CiteSeerX — Recommender Systems Based on Navigation Path Features

Based on Navigation Path Features

Wolfgang Gaul, Lars S hmidt-Thieme

{Wolfgang.Gaul,Lars.S hmidt-Thieme}wiwi.uni-karlsruhe.de

Institut für Ents heidungstheorie und Unternehmensfors hung

Universityof Karlsruhe, Germany

Abstra t:

Severalkindsoffeaturesofusernavigationpaths

(e.g., subsets of the resour es as no des of the

paths overed, subsequen es of the sequen es

used for path des ription, path fragments on-

stru ted via ombination of subsequen es and

wild ards) an b e employed to build re om-

mendersystemsdesignedfortasksasdierentas

sitep ersonalization, ross-/up-selling,andnavi-

gationassistan e. Avo abularytodes rib e dif-

ferentkindsofre ommendersystemsandgeneri

qualitymeasures for systemevaluationare for-

mulated. The onstru tion of sp e i re om-

mendersystems,esp e iallysystemsbasedonfre-

quentpathfeatures,isexplained. Inadditionto

the attemptto provide aformalframeworkfor

navigationpathbasedre ommendersystemsre-

sults on the p erforman e of dierent typ es of

su hs systemsarerep orted.

1 Introdu tion

A re ommender system is software that ol-

le tsandaggregates informationab outsitevis-

itors (e.g., buying histories, pro du ts of inter-

est,hints on erningdesired/desirablesear hdi-

mensions or other FAQ) and theira tual navi-

gational and buying b ehavior and returns re -

ommendations(e.g., based on ustomer demo-

graphi sand/orpastb ehaviorofthea tualvisi-

torand/oruserpatternsoftopsellerswithelds

of interest similar to those of the a tual on-

ta t). These re ommendations have to b e re-

ated in su h a way that they are valuable for

browsers/ ustomers/visitors as well as for site

owners. Nowadays, re ommender systems are

installedinmoreand more ommer ialsites to

assist onsumers inb etter/faster a essing use-

fulinformationbutalso site owners in onvert-

ingbrowserstobuyers,instimulating ross-and

up-sales, and in establishing ustomer loyalty

as part of the a tivities to improve ele troni

ustomer are.  Re ommender systems have

b een studied extensively sin e Resni k et al.

(1994)whousedthelab el ollaborativeltering.

Anoverviewab outappli ationsofre ommender

systemsine- ommer e an b efound inS hafer

etal. (1999,2000)andananalysisofsomere -

ommendationalgorithmsinBreese etal. (1998)

andSarwar etal. (2000).

Beside su h known approa hes to designing

re ommendersystems (based onbuyingb ehav-

ior) a new typ e of re ommender system has

emerged that aims at helping surfers in navi-

gatingtheweb. At ea h step inthe navigation

pro ess re ommendations based on the up-to-

nowknownnavigationhistoryaregiven on ern-

ingpages to visit next. Re ommendersystems

based on navigation paths thereby add to the

stati hyp erdo umentlinkingbyexpandingitto

dynami allylinkedhyp erdo uments.

Re ommender systems based on navigation

pathsareusefuline- ommer e ontextsasthey

tryto make buyingmorepleasantfor p otential

ustomers. Asmajorpartsofane- ommer esite

an onsistofpro du t atalogsandthepresenta-

tionofindividualpro du ts,linkingb etweensu h

information analsob ep erformedbytraditional

re ommendersystems. Therealstrengthofre -

ommender systems based on navigation paths

b e omes learerinmoreorlessunstru tured ol-

le tions of information, as found in, e.g., news

groups,messageb oards,web dire tories,sear h

mationwithinthose olle tions anb egathered

andjoined.

Firstro otsofre ommendersystemsforpaths

an b e found in an adaptive hyp ertext system

ofStottsandFuruta(1991)thatrequiresasp e-

ial do ument reader whi h an b e advised to

mo dify (attributes of) links already o ded in

the do uments with resp e t to usageb ehavior.

The idea of developing re ommender systems

basedonstandard HTTP-serversand theinfor-

mationintheirloglesdatesba k toYanet al.

(1996),whereaverysimple lusteringalgorithm

is used and the onstru tion of re ommender

systems is des rib ed as dynami hypertext link-

ing. Perkowitz and Etzioni (1998) build re -

ommendersystems based on o-o urren e fre-

quen iesb etween resour es and onne ted om-

p onentsof theusagegraphand allthemadap-

tivewebsites. Mobasher(2001)hasinvestigated

three dierent approa hes to ompute re om-

mendations by using sessions des rib ed as sets

ofpagesvisitedtogether(in ludingvisittimes).

His rst approa h is based on asso iationrules

for sets, for a se ond alternative session lus-

ters ( omputedvia thek-means algorithm)are

needed,thethirdapproa husesresour e lusters

( omputedbymeansofARHP(asso iationrule

hyp ergraphpartitioning)).Fromthep ointof

viewof webserver design,the problemof om-

putingre ommendationsresemblestheprefet h-

ing problem (see e.g., Bestavros (1996)), i.e.,

the predi tion of the pages requested next by

the a tive users to sp eed up server op eration

andtherebylowerwaitingtimes;theprefet hing

problemiseasierintheresp e t thatpages lose

tothere ommendationp ointarep erfe t hoi es

for predi ted requests (as they are exp e ted to

rest a shorter time in the a he).  Bo dner

andChignell(1999)ta kletheproblemfromthe

p oint ofviewoftext retrieval: they exploitthe

referen e texts of visited links and keep tra k

ofalistof relevantkey words that isfed into a

sear hengine;theresultsofthesear harelinked

fromthekeywordsfoundinthea tivedo ument.

