 8910752 75 56 43 6211564 3811320 68 88 16 2 10617 25 10 9610111811412510214 15 55 40127

Num

bero

fReachab

leActors

ActorID

In(mean)Out(mean)Sta6cin/out

0100020003000400050006000

1 2 3 4 5 6+ NotConnected

PairsofN

odes

NumberofSteps

Temporal Sta6c

•  Induced temporal direction – 2 kinds of reachability •  Static connectivity requires <= 3 steps •  Temporal connectivity requires <= 6 steps •  Ignoring time order results in complete reachability

δ= 24 hours

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§  Considera6onoftemporalsequenceresultsinmoreaccuratemeasures–  Sta6cbetweenness:theexistenceofapathissufficientforthe

calcula6onoftheshortestpath–  Temporalbetweenness:theshortestpathsneedtofollowa6me

orderedpath

§  Therankingorderchangeswhenconsideringsta6candtemporalbetweenness

Temporalbetweenness

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Betweennessranking:Sta6cvs.Temporal StaSc Temporalrank Node Betweenness Node Betweenness

1 75 0.046 75 0.067

2 80 0.022 22 0.043

3 62 0.022 81 0.035

4 82 0.019 110 0.033

5 110 0.018 52 0.028

6 22 0.018 46 0.028

7 9 0.017 89 0.026

8 27 0.016 62 0.025

9 81 0.016 9 0.022

10 46 0.015 82 0.022

11 39 0.013 60 0.021

12 52 0.013 26 0.019

13 13 0.012 39 0.018

14 94 0.012 42 0.018

15 11 0.011 38 0.016

δ= 24 hours

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Informa6onBrokerage •  SNArepresenta6oninvolvesconsidera6onoftheposi6onsoftripletsofactors

•  Brokerageopportunityisenabledbyanactor'sstructuralposi1oninanetworkofrela1onships

Quintane, E. and Carnabuci, G. (2016). How do brokers broker? Tertius Gaudens, Tertius Iungens, and the temporality of structural holes. Organization Science. 27, 1343-1360.

•  Ter1usGaudens:Brokeroffersthesolemeansofinforma6ontransferbetweentwoalters(open2-path)

•  Ter1usIungens:Brokerenablesa

connec6onbetweentwoalterstofacilitatedirectinforma6ontransfer(closed2-path)

Y

B

X

Y

B

X

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Dynamicnetworkrepresenta6on:aprocessviewofbrokerage

Y

B

X

Y

B

X

•  Brokerageopportunityininterac6onnetworksunfoldingover1me–  thebrokercontrolstheflow:bridgesmany

open(6me-ordered)2-pathscentredatB•  X->Bat6met1followedbyB->Yat6met2

–  thebrokerfacilitatesnewconnec6ons:2-pathscloseover1me

•  X->Bat6met1followedbyB->Yat6met2followedbyX->YorY->Xat6met3

Thesequenceofinterac6onsdescribesthebrokerageprocess

Spiro, E. S., Acton, R. M., and Butts, C. T. (2013). Extended structures of mediation: Re-examining brokerage in dynamic networks. Social Networks, 35(1):130 – 143.

t3

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BrokeragemeasuresbasedonprocessWe define temporal constraint as the proportion of structural holes (temporally ordered two-paths) around ego that close out of all the structural holes in ego’s network.

δ= 24 hours

Temporal Constraint = closed 2-paths/(open + closed 2-paths)

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TemporalversusStaScConstraint

δ= 24 hours

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§  Thefluidnatureofsequencesofinterac6onsindicateprocessesplayingoutin6me

§  Consideringthetemporalnatureofinterac6onsallowsustodis6nguishbetweennetworkstructureandstructureresul6ngfromrecurrentpagernsofflow–  Temporalmeasurestellusaboutanactor’sroleinthebrokerage

process–  Sta6cmeasurestellusaboutbrokerageposi6ons

§  Thedecayvariableδisanimportantconsidera6on–itsvaluedependsonthedomainbeinginves6gated

Observa6ons

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§  Experimentwithdifferentvaluesofδ–  Processesatdifferent6me-scales(e.g.dailyvsweeklyrepor6ngcycles)

–  δisassociatedwithprocessdura6on–  Specula6on:Asδincreasestemporalconstraintshouldapproachsta6cconstraint

§  Experimentwithcalcula6ngmeasuresatregular6mepointswithintheobserva6onperiod,e.g.weekly,daily,monthly– Measuresover6meindicateac6vitylevels

Futureconsidera6ons