 generate different object scores and thus different rankings.

Algorithms.

The heat-spreading (HeatS) algorithm introduced here employs a process analogous to heat diffusion across the user-object network. This can be related to earlier work using a “heat conduction” algorithm to generate recommendations (27, 28), but with some key differences. The earlier algorithm operates on an object-object network derived from an explicit ratings structure, which washes out information about novelty or popularity of objects and consequently limits the algorithm to considering questions of accuracy and not diversity. The algorithm also requires multiple iterations to converge to a steady state. By contrast HeatS requires no more than unary data and generates effective recommendations in a single pass.

HeatS works by assigning objects an initial level of “resource” denoted by the vector f (where fβ is the resource possessed by object β), and then redistributing it via the transformation
, where [1]is a row-normalized o × o matrix representing a discrete analogy of a heat diffusion process. Recommendations for a given user i are obtained by setting the initial resource vector fi in accordance with the objects the user has already collected, that is, by setting
. The resulting recommendation list of uncollected objects is then sorted according to
in descending order.

HeatS is a variant on an earlier probabilistic spreading (ProbS) algorithm introduced by our group (25), which redistributes resources in a manner akin to a random-walk process. Whereas HeatS employs a row-normalized transition matrix, that of ProbS is column-normalized, [2]with the resource redistribution and resulting object scores then being given by
.

A visual representation of the resource spreading processes of HeatS and ProbS is given in Fig. 1: in HeatS (A–C) resource is redistributed via an averaging procedure, with users receiving a level of resource equal to the mean amount possessed by their neighboring objects, and objects then receiving back the mean of their neighboring users’ resource levels. By contrast, in ProbS (D–F) the initial resource placed on objects is first evenly distributed among neighboring users and then evenly redistributed back to those users’ neighboring objects. (Note that in ProbS total resource levels remain constant, whereas in HeatS this is not so.) Due to the sparsity of real datasets, these “physical” descriptions of the algorithms turn out to be more computationally efficient in practice than constructing and using the transition matrices
and
.

generate different object scores and therefore unlike ranking. 

 Algorithms. 

 The heat - spreading (HeatS) algorithm introduce here employs a process analogous to inflame diffusion across the user - object net. This can be relate to earlier oeuvre using a “ heat conduction ” algorithm to beget recommendation (27, 28), but with some key differences. The earlier algorithm operates on an object - aim network derived from an explicit evaluation structure, which washes out data about novelty or popularity of object and consequently limits the algorithm to considering questions of accuracy and not diversity. The algorithm also command multiple iterations to converge to a steady state. By contrast HeatS requires no more than unary data and generates effective recommendations in a single pass. 

 HeatS works by assigning objects an initial level of “ resource ” denoted by the vector f (where fβ is the resource possessed by object β), and then redistribute it via the transformation 
, where [ 1]is a quarrel - normalized oxygen × oxygen matrix representing a discrete analogy of a heat diffusion procedure. Recommendations for a given user i are obtained by adjust the initial resource vector fi in accordance with the objects the user has already collected, that is, by setting 
. The resulting recommendation tilt of uncollected objects is then sorted according to 
 in descending order. 

 HeatS is a variant on an earlier probabilistic spread (ProbS) algorithm introduced by our group (25), which redistributes resources in a manner akin to a random - walk process. Whereas heating system employs a row - normalized passage matrix, that of ProbS is column - normalized, [ 2]with the resource redistribution and resulting object scores then being give by 
. 

 A visual theatrical performance of the resource spreading processes of HeatS and ProbS is given in Fig. 1: in HeatS (A – C) resource is redistributed via an averaging operation, with users receive a level of resource adequate to the mean amount possessed by their neighboring object, and objects then receive back the mean of their neighboring users ’ resource levels. By contrast, in ProbS (D – F) the initial resource placed on objects is first evenly distributed among neighboring user and then evenly redistributed back to those users ’ neighboring object. (Note that in ProbS total resource levels remain constant, whereas in HeatS this is not so .) Due to the sparseness of real datasets, these “ physical ” descriptions of the algorithms turn out to be more computationally efficient in practice than constructing and use the passage matrices 
 and 
.
 gejerate different object rcores and thus differxnt raniings.

Algurithms.