Joa hims et al. (1995) and Lieb erman (1995)

presentWebWat herandLetizia,twoagentsfor

web browsingthatare apable ofgivingre om-

mendations dep ending on the users' sear hing

b ehaviorsofar;WebWat herusessimilaritiesin

Letizia gathers additional data ab out user b e-

havior(asb o okmarkingandusageofdo uments

ondierentserversnotavailableonserver-side).

Fuetal. (2000)prop oseanotheragentthat ol-

le ts usage information of dierent users in a

entral rep ository and omputes re ommenda-

tions from sets of pages visited frequently to-

getherbymeansofasso iation rules.

2 Prerequisites for re om-

mender system evaluation

LetR b e an arbitrary set (the set of resour es

ofawebsitethat orresp ondtotheno desofthe

linkgraphofthissite). ThesetR



:=

S

n2N R

n

ofalltuplesofRis alledthesetofsequen esof

Randserves asbasisto mo deluserpaths.

A sequen e p = (p

1

;:::;p

jpj ) 2 R



des rib es

apathas sequen e of resour es of R,itslength

is denoted as jpj. Sometimes, we repla e the

tuple-notationbyjustputtingthe orresp onding

resour es oneaftertheother,i.e.,p

1 p

2

p

jpj .

Fromamathemati alp ointof view,are om-

mender system based on navigation paths is a

map

r:R



!P(R) (1)

(whereP(R)denotesthep owersetofR)andthe

setr (p)is alledre ommendationsetforp2R



.

Starting p oint for an evaluation of re om-

mender systems is a (multi)set of paths S.

Ea h path p 2 S an b e split at p osition i 2

f1;:::;jpj 1g in a history h

i

(p) := (p

1

;:::;p

i )

andafuturef

i

(p):=(p

i+1

;:::;p

jpj ). p

i

is alled

re ommendation point.

Now, a general denitionfor are ommenda-

tionquality measure an b egivenby

q:R

1

R

2

R

3

! R

+

0

(h;f;r ) 7! q (h;f;r )

(2)

whereR

1

des rib esthehistoryspa e,R

2 thefu-

turespa e,andR

3

thespa eof(setsof)re om-

mendationsr (h) derived fromh2R

1

. Various

hoi esof R

1

;R

2

; andR

3

arep ossible. We will

restri t to R

1

= R

2

= R



and R

3

= P(R), in

thefollowing.

q (h;f;r ) measures the quality of re ommen-

dations (e.g., by ho osing h = h

i

(p) and om-

paring r = r (h

i

(p)) with f = f

i

(p) for a path

p).

Simple examples of re ommendation quality

measuresare

q (h;f;r ):=jfy2r jy o ursinfgj (3)

whi h is just the numb er of re ommended re-

sour es thatalsoo urinf, or

q (h;f;r ):=

X

y 2r (h)

 q (h

jhj

;f;y ) (4)

whereq:RR



R!R +

0

des rib esamea-

sure that dep ends onlyonthere ommendation

p oint h

jhj

and evaluates the degree of onfor-

mity b etween f and a single re ommendation

y2r (h),i.e.,thequalitymeasuredo es nottake

into onsiderationany omp oundee tsas,e.g.,

preferen e ofresour es on entratedinaparti -

ular region of the site over those s attered all

overthewholesite.

Re ommendationquality antakeinto onsid-

eration the distan e b etween history resour es

andre ommendedresour es (measuredwiththe

help of the underlying site graph stru ture or,

alternatively, dened as minimalnumb erof re-

sour esb etweenre ommendationp ointandre -

ommended resour e in the a tual future of a

path), e.g.,forx;y 2Randf 2R



(e.g.,with

x =p

i

;y 2r (h

i

(p)), and f =f

i

(p) for a path

p2R



)one an dene



q (x;f;y ):=

8

<

:

u(dist(x;y )) ;ify6=x

o ursinf

0 ;otherwise

(5)

wheredistdenotesanappropriatedistan efun -

tionand umeasures theutility assignedto the

distan eb etween pairsofresour es. Themean-

ingisthat resour es inthedire tneighb orho o d

of a re ommendation p oint are easier to nd

(and, thus, to re ommend) than adequate re-

sour esfar away. Examplesforutilityfun tions

are

u:R +

0

!R +

0

(6) d7!

>

>

<

>

>

:

1 hit ount

d linears ale

logd+1 log. s ale

(d d

0 +1)Æ

[do;d1℄

(d) windowee t

with Æ

[do;d1℄

(d):=



1; d2[d

0

;d

1

℄;

0; otherwise

Uptonow,re ommendationqualitymeasures

asdepi ted in(3), (4) arerestri ted toasingle

navigationpath but,of ourse, for agiven re -

ommendersystemr ,are ommendationquality

measure q , and an underlying (multi)set S of

navigationpaths,one andene,e.g.,

Q raw

r

(S):=

X

p2S jpj 1

X

i=1 q (h

i (p);f

i (p);r (h

i (p)))

asrawre ommendations orefor rrelativetoS.

Let

Q raw

max

(S):=max

r Q

raw

r (S)

b ethe(theoreti ally)maximal re ommendation

s ore (relative to a given quality measure q );

see se tion 3 for a simple metho d to ompute

Q raw

max

(S)for a giventest set S. Then, one an

dene

Q

r

(S):=Q raw

r

(S)=Q raw

max (S)

asnormalizedre ommendations ore,whi hisa

useful hara teristi numb erforthe omparison

ofthep erforman eofare ommendersystemon

dierent test sets or of dierent re ommender

systemsonthesame(multi)setS.

