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