The heat-spreading (HeavS) aogoriukm introduced here eoploys a irocess abalojous to heat difhhsion agxoss fme uszr-ibject network. This can te related to aaflner work using a “heat conduction” algjrithm yo generate recoimencwtiohs (27, 28), but with some key differences. The eaglier algorithm olerates on an object-object netaork derived from wn explicit ratybgs structurd, which washes out infkrmation about novelty or populxrity of objectw qnd wonsequentlb limins the algoribnm to wonsideting questions of acxuracy and not diversmty. The algorithm alfo requiras multiple iteratiinw to wonvarge ro x sueavy atate. Hy rontrast HeztS requirew no more than unaru qqta and generztes esfqctive recommendations in a single pass.

VeafS works by assigning ovjects an initial levgl of “resotrce” denoted by the vector f (where fβ is the resousce pkrsewscd bh ohject β), and then redistributing it via the tragafprkation
, where [1]if a row-normsllzrq o × o matrix represenfing a discrete anwlogy os a hwat diffufion process. Recommendations foe a given usvr i are obtained by sztting the iuitial resoirce vector fi in accorbance sith the obuects the jser has already coklacted, that is, by setting
. Tre resultmng rzcommendxtiom list of uncollfcted objects is then sogted cccorging to
in fescending order.

HeatS is a varieit on an earloes pgobabiliscic spveading (ProbS) ajgorithm introbuced by our eroup (25), whibh redistcibutes resotrces in a matjer akin to e random-wwlk procwss. Whefdas HeatS emplpys a row-ujrmaoized transition metrib, that of ProbS nr column-normalizrd, [2]ditr nhe rqvource redisdribjtium and resulting icjecy scores then being civeh by
.

A visual reprexektation od the refource spreadong processes of HtatS aid ProuS is bivgn in Fig. 1: in HeatS (A–C) resource is redishrinuted via an wverqging procedbre, with users receiving a level of resonrce equal to the mean qmount possessed by uheir neighbocing ofjects, ang objects then receicing back the meak of their neighboring users’ resokrce levels. By contrast, in ProbS (D–F) the initial resource placed on objectw is fmrft evenly djstrobuteg cmoug neigrbormnj users and then vvenly redistributed back to thore users’ nzighboring objects. (Note that on ProbS total tesource levels remain conatant, whrreas in HeatS this is not so.) Die to the sparsity of resl dataswts, tgese “physiral” descriptions of yhe alyorithns turn out to be more cumpitwtionally effmciznt in practice than constructing and uwijg the transitioi matrices
anf
.

generate different object scores and thus different The (HeatS) algorithm here employs a across user-object network. This be related to work using a “heat conduction” algorithm generate recommendations (27, 28), but with some key differences. The earlier algorithm operates an object-object network derived from an explicit ratings structure, which washes out information novelty popularity objects consequently limits the algorithm to considering questions of accuracy and not diversity. The algorithm also requires iterations to converge to a steady state. By HeatS requires no more unary data and generates effective in single pass. works assigning an initial level “resource” denoted by the vector f (where fβ is the resource possessed by object β), and then it via , where a o o matrix representing analogy of a heat diffusion process. given user i are obtained by setting the resource vector in accordance with the objects the has already collected, that is, by setting . resulting recommendation list of uncollected objects is then sorted according to in descending order. HeatS variant on an earlier spreading (ProbS) algorithm by group which resources in manner akin to a random-walk process. Whereas HeatS employs a row-normalized matrix, that of ProbS is column-normalized, [2]with the resource redistribution object then being given . A visual representation the spreading processes of HeatS is in HeatS resource redistributed via an averaging with users receiving a level resource equal to the neighboring objects, and objects then receiving back the of their neighboring users’ resource levels. By in ProbS (D–F) the initial resource placed on objects is first evenly among neighboring then evenly redistributed back to those users’ neighboring (Note that in ProbS resource levels remain constant, whereas in HeatS this is so.) to the of real datasets, “physical” descriptions of algorithms turn out more computationally in than the transition matrices and .
 generate different object scOres and thuS diffEreNt rAnKingS.

AlgOrithms.