Now, theproblem to ndan optimalre om-

mender system an b e formalized as follows:

given a quality measure q onstru t a re om-

mender system r on the basis of information

froma training set S train

of paths so that the

rawre ommendations oreofronatestsetS test

ofpaths(notusedforbuildingthere ommenda-

tionsystem) ismaximal.

For thesimplere ommendationqualitymea-

sure (3)that just ounts thenumb erof onfor-

mitiesb etweenresour es ofrandf,apparently,

theoptimalre ommender systemis thesystem

that simply re ommends all resour es for any

givenhistory: forsure,thisre ommendationset

willhit allresour es inthefuture and b eof no

interestwhatso ever. Twokindsofmo di ations

arep ossibletomaketheproblemmoreinterest-

ing:

sure. Forinstan e,onemaythinkof ount-

ingthenumb erofhitsrelative tothenum-

b erofgivenre ommendations. Whileopti-

malre ommendationsforthesimplequality

measure (3) onsist of large re ommenda-

tionsets, optimalre ommendationsforthe

relativenumb erofhittingre ommendations

haveverysmallre ommendationsets: inal-

mostall asesforea h historyonlytheone

resour e withhighest follow-upprobability

issele tedandallotherresour eswithlesser

but p erhapsalsohighprobabilitiesaredis-

arded.

2. Restri t the spa e of possible re ommender

systemsbyimposing additional onstraints.

Arestri tionthat alwayseverissensiblein

pra ti e is to allow only re ommendation

setsofagivenmaximalsize(i.e.,jr (h)jn

for all h 2 R



and a given n 2 N). This

onstraintfor esarestri tiontotheb estn

re ommendations; in pra ti e, n will b e a

smallnumb er,say3upto5,ofre ommen-

dationsthatusersmayb ewillingtolo okat.

Thus,onemaysp e ializetheproblemof

ndinganoptimalre ommendersystemto

the onstru tion of an optimalone among

apredened lassofre ommender systems

(e.g.,thosewithatmostagivennumb erof

re ommendationsp erhistory).

For paths in a (sparsely linked) graph the

omputation of re ommendations with resp e t

toaqualitymeasurebasedonhits(disregarding

distan es of re ommendedresour es) will in

most ases  still result in a set of resour es

dire tly linked to the re ommendation p oint.

Here, we use the idea of Mobasher (2001) to

weightresour es farther aparthigherby ho os-

ing an appropriate quality measure dep ending

on thedistan es ofthe re ommendedresour es

(another,simplerdistan esensitivequalityfun -

tion an b e foundinCo oley et al. (1999)). Of

ourse,utilityfun tions whenused formo d-

elingdierentproblemsmaydep endonother

parameters(as,e.g.,expli itorimpli itratings)

b esidesdistan eas well.

mender systems

As the pre eding dis ussion has shown, avari-

ety of optimality riteria for re ommender sys-

tems anb edesignedonthebasisofappropriate

hoi esofqandoptionalrestri tionsforr .Here,

westartwithsomeobviousp ossibilitiestotyp e-

astre ommendersystems.

As normallythe numb er of olle ted naviga-

tionpathsisverylarge omparedtothenumb er

ofresour esoftheunderlyingsite,wemaybreak

downtheglobalproblemofndingoptimalre -

ommendersystems for a whole site into a set

ofsmallersubproblems of onstru ting optimal

systems for ea h single resour e. We split R



forx2Rinto spa es R



x

:=fp2R



jp

jpj

=xg

that onsistonlyofsequen es withxatthelast

p ositionand all

r

x :R



x

!P(R) (7)

alo alre ommendersystematresour exin on-

trastto theglobalversiondes rib ed by(1). A -

ordingly, the training set S train

for the global

systemistransformedintotrainingsetsS train

x



R



x

R



forthelo alsystemsthat onsistofall

navigation paths psplit at x (as re ommenda-

tion p oint, if p ontains x); inthe ase that a

resour exapp ears ktimesinapathp2S train

,

thenS train

x

ontainsk repli ationsof psplit at

ea ho urren e ofre ommendationp ointx. 

On e that optimallo alsystems for all x 2 R

have b een found, they an b e pie ed together

to aglobalsystem r : R



! P(R) by delegat-

ingthere ommendationtasktotheappropriate

lo al mo del, i.e., r (h) := r

h

jhj

(h), as there is

nodep enden yofthere ommendationsgivenat

onere ommendationp ointup onthose given at

anotherre ommendationp oint.

We further distinguish b etween stati and

dynami re ommender systems: stati re om-

mender systems do not take into a ount the

formernavigationhistoriesofusers andprovide

astati set ofre ommendations forallvisitors,

while dynami re ommender systems may de-

p endonthehistories andprovidedierent re -

ommendationsetsforuserswithdierenthisto-

ries.Dynami systemsmayb ebuildbyrstpar-

titionthehistoriesofthetrainingsetS train

and,

then, omputeastati systemforea h lass.

systems an b e des rib ed as (multi)sets of fu-

tures F

x

R



, extra ted fromS train

x

viaF

x :=

ff 2 R



j9h2 R



x

: (h;f) 2S train

x

g. A simple

re ommender system just ounts frequen ies of

resour es y2Rinthefuturepathsvia

freq (y ):=jff 2F

x

jy o ursinfgj

and re ommendsthen mostfrequent ones. Up

tonow,noutilityfun tionshaveb eentakeninto

onsideration. Todoso,one hastosumupthe

utilityvaluesforallresour esinthefuturepaths,

e.g., for the distan e sensitive utility fun tions

within (6)one omputesthe weightedfrequen-

ies

wfreq(y ):=

X

f2Fx

 q (x;f;y )

with qas given in(5)and, again,sele ts then

highest valued follow-up resour es. Note that

the omputationoftheweightedfrequen iesde-

p ends onthere ommendationp ointx, but the

re ommendationsetitselfdo esnot,thus,astati

re ommendersystemisgenerated. By onstru -

tionthisis theoptimalsystemamongallstati

re ommendersystems atx.

Dynami (lo al)re ommender systems make

use of a history partition C = fC

1

;:::;C

m g of

(a sup erset of)all histories h 2 R



inthe test

set(wherem2Nisthenumb erof lasses). The

test set S test

an b e partitioned into test sets

S test

j

C

:=f(h;f)2S test

jh2Cgforea h lass

C2C andastati re ommendersystem anb e

buildforea h su h lass.

While the use of ordinary partitions is

straightforward,fuzzypartitionsneedadditional

information ab out predi ted utility values for

re ommendations given by the stati re om-

mender systems for ea h lass. Now, let C =

fw

1

;:::;w

m

gb e a fuzzypartitionof the histo-

ries,i.e.,allw2C arefun tions w:R



![0;1℄

with P

w 2C

w (h) = 1 for all h 2 R



. w (h) is

alledweight ofhin lassw . Thestati re om-

mendersystemsforea h lasswhavetoprovide

apredi ted utilityvalue for ea h re ommenda-

tion,i.e.,they aremaps

r

w :R



![0;1℄

R

withre ommendationset

re (h):=f(y ;v )2R[0;1℄jv=r

w

(h)(y )>0g

h a dynami re ommender system using fuzzy

partitions,rst, omputes the weightsof h for

all lasses w , se ond, omputes for all lasses

wwith w (h) > 0the (extended) re ommenda-

tionset r

w

(h),third,adjuststhepredi ted util-

ity values bythe weightw (h) for the lass the

re ommendationsstemfrom,and,then, ho oses

thenre ommendationswithhighest (adjusted)

predi tedutility.

Atrivialexampleforadynami re ommender

systemis theonebuild up onthesingletonpar-

titionC=ffhgj9f 2R



:(h;f)2S train

g,that

we allre ommendersystembasedonnesthis-

torypartition. Asea hdieren eb etweenhisto-

riesresultsindierent lasses,thisre ommender

system extremely suers from overtting and

thereforep erformsveryp o orlyontestsets. But

b eside the fa t that the re ommender system

based onnest history partitionis atrivialex-

ample for a dynami system, it an b e useful

forthe omputationof anupp er b ound forthe

rawre ommendations ore(that anb ea hieved

by any re ommender systemon theunderlying

testset),ifitistrainedbythetestset(!) itself,

i.e.,therawre ommendations oreofthere om-

mendersystembasedonnesthistorypartition

fora test set S test

is the theoreti ally maximal

rawre ommendations oreQ raw

max (S

test

).

The omputationofQ raw

max

requiresthe build-

ing of a huge amount of stati re ommender

systems(one for ea h history), thateither may

onsumea onsiderable large amount of mem-

oryor for esseveral iterationsoverthe test set

database,i.e., annotb edonee iently. Ifrun-

timeisanissue,amorep essimisti upp erb ound

an b e omputed by ho osing the b est re om-

mendationfor ea h very history in thetest set

inasinglelo opoverthetestsetdatabase. Note

thatthismayresultindierentre ommendation

sets for the very same history and, thus, may

never b e a hieved by a real re ommender sys-

tem. But for test setswith manydierent his-

toriesandade ent qualityfun tionthis b ound

is loseenough toQ raw

max

. Asinourexp eriments

inse tion6runtimehasnotb een onsidered,we

useexa t values for Q raw

max

.  Pleasenote that

Q raw

max

isusedto omputenormalizedre ommen-

dations ores. As themainpurp ose ofnormal-

ized s ores is the omparison of re ommender

systemsondierentdatasetsorofdierentlo i

ofalo alizedsystem,itsmainappli ationsarein

resear h oriented ontexts. For tra kingre om-

mendersystemp erforman einop erational on-

textsraws oresmayb e used.

Obviously, b oth,lo alization (i.e.,the deter-

minationoflo alsystems)andtheusageofhis-

torypartitions anb eviewedasappli ationofa

lusteringte hniquetothehistoriesofpathsinS

thatbasedonanadequatesimilarity riterion

fornavigationpathsredu estheglobalprob-

lemtothehandlingofsubproblemsdes rib edby

morehomogeneoussub(multi)setsSj

C

,whereC

denotesthe lassunder onsideration ofthere-

sulting(p ossiblyfuzzyoroverlapping) lassi a-

tion. Of ourse,one an ombinethesep ossibili-

tiesandapplyhistorypartitionstoS

x

,resulting

inS

x j

C

, or split Sj

C

into subsets ofnavigation

paths with same re ommendation p oint x, re-

sulting in(Sj

C )

x

. Note, that whilelo alization

usesveryintuitive lassesthatdonothavetob e

omputed,thehardpartofhistorypartitionsis

the omputationof the partitionitself. There-

fore,onerstapplieslo alizationandafterwards

omputes history partitions for ea h lo al sys-

tem. An overview ofthe ar hite ture of su h a

omplexsystemisgiveningure1.