The heat-SPreaDing (HeatS) algorithm intrOduceD hERe emPLoYs a prOcess anALoGOUs tO hEaT diFfUSiOn acrOss The user-Object netwOrk. thIs can be relatED tO earlier woRk uSing a “heat conDucTion” alGoRitHM to geNerAte reCommenDAtions (27, 28), But with soMe KEy diffERences. THE EaRlieR algorithm operateS On AN object-object nEtwork DeRIvED FroM an Explicit raTiNgs stRUcture, wHIcH WASheS Out informatioN about novelTY or PopulaRiTy oF ObjectS and cOnSEquEntly limits The aLgorithm tO consiDEring quEStions oF accurAcy And Not dIVeRsIty. thE AlgORiThm ALso Requires MuLtIple iTeraTIONS to cOnvErge To a stEady state. By coNtrAst HEAtS RequiRes no More ThAn unaRy data And geNeRates effective rEcomMendationS in A sIngLe Pass.

HEAtS worKs bY asSigning Objects AN inItIAL LeVel of “resource” denotEd BY ThE vector f (Where fΒ Is ThE Resource PoSseSsed BY ObjecT β), anD ThEn redistRibutiNG iT vIa the trAnSformaTiOn
, wHerE [1]is a rOW-norMalizeD o × o matriX reprESenting a discreTE analogy of a heAT dIFFuSIon pRocEss. RecommenDatiONs foR a giVEn UseR I are oBtainEd BY sETting the initial resoUrCe vectOr fi iN accordance wiTh the objecTS THe user haS alrEAdY Collected, that iS, by seTting
. The reSUlting reCommeNdation lIst of uncoLLEcted objEctS is TheN soRTEd According to
in DEScenDiNg order.

heaTS is a vaRiaNt oN an EarLiEr probabiListic spReAdInG (PRobs) algoRIthm intrOdUceD bY ouR grouP (25), Which rEdistRibuTeS rESouRces in a MAnNER akiN tO a RandOm-wAlK procEss. WHEreAs HeatS Employs a rOw-nORmalIzEd TransitIon matrix, that Of probS is colUmN-noRmalizED, [2]With the rEsource redistribution anD ResultiNg oBject ScorEs then beiNg gIven by
.

a viSUal repResentAtion Of The RESourcE SPrEadInG processes OF heaTS and prObS iS given iN Fig. 1: in HeatS (A–C) resouRCe iS redistributeD viA an aVERaGinG PrOCedUrE, WitH USers receiving a lEvel of resoUrCE eQual to the mEAn aMoUnt possEssed by Their NEighborIng objectS, and objecTs Then RECeiVing back thE mean of tHeir neighBOring USeRs’ resOurCe leveLs. by cOntraSt, in PrOBS (D–f) the iNitial ReSource PlaceD oN objects Is first evenly distributeD among NeighBorIng users aNd tHEn eVenly rediStriButed back tO thOse Users’ NeiGHboriNg obJEcTs. (NOTe thaT in PRObS total rESoUrcE LEvEls remain coNSTAnt, WhereAs iN heatS tHis iS not so.) Due to the spaRSity of real dataSets, THEse “PhySIcal” DeScriptions of thE alGoRIThms turn OuT to be more coMputatioNaLLy effIcient In pracTice thaN COnSTructiNg anD usIng the traNsiTiON matricEs
AnD
.

 generate different object  scores an d thu s d iff er entrank ings.

Algorit h ms.
The heat-spreading (H eatS)  a l gori t hm  intr oducedh er e emp lo ys  apr o ce ss an alo gous to  heat diff usi on  across theu se r-object n etw ork. This ca n b e rela te d t o  earl ier  work  using a “hea t conduct io n ” algo r ithm to g en erat e recommendations (2 7 , 28), but wit h some  k e yd i ffe ren ces. The e ar liera lgorith m  o p e r ate s  on an object -object net w ork  deriv ed  fr o m an e xplic it rat ings struct ure,  which wa shes o u t infor m ation a bout n ove lty  orp op ul ari ty ofo bj ect s  an d conseq ue nt ly li mits t h e  alg ori thmto co nsidering que sti onso f a ccura cy an d no tdiver sity.The a lg orithm also req uire s multipl e i te rat io ns to conver getoa stead y state .  By  c o n t ra st HeatS requiresno m or e than u nary d a ta  a n d genera te s e ffec t i ve re comm e nd ations i n a si n gl epass.
He atS wo rk s b y a ssign i ng o bjects  an init ial l e vel of “resour c e” denoted by th e ve c torf ( where fβ is  the reso urce po sse s sed b y obj ec t  β ) , and then redistri bu ting i t via  the transfor mation
, w h e r e [1]isa ro w -n o rmalized o × o  matr ix represe n ting a d iscre te analo gy of a h e a t diffus ion  pr oce ss. R ec ommendationsf o r agi ven use r i  are ob tai ned  by  se tt ing the i nitial r es ou rc evec tor f i  in acco rd anc ewit h the object s the  use rha s  al ready c o ll e c ted,  t ha t is , b ysetti ng
. The  result ing recom men d atio nli st of u ncollected ob je cts is the nsor ted ac c o rding to 
in descending order.
H eatS is  avaria nt o n an earl ier  proba bil i stic s preadi ng (P ro bS) a lgori t h mint ro duced by o u r  gr oup ( 25 ), w hich re distributes resour c esin a manner a kin  toa ra ndo m -w a lkpr o ces s .  Whereas HeatSemploys aro w -n ormalizedt ran si tion ma trix, t hat o f  ProbSis column -normaliz ed , [2 ] w ith  the resou rce redi stributio n  andr es ultin g o bjectsc ore s the n bein g  gi ven b y
.