4 Path features

Paths that users have taken on a site b elong

to the most valuable information that an b e

gained. But paths as sequen es of resour es of

dierentlengths are omplex obje ts whi h are

not that easy to ompare and to use in data

mining algorithms. Thus, one is interested in

determiningsetsofsimplerfeaturesforpathde-

s ription(featureextra tion).

A substru ture spa e of R



is dened as pair

(A;) of a set A and a relation on AR



where a2A is alled substru ture of p2R



if

ap.

Æ

a : R



! f0;1g

p7!



1; ifap

0; otherwise

is alled indi ator fun tion of substru ture a.

Examplesofsubstru tures are:

1.sets(P(R);)ofresour es,whereasetofre-

sour esa2P(R)isdenedtob easubstru ture

ofapathp2R ifallresour es x2a o urin

pathp,

2.sequen es(R



;

t

),whereasequen ea2R



is dened to b e a substru ture of a path p if

it is a ontiguous subsequen e, i.e., it exists

i

0

2 f0;:::;jpj jajg with a

i

= p

i0+i for all

i=1;:::;jaj,

3.generalizedsequen es((R[fg)



;

g en ), i.e.,

sequen es onsisting of elements of R and an

additionalsymb ol used as wild ard, where a

generalized sequen e a 2 (R[fg)



is dened

to b e a substru ture of apath p ifit is a gen-

eralization of a ( ontiguous) subsequen e of p

andgeneralizationmeansthatarbitrarypartsof

thesequen e mayb e repla edbywild ards(see

Gauland S hmidt-Thieme (2000) for an exa t

denition),

4.simplegeneralizedsequen es(R



;

n t

),where

a(simplegeneralized)sequen ea2R



isdened

tob easubstru tureofapathpifitisanon on-

tiguoussubsequen ewiththefollowingmeaning:

It exists j : f1;:::;jajg ! f1;:::;jpjgstri tly

in reasingwith a

i

= p

j(i)

for all i = 1;:::;jaj,

i.e., inthe ontext of generalized sequen es, if

a

1

a

2




a

jaj



g en p.

Thespa eofsimplegeneralizedsequen es an

b eviewedasasubspa eofthespa eofgeneral-

izedsequen es where awild ardisintersp ersed

b etween ea h two resour es. Noti e that in

pra ti alappli ationsonlygeneralizedsequen es

withouta wild ardat therst and/or last p o-

sition (i.e., a 2 (R[fg)



with a

1

;a

jaj 2 R)

areofinterest. These sequen es are alled path

fragments.

For any substru ture spa e A the symb ol ;

des rib estheemptysubstru ture(i.e.,theempty

setortheemptysequen e, resp e tively) andjaj

thesubstru ture omplexityof a2Adened as

ardinality(forsets)orlength(forsequen es).

Now,we andeneapathfeaturetob eapair

(;'),whereisanarbitraryset alledfeature

spa eand':R



!thefeature mapmapping

paths to features. For a path p 2 R



we all

'(p)the'-featureof p.

Trivial examples for path features are its

length(':R



!N;p7!jpj)anditsentryp oint

(':R



!R;p7!p

1

). Moreinterestingfeatures

areobtainableviasubstru tures.

From anarbitrary substru ture spa e (A;)

re ommendations

for lass1

re ommendations

for lass2

.

.

.

re ommendations

for lass...

delegateon

lassweights

weightand

sele tb estn

dynami lo alre ommendersystematA

{ { { { { { { { {

==

f f f f f f f

22

C C C C C C C C C

!!

3 3 3 3 3 3 3

 R R R R

))       

EE

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

                                   

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ooo ooo oo ooo ooo ooo

ww

          

                   



.

.

.

re ommendations

for lass1

re ommendations

for lass2

.

.

.

re ommendations

for lass...

delegateon

lassweights

weightand

sele tb estn

dynami lo alre ommendersystematX

{ { { { { { { { {

==

f f f f f f f

22

C C C C C C C C C

!!

3 3 3 3 3 3 3

 R R R R

))       

EE

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

                                   

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

delegateon

re ommendation

p ointhjhj

re ommendation

setr (h)

dynami globalre ommendersystem

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

                                                                                                 

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

//

historyh

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



HH

forh

jhj

=A

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3



.. . . ..

..

..

..

. . ..

..

.. . . . . ..

.. .



forh

jhj

=X

DD

Figure1: Ar hite ture ofadynami globalre ommendersystem

we derive itsasso iatedpathfeature

': R



! f0;1g A

p7!



Æ: A! f0;1g

a7! Æ

a (p)



i.e.,afeature spa ethat forevery path p

ontainsa binary ve tor indi atingwhether an

elementa2Aisasubstru tureofpornot.

Feature spa es based on substru tures turn

out to have the disadvantage of high dimen-

sionality: the feature spa e build from subsets

hasdimension2 jRj

,theonebuildfromnitese-

quen es(ifsubsequen es arerestri tedtolength

n)hasdimension P

n

i=1 jRj

i

. Therefore given

an underlying (multi)setof navigation pathsS

that has to b e analyzed  one is lo oking for

interesting subsets of substru tures that result

into a smaller numb er of dimensions but still

arries as mu h information as p ossible for a

des ription of the obje ts of (multi)set S (fea-

turesele tion). We alladimensionsparsewith

respe t to S if the orresp onding entry in the

binary ve tor is zero for almost all paths of

(multi)setS. Inappli ations,oneoften androp

avastnumb erofsparsedimensionsandrestri t

tothosedimensionsforwhi h thep er entage of

non-zeroentries inthebinary ve tors ex eedsa

lowerb ound.

Dep endent on this b ound alled minsup fre-

quentsubstru tures ofthepathsofS anb ede-

terminedb eforehand. Forasubstru turea2A

onedenesitsrelative frequen y

sup

S (a):=

jfp2Sjapgj

jSj

assupportofainS. Thetaskto omputeallfre-

quent substru tures, i.e., theset 

(S;minsup) :=

fa 2 Ajsup

S

(a)  minsupg of all substru -

tures with at least a given minimum supp ort

minsup2 R +

0

, is wellknown and a omplished

Srikant(1994)),sequen es(AgrawalandSrikant

(1995)) and generalized sequen es (Gaul and

S hmidt-Thieme(2000)),resp e tively. Building

the feature spa e from the frequent substru -

tures 

(S;minsup)

only instead of using all sub-

stru tures of A an redu e the dimensionality

dramati ally(dep ending on the minimumsup-

p ort and the stru ture of S, of ourse). We

all this feature spa e path features based on

frequent substru tures in general, and, in par-

ti ular, path features based on frequent subsets,

subsequen es, (simple)generalized sequen es or

pathfragments, et .

5 Re ommender systems

based on frequent sub-

stru tures of navigation

histories

Frequent substru tures of thehistories at a re-

sour ex anb eusedtobuildfuzzypartitionsfor

dynami lo alre ommendersystems.Foralo al

trainingsetS train

x

R



x

R



atare ommenda-

tionp ointx2RletH

x

:=fh2R



x

j9f 2R



:

(h;f)2S train

x

gb ethe(multi)setof orresp ond-

inghistories. Let

x :=

(Hx;minsup)

denotethe

setoffrequentsubstru turesofH

x

. Forea hfre-

quentsubstru ture a2

x

we builda lassrep-

resentedbytheweightfun tionw

a :R



x

![0;1℄.

Note, that the empty substru ture ;o urs in

everyhistorybydenitionand,thus,isfrequent

inany ase;itservesas lassforhistorieswithout

any frequent (non-empty) substru tures. For

h2R



x

let(h)denotetheset offrequentsub-

stru tures o urring in h (in luding ;). The

lassweightfun tions w

a

arethendenedas

w

a (h):=

(

(a)

P

b2(h)

(b)

; fora2(h)

0 ; otherwise

where :

x

!R +

0

isafun tionthatmeasures

how sp e i or interesting afrequent substru -

tureis. mightb eset onstantlyto1(notmak-

ing any dieren es b etween dierent substru -

tures and, thus, weighting themall equally),it

might onsider the frequen y of a substru ture

andb esettotheinverseofthesupp ort 1

sup

Hx (a)

,

substru tureandb esettothe omplexityjajof

afrequentsubstru ture. Wewillusethese ond

variantinourexp erimentinse tion6.

Alternatively,one anassignhistoriesonlyto

lassesrepresentingmaximalsubstru tures. Let

 max

(h) b e the set of maximal frequent sub-

stru tures of h (esp e ially f;g, if there are no

frequent substru tures at all)and ompute w

a

with max

(h)instead of(h).

Partitions based on frequent stru tures tend

tob e omeratherlargeforsmallminimumsup-

p ort values. A pruning step an de rease the

numb erofinterestingfrequentsubstru turesb e-

fore the partition is formed and stati re om-

mendersystemsforea h lassarebuild. Forany

frequent substru ture a 2

x

, all moresp e i

substru tures b 2 

x

(e.g., sup ersets, sup erse-

quen es,et .),that have thesamesupp ort, an

b eremoved:asidenti alsupp ortvalueswilllead

tothe same lass weightsand as b oth frequent

substru tures a and b have the same training

sets, i.e., lead to the same stati re ommender

system,theywould reatetheverysamere om-

mendations, and, onsequently, the system for

b is sup eruous. The reader a quainted with

thenotionof losedsubsets inthetheoryof set

based asso iationrules willsee the analogyb e-

tween this pruningstep andthe sear h just for

losedsubsets(see,e.g.,ZakiandHsiao(1999));

butnote,thatwhile losedsubsetsarethemost

sp e i ones amongallsubsets with samesup-

p ort, here  by ontrast  we keep the most

generalonesamongallsubstru tures withsame

supp ort.

6 Examples andexperiments

Afterthetheoreti aloutlinewe presentaseries

of small examples to illustrate the apabilities

and short omings of the dierent kinds of re -

ommender systems based on navigation paths

b efore an exp erimental evaluation is des rib ed

withtheintentiontoprovideafeelingforthead-

vantages obtainable by appli ationof pathfea-

tures.

Figure 2shows thelinkgraph of asmallsite

of seven resour es R = fA,B,C,D,E,F,Gg and

threeexamples ofnavigationpathsthat areall

analyzedatre ommendationp ointC.Thepaths

??

A

OO 

~~ ~~ ~~ ~~ ~~ ~~ __

@ @

@ @

@ @

@ @

@ @

@ @

B

oo //

OO 

C

oo //

OO 

??