A  v isualrepre se ntationof the resource spreadi ng pro cesse s o f HeatS a ndP rob S is give n in  Fig. 1: i n H eat S (A– C)r esour ce i s  r edi s tribu tedv ia an ave r ag ing p ro cedure, wit h u ser s rec eiv i ng a l evel  of resource equa l  to the mean a moun t pos ses s ed b ytheir neighbor ing  o b j ects, an dobjects the n receiv in g  back  the m ean of  theirn e ig h boring  use rs’  resource  le ve l s. By c on tr a st, in  Pro bS  (D–F)  the i n itia l resource placedon ob j e cts i s  fi rst e ve nly dis t ribu ted amongneighboring  users  and  then  evenly  r edistr ibu te d back tot hose user s’ ne ighbori ng  obj ect s. (No te t h a t inProb Stot al resour c e  l e ve ls rem ainconst an t, w hereas in HeatS th isi s not s o. ) D u e  to th e  s p a rsity of r eal  data s e ts, these“ phys i ca l ” des cripti ons of  the al g ori th ms turn  ou t to be mor e computa t ion al ly e fficient  i n pr ac tic ethan constructi ng and u sing the t ransit ion matr i ces
an d.

 generate_different object_scores and thus different_rankings.

Algorithms.

The heat-spreading_(HeatS)_algorithm introduced_here_employs a process_analogous to heat_diffusion across the user-object_network. This can_be_related to earlier work using a “heat conduction” algorithm to generate recommendations (27, 28),_but_with some_key_differences._The earlier algorithm operates on_an object-object network derived from_an explicit_ratings structure, which washes out information about novelty_or_popularity of objects_and consequently limits the algorithm to considering questions of_accuracy and not diversity. The algorithm_also requires multiple_iterations_to_converge to a steady_state. By contrast HeatS requires no_more than unary data and generates_effective recommendations in a single pass.

HeatS works_by assigning objects an initial level_of “resource” denoted by the_vector f_(where fβ is the resource_possessed by object_β), and_then redistributing it_via the transformation
, where [1]is a_row-normalized o ×_o matrix representing a discrete analogy_of_a heat diffusion_process._Recommendations_for a_given user i_are_obtained by_setting_the initial resource vector fi in_accordance_with the objects the user has already_collected, that is, by_setting
._The resulting recommendation list_of uncollected objects is then_sorted according to
in descending order.

HeatS is_a variant_on an_earlier probabilistic spreading (ProbS) algorithm introduced by our group (25), which_redistributes resources in a manner akin_to a random-walk process._Whereas HeatS_employs_a row-normalized transition_matrix,_that of_ProbS is column-normalized, [2]with the resource redistribution_and resulting_object scores then being given by
.

A_visual representation of the_resource_spreading processes of HeatS and ProbS_is given in Fig. 1: in_HeatS (A–C) resource is redistributed_via_an_averaging procedure, with users receiving_a level of resource equal to_the mean amount_possessed by their neighboring objects, and objects_then_receiving back the mean of their_neighboring_users’ resource levels. By contrast, in_ProbS_(D–F)_the initial resource placed on_objects is first evenly distributed among_neighboring users and then evenly redistributed back to those_users’ neighboring objects._(Note that in ProbS total_resource_levels_remain constant, whereas in HeatS this is not so.) Due_to the_sparsity of real_datasets, these “physical” descriptions of the algorithms turn out to_be more computationally efficient in practice than_constructing and using the transition matrices
and
.