D

~~ ~~ ~~ ~~ ~~ ~~ OO 

E

oo //

^F

oo //

^G

a)sitegraph

utility u

1

: 1 2 3 4

u

2

: 1 1.7 2.1 2.4

path history future

p

I

: AC D F G

p

II

: ABC F E B

p

III

: AC F D G D

p

IV

: ABC D G F E

b)pathsofexample1

utility u

1

: 1 2 3 4

u

2

: 1 1.7 2.1 2.4

path history future

p

V

: BAC D F G

p

VI

: ABC F E B

p

VII

: BAC F D G D

p

VIII

: ABC D G F E

)pathsofexample2

utility u

1

: 1 2 3 4

u

2

: 1 1.7 2.1 2.4

path history future

p

IX

: BAC D F G

p

X

: ADFEBC F E B

p

XI

: BAC F D G D

p

XII

: ABC D G F E

d)pathsofexample3

Figure 2: Sitegraphand samplepathsofexamples1,2,and3

dier with resp e t to theirhistories while their

futuresremainun hangedinthedierentexam-

ples. Forsimpli ity,atmostasinglere ommen-

dation(n=1)isprovidedfor omparisons.

We start with example 1 and assume that

the underlying system is stati and that S =

fp

I

;p

II

;p

III

;p

IV

g is the (multi)set of naviga-

tion pathsunder onsideration. Without using

utility fun tions one just ountso urren es of

resour es followingC:Fistheonlyresour eo -

urring in all four futures and, thus, a re om-

mendersystembasedonmerefrequen ieswould

re ommend F. Now, we add a utility distan e

u

1

(d):=dor u

2

(d):=lnd+1,resp e tively; the

utility values for the resour es followingC are

given inthe upp er two lines of the tables. Us-

ingutilitysums,there ommendersystembased

on weighted frequen ies now omputes the u

1 -

utilitysum7forFandthe u

1

-utilitysum8for

G or the u

2

-utility sum 5.8 for F and the u

2 -

utility sum 5.9 for G and  in b oth ases 

would re ommendGinstead of F.Ifaresour e

o ursmorethanon einthefuture path(asD

inpathp

III

)onlythersto urren e (shortest

empiri aldistan e)is ounted.

There ommendersystembasedonnesthis-

tory partitionre overs twohistory lasses fAg

andfABg. For lass fAg the re ommendation

Gwith u

1

-utilitysum 6(or u

2

-utility sum4.2)

is omputed, for lass fABg re ommendation

E,also withu

1

-utilitysum6(or u

2

-utilitysum

4.1),is found, resulting inatheoreti ally max-

imal p ossible re ommendation s ore of 12 for

u

1

(or 8.3 for u

2

). Thus, the normalized re -

ommendations ore ofthere ommendersystem

basedonmerefrequen iesis 7=12=0:58foru

1

(or5:8=8:3= 0:70for u

2

) while for thesystem

basedon weightedfrequen ies it is8=12=0:66

foru

1

-utilitysummation(or 5:9=8:3= 0:71for

u

2

-utility summation). This part of our small

examplewasdesignedtoshowthatthein orp o-

rationofutilitydistan es(otherutilityfun tions

are thinkable) leads to the re ommendation of

resour e Gfarther apart fromre ommendation

p ointCinstead ofresour e Fdire tly linked to

C.

Nowweapplythere ommendersystembased

onfrequentsubsetswithminsup=0.5toexam-

ple1. Thetwofrequentsubsets fAg withsup-

p ort1andfA,Bgwithsupp ort0.5arefoundand

leadtoafuzzy partitionwith3 lasses. Forthe

stati re ommender system based on weighted

frequen ies that takes into a ount allhistories

ontainingfAgallfour pathsareusedand on-

sequently G is omputedas b est re ommenda-

tion(thesameasforhistories ontaining;),for

the samesystem that, now, takes into a ount

allhistories ontainingfA,Bgonlythepathsp

II

andp

IV

areused,sothatEis omputedasb est

re ommendation. Thus, the re ommender sys-

tembasedonfrequentsubsetsa hievesare om-

mendations oreof 1.0.

Next,wetakeexample2whi hisaslightmo d-

i ationof example1: pathsp

V andp

VII now

startwithBA(insteadofAasp

I andp

III did).

Consequently, the re ommender system based

onfrequentsubsetsre oversonlyone lassofhis-

toriesfA,Bg and, thus, willa hieve there om-

mendations oreof0:70only. There ommender

systembased onfrequentsubsequen es still an

distinguish b etween subsequen es AB and BA

ofthe historiesanda hieves are ommendation

s oreof1.0again.

If we, now, lo ok at example 3in whi h only

thehistoryofpathp

X

was hanged omparedto

pathp

VI

ofexample2byinsertingasmalldevia-

tionDFEb etweenAandBCinthehistory,the

extra tion of frequent ontiguous subsequen es

would result in BA only and a distin tion b e-

tweenthetwo lassesfoundinexample2would

not b e p ossible. But the re ommender system

based on simple generalized subsequen es (or

pathfragments)isableto separatethetwofre-

quent simple generalized sequen es A ? B and

BAinthehistoriesofexample3and an,thus,

give b etterre ommendations.

Finally, our ndings have b een he ked on

larger (multi)sets. Table1 shows the result of

an exp erimental evaluationof some of the dif-

ferent re ommender systems based on naviga-

tion paths as des rib ed earlier. We mo deled

fourdierent lasses ofusers navigatingasmall

siteof20resour es(A-T)withseveral rossings.

On the basisof an abstra t des ription of user

lasses (p er entage of total users, templates of

navigation b ehavior, distributions of variations

et .) we reatedadatabaseof10.000navigation

paths. 90%ofthepathswereusedastrainingset

S train

tobuildthemo dels,theremaining10%of

thepathsastestsetS test

toevaluatethequality

resour esABCDEFGHIJKLMNOPQRST weights0.100.150.180.070.060.050.050.070.080.050.050.050.050.050.040.050.050.050.060.05 globallo al stati re ommendersystems: freq0.410.210.310.570.420.360.580.160.670.580.510.140.180.520.570.320.500.150.530.170.50 wfreq0.640.880.690.570.660.730.580.580.670.640.650.670.610.520.600.520.560.550.610.670.54 dynami re ommendersystemsbasedonfrequentsubstru tures: sets0.650.870.710.550.660.730.580.610.670.640.640.690.550.580.690.620.560.640.640.690.56 seq0.690.880.710.580.680.780.730.710.670.740.660.710.630.650.740.590.680.670.620.710.63 sgseq0.760.960.770.630.680.770.800.820.730.780.760.760.730.680.820.700.730.790.790.830.73 frag0.770.960.770.660.670.780.800.830.730.790.800.770.750.690.810.730.740.800.790.860.73 Table1:Experimentalevaluationofdierentre ommendersystemsonasmallsitewith20resour es(AT).Therowweightsdes ribes theper entageofo urren esofea hresour einthepathdata.The olumnglobalgivestheglobalre ommendationqualityforea h re ommendersystem.Furthermore,forthelo alpartsofea hre ommendersystemwithresour eAuptoresour eTasre ommendation pointthequalityisgiven.Thefollowingre ommendersystems(r.s.)areevaluated:r.s.basedonfrequen iesofresour esfollowingthe (a tual)re ommendationpoint(freq),r.s.basedonweightedfrequen ies(wfreq),r.s.basedonfrequentsubsets(sets),r.s.basedonfrequen subsequen es(seq),r.s.basedonfrequentsimplegeneralizedsubsequen es(sgseq),andr.s.basedonfrequentpathfragments(frag);all re ommendersystemsbasedonfrequentsubstru turesusedaminimumsupportof0.2.

ofthemo delsand omputethere ommendation

s ores. We used theutilityfun tionu

1

and al-

lowedonlyasinglere ommendationatea h re-

sour e (n =1). As exp e ted, the useof simple

frequen ies(mo delfreq)resultedinalowglobal

re ommendations oreof0.41,forthelo alver-

sionsthequalitydropp ed b elow0.15atsomeof

theresour es. Usingthisasbaseline,themo del

basedonweightedfrequen ies(wfreq)thattakes

into onsideration the sp e ial formof the util-

ityfun tiona hieves anoverallimprovementin

global quality of over 50% (global re ommen-

dations ore 0.64). Aswe put strongsequential

ee tsinthenavigationpatternsofthedierent

user lasses, dynami re ommendations based

onfrequentsets (sets)donotresult inab etter

globals ore(0.65),butusingsequen es(seq)or

evenb ettersimplegeneralized sequen es(sgseq)

orpathfragments(frag)furtherimprovementsof

the globalre ommendation qualitywere p ossi-

ble(byanother32%withresp e t tothebaseline

s ore (see the globalre ommendations ores of

0.69,0.76,and0.77,resp e tively)). Inall ases,

thefrequentsubstru turesforthedynami mo d-

elshaveb eenextra tedwithaminimumsupp ort

of 0.2 (the smallest exp e ted user segment size

forthedataset).

The lo alqualitys ores showthat notin all

ases (i.e., at all re ommendation p oints) the

samerankingas fortheglobalre ommendation

qualityvalues an b e observed. This is due to

thefa tthatatsomere ommendationp ointsse-

quential ee ts were not strong enough in the

data so that lo al re ommender systems based

onpathsubstru tures ouldnottake advantage

ofsomeofthepathfeatures, and/ordueto the

hoi e of thesameminimumsupp ort for alllo-

alre ommendersystems,thatisresp onsiblefor

smalloverttingee tsinsomeofthelo alsys-

tems.

Of ourse,ndingsdep endonthestru ture of

theusersegments. Ifonlyfewsequentialee ts

are in the data, the more omplex mo dels an

notshowtheirstrengthsandgivesimilarresults

than the other mo dels. In another exp eriment

with 10.000 users in 10 segments (of size 10%

ea h) onasmallsite with100resour es, where

rossingsofnavigationpathso urredby han e

0.27 0.37 0.40 0.40 0.40 0.41

Table 2: Exp erimental evaluation of dierent

re ommender systems on a smallsite with 100

resour es. andonlyfewsequentialee ts inthe

data

7 Outlook

Aframeworkforre ommendersystemsbasedon

navigationpathshasb eenpresentedandthein-

uen e ofdierentpath featuresonre ommen-

dation quality onsiderations was theoreti ally

dis ussedandempiri allydemonstrated. Wede-

velop edageneri metho dtomeasurethequality

ofre ommender systemsintermsofathe(nor-

malized)re ommendation s ore, so that dier-

entsystems aneasilyb e ompared,andgaveex-

amplesforre ommendersystemsbasedonnavi-

gationpathsthatmadeuseoffrequentsubstru -

turesinthepathhistories.

Futureworkshouldaddressquestionsasprun-

ingbased onsubstru ture partitions,automati-

allyndingoptimalsupp ort values,as well as

omparingourresults to thoseofhistory parti-

tionsobtainedbyapproa hesdierentfromfre-

quentsubstru tures. Besidetheoreti alworkon

mathemati almo delingofre ommendersystems

an empiri al evaluation of how they are used

(andliked)by site visitors is one ofthe urgent

questionsintheeld.

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